UBC Theses and Dissertations

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UBC Theses and Dissertations

A study of the sequence category Gentle, Ronald Stanley 1982

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A STUDY OF THE SEQUENCE CATEGORY by RONALD STANLEY GENTLE B . S c , U n i v e r s i t y o f T o r o n t o , 197 1* M . S c , U n i v e r s i t y o f B.C., 1 9 7 6 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department o f Mathematics) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d . THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r , 1 9 8 2 © Ron a l d S t a n l e y G e n t l e , 1 9 8 2 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of mATHEmATiCS  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date ABSTRACT For a given a b e l i a n category &\ , a category ct i s formed by c o n s i d e r i n g exact sequences of Crf. I f one imposes the c o n d i t i o n that a s p l i t sequence be regarded as the zero o b j e c t , then the r e s u l t i n g sequence category i s shown to be a b e l i a n . The i n t r i n s i c a l g e b r a i c s t r u c t u r e of £/j> i s examined and r e l a t e d t o the theory of coherent functors and funct o r r i n g s . t/£ i s shown to be the n a t u r a l s e t t i n g f o r the study of pure and copure sequences and the theory i s f u r t h e r developed by i n t r o d u c i n g repure sequences. The concept of pure semi-simple categories i s examined i n terms of . L o c a l i z a t i o n w i t h respect to pure sequences i s developed, l e a d i n g t o r e s u l t s concerning the existence of a l g e b r a i c a l l y compact o b j e c t s . The f i n a l t o p i c i s a study of the simple sequences and t h e i r r e l a t i o n -ship t o almost s p l i t exact sequences. TABLE OF CONTENTS Chapter Page Acknowledgements Iv Introduction 1 1 The Sequence Category 3 2 Basic Facts Concerning tls& 6 3 Embedding of o/ into £/4 and Projectives. 16 4 The Punctorial Approach 29 5 Pure and Copure Subcategories 38 6 Pure Semisimple Categories 50 7 The Repure Subcategory 71 8 L o c a l i z a t i o n and Colocalization in %Jj3 78 9 Functor Category Techniques 102 10 Simple Sequences 120 Bibliography 146 i i i i v ACKNOWLEDGEMENTS My p r i m a r y acknowledgement i s t o S t a n l e y Page f o r the d i f f i c u l t and i n t a n g i b l e t a s k o f s u p e r v i s i n g me; p r o -v i d i n g me w i t h h i s f a i t h , p a t i e n c e and encouragement. I am i n d e b t e d t o many m a t h e m a t i c i a n s , but s i n g l e out M. A u s l a n d e r and P. F r e y d whose works s t a r t e d the b a l l r o l l i n g . Ms. C h e r y l McKeeman the t y p i s t [ t e m p t r e s s ] gets more th a n the u s u a l kudos f o r the amazing f o r t i -tude r e q u i r e d t o d e c i p h e r [ g r a t i f y ] my s c r a w l i n g s [ c r a -v i n g s ] . Thanks a l s o goes t o C h e r y l f o r s t a n d i n g by me, g i v i n g e s s e n t i a l p e r s o n a l s u p p o r t and l o v e . F i n a l l y , I acknowledge J e n i f e r , C a s s i d y and Morgan f o r b e i n g t h e r e w i t h me. 1 ' INTRODUCTION Category t h e o r y has d r a s t i c a l l y a l t e r e d t h e f a c e o f r i n g t h e o r y . However t h e r e i s s t i l l a g r e a t r e s i s t a n c e t o c a t e g o r y t h e o r y as a bona f i d e b r a n c h o f mathematics. One even has the a l g e b r a i s t ( m i s ) u s i n g M i t c h e l l ' s [ 2 1 ] embedding theorems o f a b e l i a n c a t e g o r i e s i n t o module c a t e -g o r i e s t o d i s m i s s a b e l i a n c a t e g o r y t h e o r y as e s o t e r i c r i n g t h e o r y . Given s u i t a b l e knowledge o f a b e l i a n c a t e g o r i e s , the stu d y o f r i n g s by ex a m i n i n g t h e module c a t e g o r y g i v e s a f i r m f o u n d a t i o n and s t r u c t u r e t o much o f the e x i s t i n g t h e o r y and s t i m u l a t e s f u r t h e r r e s e a r c h . However i f a b e l i a n c a t e g o r y t h e o r y were o n l y s l i g h t l y g e n e r a l i z e d r i n g t h e o r y , then t h i s approach would be p u t t i n g the c a r t b e f o r e the h o r s e . There are two key f e a t u r e s t o t h i s t h e s i s which the r e a d e r s h o u l d keep i n mind. The f i r s t i s t o r e g a r d t h e t h e s i s as a d e m o n s t r a t i o n o f a p p l i e d elementary a b e l i a n c a t e g o r y t h e o r y . A s p e c i f i c c a t e g o r y , the sequence c a t e -g o r y , i s i n t r o d u c e d , and i s examined as an a b e l i a n c a t e g o r y . To some e x t e n t , module t e c h n i q u e s /can be mimicked, however attempts t o r e p r e s e n t t h i s c a t e g o r y as a module c a t e g o r y f a i l . I n d eed, the sequence c a t e g o r y w i l l not have a g e n e r a t o r and w i l l not be l o c a l l y s m a l l ( i . e . , o b j e c t s w i l l n o t have j u s t a s e t o f s u b o b j e c t s ) except i n very s p e c i a l c i r c u m s t a n c e s ; i n p a r t i c u l a r , i t i s not a G r o t h e n d i e c k c a t e g o r y . The second f e a t u r e w i l l be the d i r e c t a p p l i c a t i o n o f t h e r e s u l t s , g l e a n e d from the sequence c a t e g o r y , t o the s t u d y o f r i n g s , accompanied by i n d i r e c t l y r e v e a l i n g t h a t a s u i t a b l e framework has been e s t a b l i s h e d i n t o which v a r i o u s r i n g t h e o r e t i c a l problems may be posed and s o l v e d . Hope-f u l l y f e r t i l e ground has been exposed. The f o l l o w i n g i s a q u i c k breakdown o f the c o n t e n t s . Chap-t e r one i s a s h o r t i n t u i t i v e i n t r o d u c t i o n t o t h e o b j e c t o f 2 study : the sequence category ~tjj^ . In chapter two, chain homotopy between exact sequences becomes the c r u c i a l factor i n defining morphisms for the sequence category. The major resul t i s to demonstrate e x p l i c i t l y that i s an abelian category. The chapter concludes with propositions intended to give the 'flavour' or ' f e e l ' f o r the algebra involved i n working with sequences as objects, so that the reader w i l l be comfortable with the mechanisms of t h i s s p e c i f i c abelian category. In chapter three, the basic li n k between the underlying category and the sequence category i s established by examination of the projectives i n Chapter four places the sequence category within the f a m i l i a r ground of coherent functors. The equivalences established i n t h i s chapter should be kept In mind, so that any r e s u l t concerning sequences can be formulated into functors. A torsion theory for i s introduced i n chapter f i v e . The f a m i l i a r concept of purity enters as the torsion free part of this theory. In chapter s i x the problem of characterizing those module categories, i n which every object i s a dir e c t sum of f i n i t e l y generated objects, i s examined i n the context of The repure category i s introduced and studied i n chapter seven, as the torsion free part of a torsion theory, now using pure sequences as torsion. ( C o ) l o c a l i z a t i o n i s the major topic of chapter eight using the torsion theories of chapters f i v e and seven. The major resu l t shows the existence of the category of additive fractions with respect to pure sequences. This category turns out to be a functor category, and consequences of this fact are examined. Chapter nine i s somewhat of a diversion, r e l a t i n g the the-ory of functor rings to the functor category a r i s i n g i n Chap-ter eight. In chapter ten, the simple objects of are characterized and compared to almost s p l i t exact sequences. 3 CHAPTER 1 THE SEQUENCE CATEGORY I n the st u d y o f an a r b i t r a r y a b e l i a n c a t e g o r y 0} the s u b c a t e g o r i e s ^ 3 and \} , p r o j e c t i v e and i n j e c t i v e o b j e c t s , f i g u r e p r o m i n e n t l y . I f one e s t a b l i s h e s an adequate know-a l e d g e o f e i t h e r c l a s s , f o r example i n c e r t a i n module categor-i e s e very p r o j e c t i v e i s f r e e , t h e n i n s t u d y i n g the s t r u c t u r e o f g e n e r a l o b j e c t s one would l i k e t o d i s p e n s e w i t h ' p r o j e c t -i v e n e s s ' o r 1 i n j e c t i v e n e s s • ( t h e s e terms t o be ta k e n i n t u i -t i v e l y f o r the moment). The method o f t h i s d i s p o s a l w i l l be t o pass t o the a d d i t i v e q u o t i e n t c a t e g o r i e s &1/(P and Of/^J-( C h a p t e r 3 ) • However these q u o t i e n t c a t e g o r i e s are r a r e l y a b e l i a n ( 3 - 9 ) , and t h i s i s a major s t u m b l i n g b l o c k . Another way t o study Of i s t o c o n s i d e r the c a t e g o r y o f e x a c t sequences o f OJ . ( f * has o b j e c t s ~ .'• e x a c t s e -quences and a morphism i s a t r i p l e o f morphisms o f CfJ making the f o l l o w i n g diagram commutative : 0 — * A — * B — * C — * 0 = E, 0—• A'—*B'—• C—*0 = E 2 A l t h o u g h £ n a t u r a l l y i n h e r i t s an a d d i t i v e s t r u c t u r e , i t i s n e v e r a b e l i a n e xcept t r i v i a l l y when Of =0. We w i l l e l a b o r a t e on a p r o o f o f t h i s s t atement (Maclane [ 11 ] , page 3 7 5 ) . because i t w i l l g i v e some i n s i g h t i n t o what f o l -lows . 1.1 i s not a b e l i a n , u n l e s s Bf= 0. P r o o f a map o f the form 0 ••A'-—*B' >C«—*Q w i l l be e p i i n £ i f B—*>B' and C — » C are e p i i n OJ . F o r suppose 0 —*• A *B •> C »0 i * i 0 * A * — » B ' — » C +0 I i i 0 • A " — * B " — » C " — P O g i v e s the zer o map. Then s i n c e both C—*>C—*C" and B—»>B'—*B" are z e r o , the e p i s i n CSf c a n c e l t o g i v e C — * C" = 0 and B ' — ? B " = 0. But then A' •—»B' A' 0 g i v e s if I s the z e r o map, A">->B" A'> •B" and the monic can be c a n c e l l e d , so A'—*A" i s a l s o the z e r o map. D u a l l y , a map o f the form i s monic i n (ft . Hence the map 0 — > 0 — » B * » B — * 0 J II b 0 — , B * B » 0 »0 i s b o t h monic and e p i . But t h e r e i s on l y one map from the lower sequence t o the upper, the z e r o morphism 0—-»B—r* B •>© » 0 > B — B •> 0 — [o jo i o 0 — * 0 9 B » B * 0 so t h e r e can be no i n v e r s e map. Hence ct cannot be a b e l i a n . / / The p r o o f s u g g e s t s b o t h sequences o f the form and 0-—* 0 => B " »B => 0 be c o n s i d e r e d as z e r o o b j e c t s , and so the n a t u r a l sum 0 — * A — ? A @ B — * B — 9 0 , the c a n o n i c a l s p l i t sequence, s h o u l d 5 a l s o be the z e r o o b j e c t . Thus one I s l e d t o c o n s i d e r the q u o t i e n t c a t e g o r y where ^ i s the s u b c a t e g o r y o f s p l i t sequences. O b j e c t s o f zu are those o f £ , but ( E 1 , E 2 ) = H 0 M < f ( E l » E 2 ) / ^ ^ E i » E 2 ) where ^ / ( E 1 , E 2 ) i s the subgroup o f morphisms t h a t f a c t o r through a s p l i t s e -quence ( t h e z e r o o b j e c t ) . P r o p e r t i e s o f t h i s subgroup w i l l be g i v e n i n Chapter 2 ( P r o p . 2.A). i s an a b e l i a n c a t e g o r y ( C h a p t e r 2, Thm. 2.5), and i f of has s u f f i c i e n t p r o j e c t i v e s ( i n j e c t i v e s ) then t h e r e i s a f u l l embedding of/'p < = - — > g ^ J ( & / / ^ c — > • S / J ) a s s i g n i n g t o each o b j e c t X a p r o j e c t i v e p r e s e n t a t i o n 0 — > K — > P — » X — > 0 ( i n j e c t i v e c o - p r e s e n t a t i o n 0-^X-^I—*N-*0) (Thm. 3.6). Under t h i s embedding, Of/P (Of/**) be comes a ( c o ) - r e s o l v i n g c l a s s o f p r o j e c t i v e s ( i n j e c t i v e s ) (see Chapter 3). One then has t h i s c u r i o u s p r o c e s s o f e l i m i n a t i n g p r o -j e c t i v e n e s s from oj v i a the passage b u t th e n embedding CfjfP as a r e s o l v i n g c l a s s o f p r o j e c t i v e s i n the l a r g e r a b e l i a n c a t e g o r y Thus i n some ways the d i f f i -c u l t y o f n o n - a b e l i a n n e s s o f °lJP i s somewhat overcome i n the embedding, and the embedding i s q u i t e e f f i c i e n t because the s t u d y o f p r o j e c t i v e o b j e c t s i s a t r a c t a b l e one. 6 CHAPTER 2 BASIC FACTS CONCERNING ll<&> In t h i s chapter we show i s abelian and investigate various consequences of t h i s . The aim i s to work i n and translate results to Of , so we w i l l develop the algebra of £/<5 , e x p l i c i t l y i l l u s t r a t i n g the abelian concepts of kernels, cokernels, sums, products, int e r s e c t i o n s , etc. The following three lemmas are recorded for reference (they arose i n the study of homological algebra, but i n essence are statements r e f l e c t i n g the abelian structure of Lemma 2.1 (Hilton and Stammbach [ » 5 ] , page 83) Given ^ C - ^ A A ( o t , ^ (->./ then (0 >) C * A Q B »D [—*0] i s exact i f f the square i s a (pull-back),[push-out]. Lemma 2.2 (Hilton and Stammbach [ 2 1 ] , page 84) I f 0 —9B » E ' >A' =»0 II I i 0 >B »E »A * 0 i s commutative with exact rows, then the right-hand square i s a pull-back and a push-out. Lemma 2.3 ( M i t c h e l l [ ], page 163) Any A — » B i n £ has a fa c t o r i z a t i o n 0 1 A"-—-> A *Af. »0 A That i s , A — » B can be f a c t o r e d as a push-out ( o f A"—>A ) A' B f o l l o w e d by a p u l l - b a c k ( o f B ). T h i s w i l l be the epi - m o n i c f a c t o r i z a t i o n o f A g i v e n morphism i n . R e c a l l t h a t H 0 M ( A j B ) = H0M '(A,B) / A (A,B) c / o — — c ~ ~ ~~ ~ where yS (A,B) was the -subgroup o f morphisms f a c t o r i n g t h r ough s p l i t sequences. The next p r o p o s i t i o n g i v e s the b a s i c f a c t s c o n c e r n i n g such morphisms. S i n c e the o b j e c t s o f £ , b e i n g e x a c t sequences, can be thought o f as ( s h o r t ) c h a i n complexes, the n o t i o n o f homotopy n a t u r a l l y a r i s e s . P r o p o s i t i o n 2.k G l v e n A : 0 > A » - ^ A - ^ * A ' » 0 h £ 1 f " l I'-'iL*' Lt* B 0 •> B " — * B* »B '— -* 0 the f o l l o w i n g are e q u i v a l e n t : ( i ) t h e r e e x i s t s g such t h a t g °c = f " ( i i ) t h e r e e x i s t s h such t h a t /3'h • f ( i i i ) t h e r e e x i s t s g and h such t h a t y#g + hoc' - f ( i v ) f f a c t o r s through a s p l i t e x a c t sequence (v) F i s c h a i n homotopic t o z e r o ( f - ~ 0 ) . NOTE : ( i ) £=»(ii) £=Kiii) F i e l d h o u s e [ 6 ] . ( i ) 4=£ ( i v ) F r e y d [ 8 ] . P r o o f I t s u f f i c e s t o show (i)<S=>> ( i v ) f o r the n ( i i ) <=>(iv) i s pr o v e d d u a l l y , and ( i ) w i t h ( i i ) ^=0 ( i i i ) i s c l e a r , f i n a l l y (1)^(11)^,(111) c o n s t i t u t e ( v ) . (iv)=Mi) 0—»A"—>A —»A'—>0 0 — » C —> OJD—*D » 0 0 — — ? B — * B ' ~ * 0 The r e q u i r e d g i s a c h i e v e d v i a the p r o j e c t i o n C Q D (i)=^(iv) Consider 3 B " where the square i s a push-out. K exists to give a commutative diagram. Hence B " — > E i s s p l i t monic and result now follows by Lemmas 2.2 and 2.3 .// The following theorem i s the major res u l t of this section. The proof i s adapted from Freyd [ 8 ], Thm. 3.3. However, we wish to work i n t e r n a l l y i n , and for our purposes we need the e x p l i c i t calculation of the kernel and cokernel of a morphism and Its canonical f a c t o r i z a t i o n for further propositions. i s abelian. <T i s additive Theorem 2.5 Proof <£/><& i s additive because T i s additive ( a d d i t i v i t y e a s i l y seen to be preserved under quotients). Hence i t w i l l s u f f i c e to prove that every morphism f has a kernel and cokernel, and a f a c t o r i z a t i o n f - (gh where h i s a cokernel and £ a kernel (Stenstrom [ 2 . 5 ] , page 8 7 ) . Given f : A—*B we w i l l show 0-0' 0 0 0 -> A" -9 A" • -> B" II -»B'-B'$A-I —> A — 1 — 9 E -B— I -»E--» A»' II -,A!-l -»B'-II ->B»--> 0 -» 0 -*0 -»0 k h £ 1 represents 0~»ker f >—•> A -—>> im f ~>—*B—=» coker f — * • By Lemma 2.3, f =g_h . The exact sequences at top and bottom r e s u l t from Lemma 2.1, using Lemma 2.2 and i t s dual to show they are exact. 0. We prove (a) k = k e r f (b) £ • k e r 1 th e n d u a l l y ( a , y ^ „ c o k e r ^ ( b , y ^ = c o k e r k < (a) ( 1 ) k i s monic x „ ^ x ^ x , i ?'l I -A"—»B"($)A »E W i J' k A" * A A' I f k x = 0 the n 0 e x i s t s by P r o p o s i t i o n 2.k. The same 6 t h e n shows x " 0 . ( i i ) h k = 0 A „ *, B'^?A I* ^ ' Take ^ t o be A" , t h e p r o j e c t i o n . B" ( i i i ) Suppose h x • 0 X" >X »X' I v'\ i A" — A * A' 1 / I ) B" * B * B' so t h a t 0 e x i s t s w i t h the p r o p e r t i e s o f Prop. 2.4. Then X" => X »X' I i I A"~* B'£& »E II I 1 A" > A >A' g i v e s a f a c t o r i z a t i o n o f x through k. ( i ) , ( i i ) and ( i i i ) e s t a b l i s h (a) k • k e r f . (b) (i)_£ i s monic, p r o o f same as f o r k. ( i i ) 1 g - 0 B" > E 1) /& B" r ' I t t a k e 9to be i d e n -E ^ t i t y . 10 ( i l l ) Suppose 1 x = 0 . X" I <• 1 / X *X» I I B * B' J II >B^A'—*B» Then # e x i s t s as i n Prop. 2 . 4 . Let x° be the composite i n £ . X" I B" II Bn-> X -*E i -* X' -> B-v ->A' B» Then (x because l e f t side of - O ~ 0 x - x° Is the zero map. Hence i n <£/<5, x »• x° and x eaii- be factored through g. ( 1 ) , ( i i ) and ( i i i ) e s t a b l i s h (b) £ = ker 1 .// 2 . 6 Subobjects and Quotients We use the f a c t o r i z a t i o n of a morphism, and the construction of cokernel and kernel to next investigate the concepts of subobject and quotient object. Suppose 0 > A" A »A' * 0 \ 0 •B" -> i -» B -> A '-i -* B»- 0 represents a monic. Factoring this monic as i n the theorem 0 0 0 A" » A »A'-I P.O. i II _>B" »E »A'-II i P . B . I -> 0 - » B , ! - -•» B- -> B' •* 0 establishes an isomorphism 0 -0--A" 1 B " - -9 E • -» 0 -> A' -> 0 So, without loss of generality, one can assume each subobject of a sequence results from a pull-back, and dually each quotient from a push-out. 11 2 . 7 K e r n e l s and C o k e r n e l s G i v e n a map o f sequences —»E_ 2 , the k e r n e l £ i s d e t e r -mined by a push-out as f o l l o w s : 0 -> A , » A JjB.. — > E — - > 0 t< l i 1 21 1 I T 0- > A. > B- >C. * 0 E N i 1 i J 0 > A 2 * B 2 »- C 2 * 0 E 2 where E i s t h e push-out o f A, — > B.. 1 D u a l l y , t he c o k e r n e l i . I s a l s o d e t e r m i n e d by E as a p u l l - b a c k : 0 » A-. > B. — ~ > C. — * 0 ET 1 1 I1 I J1 0 > A 0 > B~ > G 0 » 0 E 0 12 JI2 T 0 * E * ci@B2 * C 2 — * 0 -where E i s a l s o the p u l l - b a c k o f C, I B 2 > C 2 Suppose 0 > AT — * B. — * C. » 0 i s monic. J 1 J 1 4 , 1 0 > A 2 •> B 2 » C 2 > 0 Then i n f a c t o r i n g t h i s map i n t o e p i - m o n i c the e p i i s an i s o -morphism : 0 • A, •> B., —-» C. * 0 I 1 I1 I I 1 0 » A 2 y E 9 C-L 9 0 We e x h i b i t t h e i n v e r s e as an i l l u s t r a t i o n o f t e c h n i q u e s used i n : one has the k e r n e l 0 * A, — * A„@BT > ^  » 0 II1 2 l \ " *> 0 f k^ 9 B 1 » C 1 — * 0 b u t t h i s s p l i t s , so by Prop. 2.4 E — f a c t o r s o v e r B^ ( t h e map E — i s the composite E-^A^B^—>*• B^ the s p l i t t i n g map f o l l o w e d " : by the p r o j e c t i o n ) . Thus one can form the diagram : 12 0 — > A 0 — » • E —»C,—•» 0 ^ I II1 ° - * A i - * B r ^ c i - ^ ° • where A2—^A.^ Is the map Induced on kernels. This gives the required inverse of the sequence morphism. We remark that A 2 — » A 1 i s not the component of the s p l i t t i n g A 2 @ B l — > A r Dually, i f 0 — % A, — » B n — > C, —». 0 0 —s> A 2 > B 2 —> C 2 — » 0 i s e p i , then 0 — » A 2 —> E » C., — » 0 j-i ^ v -» 1 ^ 0 — > A 2 .—* B 2 — * C 2 — > 0 i s an isomorphism. For the inverse, form the cokernel 0 — » A„ — » B 0 — » C 0 — * 0 l V ' ' A 2 II2 0 — , E — > C . ] © ^ — » C 2 — * 0 which s p l i t s so A 2 — » E factors over B 2 ( B 2 — * E i s the composite B 2 —> C^B^—>E the natural i n j e c t i o n followed by the s p l i t t i n g ) . So forming the diagram 2 - ' f2 9 y 2 0 —> A „ — » B„ s CU »0 II2 I 2 0 — * A 2 j> E > where C 2 — » i s map induced on cokernels, gives the inverse. I f E^—»E_ 2 i s an isomorphism, the two inverses above can be combined to give an e x p l i c i t inverse, However there i s another way of viewing this isomorphism which lacks rigour but gives some insight into the character of , and how i t d i f f e r s from <T by regarding s p l i t sequences as zero. I f E^->E_2 i s an isomorphism then both 0 '—e>A^ —->> B^A^—»'E"'—>0 and 0 — * E — 9 C ^ $ B 2 — * C 2 — * ° s p l i t * s o C ^ O 0 - ^ A ^ -*B 1©(C 1®B 2)-»C 1^C 2 - ^ > 0 and 0-^A 2 -5B 2 -^> C2—> 0 = 0^A 2@B 1-»B 2@(B ; ]fSC 1)-^C 1@C 2--->'0 and these sequences are isomorphic i n the category £ . That i s , by adding suitable s p l i t sequences an isomorphism can be l i f t e d ' t o £ . 13 P r o p o s i t i o n 2.8 I f (Tf i s ( c o - ) c o m p l e t e , t h e n so i s . P r o o f S i n c e Iff i s a b e l i a n , i t s u f f i c e s t o show d i r e c t sums e x i s t f o r co-completeness. The obv i o u s c h o i c e works. G i v e n 0 - * A ± — * » B i — * C±-^* 0 , form 0 ~>(S>k^~> <$B^ —^OC^—*0 . T h i s sequence w i l l have the u n i v e r s a l p r o p e r t y ; t h e o n l y n o n - t r i v i a l i t y i s uni q u e n e s s . Suppose 0 — * @A. — » ^ B . — » ® C . —> 0 X1 I1 4 1 0 —* X •* Y » Z — * 0 has the p r o p e r t y t h a t 0 —> A.^—> B i — > —> 0 •I \ •if 0—> © A , - * g t e , - 5 » ( ^ C , ^ — * 0 * I J 0 — * X * Y — * z —>6 i s z e r o . Then each C ^ - H * ( § G ^— > Z f a c t o r s o v e r Y and hence QjC^—^Z f a c t o r s , o v e r Y by t a k i n g t h e sum o f the i n d i v i d u a l f a c t o r i z a t i o n s . Thus the l o w e r sequence map i s z e r o . U s i n g a d d i t i v i t y , t h i s w i l l i m p l y uniqueness o f th e two maps i n d u c e d from t h i s sum sequence a g r e e i n g on the n a t u r a l i n j e c t i o n s . / / 2.9 Example P o r d i r e c t sums and p r o d u c t s , the pro c e d u r e i s t o form them i n £ and pass t o • T h i s method f a i l s t o form g e n e r a l l i m i t s - a n d c o l i m i t s . To i l l u s t r a t e t he d i f f i c u l t i e s , l e t 0/= Ab , a b e l i a n g r o u p s , and c o n s i d e r the n o n - s p l i t sequence 0—»K—'Pr-^^-^O where F i s a f r e e a b e l i a n group, ^ th e r a t i o n a l s . Now Q i s a d i r e c t l i m i t o f i t s f i n i t e l y g e n e r a t e d subgroups G 1 . Forming p u l l - b a c k s 0 — i K —>E. —>G, 0 —•> K —°> F -—? , one has t h a t i n & the sequence 0—? K—> F — — * 0 i s a d i r e c t l i m i t o f 0--*K--^E i—*G 1~»0. But a l l f i n i t e l y g e n e r a t e d subgroups o f Q a r e i s o m o r p h i c t o Si. Hence each 0-%K-J»Ei—»Gi—=>0 s p l i t s and i s z e r o i n and , t h e d i r e c t l i m i t i n tj^ w i l l t h e n be z e r o . 14 2.10 Sums of Subobjects Let X^*1-»X be subobjects l n an abelian category. Then the sum of these subobjects i n X i s the image of the induced map ®C i—3>X. Applying this procedure to , l e t A—a B. —*C. // I1 I A —>B —> C represent a set of subobjects. The sum map i s 0 — * ® k — ^ C j B , — » ® C , — * 0 0 —s> A —» B » C — * 0 and i t s image i s 0 A -» E —»3>C. — 0 II I X 1 0 -» A —* B —» C — * 0 ; that i s , the sum i s achieved by taking the pull-back of the sum map — > C with the given epi B—*C—>0. 2.11 Intersection of Subobjects I f X^^^X then the int e r s e c t i o n of the X i equals the kernel of the map X —>7T X/X^ In £/>S the quotients are O ^ B j ^ — • C 1 © B —^C—»0 , where B^ i s the pull-back of . So the map to B * G the product i s 0-—-">A — B *C *Q i I I 0->TTB1—* "/HC^B) 9-TTC "?0 which has kernel 0—>A—> BQTTB^—s>N—>0 . There i s another more i n t u i t i v e way to construct the inte r s e c t i o n : one has 0 —T> A —*7TB. — > M —=s> 0 H i 1 i 0 =>A => B± > C±—J>0 , where M = cokernel, and M — i n d u c e d from M. So 0 -—»A —>7TBi—5» M —*0 i s ' contained i n the in t e r s e c t i o n . But also, 0 —=> A — » B©(7x B.) — * N —5>0 |j ], ^ middle map 0 — » A —*? TB± » M—^0 projection shows t h a t the i n t e r s e c t i o n I s c o n t a i n e d i n 0— » A —T T T B ^ — * • M — » 0 . Hence t h i s i s t h e i n t e r s e c t i o n . To e x h i b i t an e x p l i c i t i n v e r s e t o t h i s i s o morphism : 0 — > A — * 7 T B , —> : M—* 0 1/ i i 0 — » A -^B©(7TB 1)->N—»0 where middle map i s the sum o f t h e i d e n t i t y on 7TB^ and -rf B^ —» B j —* B f o r any B j , where B j — > B r e s u l t from f o r m a t i o n o f B, as p u l l - b a c k o f B. — > C. ' i v B — * C CHAPTER 3 ^ n m w n nw a/' INTO t U ' AND P R O J E C T I V E 3 16 We now investigate the intimacy of of with i t s associated sequence category. In some respects, the s i t u a t i o n i s simi-l a r to the Yoneda embedding :.~ A—>HOM(-,A ) , which embeds of as a resolving set of projectives i n the functor category. For each A, choose a projective presentation 0 — » K — P — 3 > A — * 0 (we assume of has s u f f i c i e n t p r o j e c t i v e s ) . Proposition 3.1 The assignment of projective presentations constitutes a functor If1 of * Any two such functors are naturally equivalent. Proof Given a morphism f : A —>B , there i s an induced morphism between projective presentations 0 — * K — » P — » A — > 0 7T(f) c, v> I 0 — > L — = > Q — » B — * 0 f I f two d i f f e r e n t sequence maps both induce A >B , then the difference i s nomotopic to zero since right side i s the zero, map A 0 £ Hence by Prop. 2.4 this constitutes , the zero map i n Hence th i s i s a well-defined assignment which i s then c l e a r l y a functor Of—> <£/J'. I f IT'were defined using d i f f e r e n t presentations then 0 —=> K — » P — » A ir(A) U fl H 0-> K'-9P'-)A- U*/(A) constitutes natural transformations % 0 } and (1 - Qfi ) ~ 0 since the right side of (1- % &A ) i s the zero map A—^A . Hence i n £/<5 , and & A are mutual inverses determining a natural equivalence between /Tand ' /T /(in p a r t i c u l a r d i f f e r e n t projective p r e s e n t a t i o n s o f A are i s o m o r p h i c i n Lemma 3.2 A p r o j e c t i v e p r e s e n t a t i o n 0 — » K —*> P — ^ A—>0 i s a p r o j e c t i v e o b j e c t o f P r o o f L e t 0 — » K —•» P —"^A —">0 be a g i v e n morphism. 0—>» B ' — * B — * B"—*-0 I f E — » > B —PO I s e p i , w i t h o u t l o s s o f g e n e r a l i t y assume B i s a q u o t i e n t o f E. Then t h e g i v e n morphism can be f a c t o r e d as 0 — » K — * P »A *0 0 s»X—»Y — * B " — " 0 E J * « * 0 — > B ' — * B —*>B"—^0 B Hence 0—* K — » P — > A — > 0 has the p r o j e c t i v e p r o p e r t y . // P r o p o s i t i o n 3.3 The image o f i s a r e s o l v i n g s e t o f p r o j e c t i v e s . P r o o f -0 —> K — * P ">A •—»0 i s e p i , so r e s u l t : • II Q ^ ^ A f o l l o w s w i t h Lemma 3.2. // 3.4 The P r o j e c t i v e Homotopy Category The a d d i t i v e f u n c t o r TT -' °/ —> <5/^5 has k e r n e l t h o s e o b j e c t s A o f of whose p r o j e c t i v e p r e s e n t a t i o n i s z e r o i n ?>U , i . e . i t s p l i t s and so A i t s e l f i s p r o j e c t i v e . G i v e n f : A — * B , then 1T(f) » 0 i f K —9 P — * A i s the z e r o map,, and t h i s o c c u r s L — * Q — » B i f and o n l y i f A - — » B f a c t o r s o v er Q—>>B by Prop. 2.4. P r o p o s i t i o n 3.5 F o r f : A —*B , th e f o l l o w i n g are e q u i v a l e n t ( H i l t o n [ 1 3 ] , page 131) : ( i ) f can be f a c t o r e d t h r o u g h some p r o j e c t i v e ( i i ) f can be f a c t o r e d t h r o u g h a p r o j e c t i v e Q , such t h a t Q-*7B (IT'(f) = 0). ( i i i ) f can be f a c t o r e d t h r o u g h any C - » B . P r o o f ( i ) " ^ ( i i ) I f f : A -^B = A—* P—^ B t h e n T f ( f ) : If (A) — T > lf(?) > TT(B) i s z e r o s i n c e 18 1f(P) » 0 . ( i i ) = y > ( i i i ) L e t . C->)B. T h e n t h e r e i s a map 0 K — > P —^>A 7Y(k) —>E 0 —==> L — C —>> B — > 0 = E w h i c h f a c t o r s a s •7t(f) TT ( A ) ^ ( 8 ) — > > E a n d h e n c e i s z e r o , s o b y P r o p . 2.4 k-pB f a c t o r s o v e r C . / / We h a v e now c h a r a c t e r i z e d t h e k e r n e l o f /Y? , a n d c a n f o r m a q u o t i e n t c a t e g o r y tf/(P . T h e o b j e c t s o f tf/<P a r e t h o s e o f &i , b u t H 0 M ^ ( A , A ' ) « H0M^ ( A , A » ) / tf°(A,A») w h e r e tfP ( A , A ' ) i s t h e s u b g r o u p o f m o r p h i s m s f a c t o r i n g o v e r a P r o j e c t i v e . C o m b i n i n g r e s u l t s g i v e s T h e o r e m 3.6 ^ -^-p- t/4 l s a f u H e m b e d d i n g o f \y y a s a r e s o l v i n g c a t e g o r y o f p r o j e c t i v e s . We w i l l now d e n o t e /{?c-*? £ / b y IT . P r o p o s i t i o n 3 . 7 ?//P h a s weak k e r n e l s . P r o o f L e t A — > B i n &f/(P y p a s s t o , a n d l e t K > - > i r ( A ) — 9^(B) b e e x a c t i n • ' I f ' l r * ( K ) —;>?K t h e n I r ^ K ) - » 1r"(A) — ->TT(B) i s a weak k e r n e l , f o r i f t T ( x ) —9 TTCA) - * t r ( B ) i s z e r o ff(K) » K > > 1 T(A) * 7T(B) t h e r e i s a n i n d u c e d map i n t o t h e k e r n e l K , a n d s i n c e T f ( X ) i s p r o j e c t i v e , t h i s f a c t o r s o v e r 1 T ( K ) - » K . / / P r o p o s i t i o n 3 . 8 I f 0—-?P —>k —>B—»0 i s e x a c t i n cr/ , a n d P p r o j e c t i v e , t h e n A —>B i s m o n i c i n P r o o f L e t Q - * > B , Q p r o j e c t i v e , f o r m i n g p u l l - b a c k 0 — * P » E ~ > > Q — * 0 II i i 0 —-* p — , A —»> B —=> 0 19 Top row s p l i t s s i n c e Q i s p r o j e c t i v e , hence E i s p r o -j e c t i v e . I f X — » A — » B i s z e r o i n ef/P, i t f a c t o r s o v e r Q-»>B by Prop. 3.4; hence X---^E i n d u c e d i n t o p u l l - b a c k v 'A — so X —>A f a c t o r s o v e r the p r o j e c t i v e E and i s z e r o i n c//<f . So X—*A—•> B z e r o i m p l i e s X — » A I s z e r o and by d e f i n i t i o n A — * B i s monic. // 3.9 Example w i l l n o t , i n g e n e r a l , be a b e l i a n . Take tf= Ab, It = P - f r e e s . Then i n Ab / ? T , <Q —> <Q / ~2. i s monic and e p i but not an isomorphism. P r o o f HOM^ = 0 i m p l i e s HOM^(#,F) = 0 f o r F f r e e . Suppose — » X i n Ab, the n t h i s remains e p i i n Ab/9^ . F o r i f <B — * X—*A i s z e r o i n Ab/9T , i t f a c t o r s o v e r some f r e e $-*X—->A but (D —^F = 0 i m p l i e s *•> X—">A i s z e r o , f u r t h e r i m p l y i n g X •—9 A i s z e r o . I n p a r t i c u l a r i s e p i and by Prop. 3.8 i t i s monic, but t h i s c o u l d n o t be an isomorphism because HOM^C^ / Z , <Q) = 0 i m p l i e s a l s o HOM^^, ( ( $ / Z > & ) a 0 . / / P r o p o s i t i o n 3.10 L e t S be a f u l l s u b c a t e g o r y o f r e s o l v i n g p r o j e c t i v e s o f an a b e l i a n c a t e g o r y £ . Then the i n c l u s i o n ^^""^ & p r e s e r v e s k e r n e l s . P r o o f Suppose K—^C i s the k e r n e l o f G —>D i n . C l a i m -K—>C i s mpnic i n 0 . F o r i f N—=>K—»C i s z e r o l e t B-^>N, B i n & , t h e n B—v?N—*K —*C • 0 i m p l i e s B-*>N—?K - 0 f u r t h e r i m p l y i n g N -—>K = 0. L e t L — » C be the k e r n e l o f G—>D i n C and l e t B—>?L, B i n 6 . 20 Lc .K 5>C ->D '3 / - v • / t B > > L g exists since L = ker C—>D i n & , and K — » C monic implies g i s monic. ~k exists since K = ker C—*D i n e. C K > C C / • / 3 B *L and the monic L—>C can be cancelled hence B » L i s commutative, implying that g i s also epic, g i s then an isomorphism and K—>C i s also the kernel of C—>D i n C . // Corollary : "TV • preserves kernels. Proposition 3.11 (A remark of Freyd [ 8 ] , page 88) I f oijIP has kernels, the projective dimension of £/A - 2. Proof For N i n Z/J , choose 1f(B) — * "ft-(C)—* N — >0 exact, B,C i n fff, by Prop. 3.10 , 0 — * IT (A) » 1T(B) — » 1 ^ ( O — N > 0 for some A i n &j , since IT(A) i s proj e c t i v e , p.d.N ^ 2. // Lemma 3.12 I f ft (A) = Tf(B) © X then X - 1 T ( B « ) for some B'. Proof . I f 0 — * X — ^ ( A ) — > ^ (B) — * 0 i s an exact s p l i t t i n g , then the map Tf(A) — » fl* (B) arises from a map A—>B, and corresponds to the commutative diagram 0 K — * P — * A — » 0 I I i 0—=> L *» Q *B ^0 Forming the pull-back B' *A , Q * B 21 the sequence 0 — » B ' — » A ( ± ) Q — > B — = > 0 i s the coker of Tt ( A ) — 2 > 7 T ( B ) , which Is zero, and hence s p l i t s . Then X ^ ker: 7T(A)-3>7r (B) • kerTT (A©Q) —s»TT (B) » 7T(B') since ft* (A) = 7 T(A©Q). // The image of IT i s a resolving class of projectives, but are there others ? To answer t h i s , we mimic a res u l t of H. Bass, replacing free modules by elements of the image of nr. Assume co-complete, i f X i s projective i n then X © X1 = IT (A).* Then i f I i s a countable index set X(+)1T ( © A ) = X ( 9 © 1T(A) by 2.8 I I = x<£> (x« © x ) © (x« €)x) 6) . . . (x(±)x') Q (x<-px«) (£). . . = <£>7T(A) = TT ( O A) by 2.8 I I Proposition 3.13 I f '&f i s co-complete, then every projective i s of the form 7T(A) for some A. Proof Given X p r o j e c t i v e , one can determine a G such that X(±)TT(C) = 1T(C) by above. Now apply Lemma 3.12; // 3.14 The Syzygy Functor The functor was defined by choosing s p e c i f i c projective presentations f o r each object of , d i f f e r e n t choices giving r i s e to a functor naturally equivalent to IT . Associ-ated with / i f , define Z(A) by 0 — » Z ( A ) — » P — > A — » 0 = IT ( A ) . Z i s not a functor from */ to fff ; however i f the target i s OJf(P , then Z i s a functor. .... De f i n i ng Z on morphisms by 0 — » Z ( A ) — - » P > A » 0 A i I f 0 * Z(B) » Q *B ' 0 i f Z( f) i s well-defined, then i t w i l l c l e a r l y be an additive functor Cff—* /{P . To do t h i s , i t w i l l s u f f i c e to consider the case f = 0 , and prove Z( f ) = 0 ; that i s , Z( f ) factors over a.projective. However even more i s true : i f f factors (*for'some A since image resolves.) 22 over a projective ( f = 0 i n tfjP ) then so does the induced wap Z(A)—*Z(B). In f a c t , i f f factors over Q i n the above diagram, then by Prop. 2.4 Z(A)—>Z(B) factors through P . Hence Z i s a functor and factors of Z > OjJP It w i l l be more convenient to i d e n t i f y Z with the functor Of/p _ > Orf/cP . Suppose Z and Zf are d i f f e r e n t Syzygy functors, a r i s i n g from d i f f e r e n t presentations chosen. Consider K1 = K» I 1 K — * N » P' II X I K * P * A Let ,'.'.hA : K —> K © P' = K' (±) P — » K ' . Then h A i s an isomorphism i n Of/(P and determines a natural equivalence between Z and Z'. The n t h Syzygy functor i s then defined by Z n(A) = Z C Z ^ U ) ) , where Z i s now regarded as a functor GJjP—^tfjP . One can now extend Z to a functor — * £//6 using 3.15 (Freyd [ 8 ] , Prop. 1.2 ) For any abelian category C , i f e i s a f u l l subcategory of resolving p r e j e c t i v e s , then any functor ,9 abelian, has a unique right exact expansion ^ . E x p l i c i t l y for each C*C , choose B » — > B — » C — * 0 , B,B' i n & and define G(C) = coker : G(B') *G(B). Theorem 3.16 Let A = 0 — » A " — » A — * A ' — * 0 be exact i n Then there exists an exact sequence i n (columns are elements of Z/£ ), where P, P',P" are projective : w K" = K" — » K —*K'- > A" "> 0 x x x x x K v P" — »P'. * A » 0 Jr J , J , I 1 C v >f !? p ,K» ?A" > A »A» ^==r A' > 0 23 -w B e f o r e t h e p r o o f , some c o r o l l a r i e s . C o r o l l a r y 3.17 I f A = 0 — * A"-—» A-—» A'-—* 0 i s e x a c t i n ^ then 0 — * Z ( A " ) — » Z ( A ) — * Z ( A ' ) — * ; 0 i s e x a c t i n %5 ( u s i n g the embedding of/d* P r o o f U s i n g n o t a t i o n o f theorem, K = Z(A) , K» * Z(A') and K" = Z(A") i n tf/tP, hence i n &/d .// C o r o l l a r y 3-18 (Remark o f F r e y d [8 ] , page 109) ( i ) The e x t e n s i o n o f t h e Syzygy f u n c t o r t o t/S i s g i v e n by Z'(A) - 0 - * Z ( A " ) — » Z ( A ) — * Z ( A ' ) — > 0 . . ( i i ) 0 — » Z ( A ) — » - 3 r ( A " ) 7T(A)—» ^(A')'-"*£-*0 i s e x a c t i n P r o o f Prom Thm. 3.16 (A" ) —> "TT(A) —> V( A' ) —* A —»0 i s ex a c t f o r any e x a c t sequence A. By d e f i n i t i o n o f the e x t e n s i o n f u n c t o r Z(A) = c o k e r ( ^ ( Z ( A ) ) —* 7T(Z(A' ))) = cok e r (7T(K) —*>TT(K')) = 0—*K"-^ K—?K' — 0 (by theorem a p p l i e d t o K = -^K-* K' -*0 ) = 0 —» Z(A") ~-* Z(A)—» Z(A' ) — * 0 i n Z/d . // C o r o l l a r y 3.19 (i)1T i s a h a l f - e x a c t f u n c t o r °f—9 £/?S . ( i i ) '/T'ls r i g h t e x a c t i f f Z/d=0 i f f a l l s h o r t e x a c t sequences s p l i t , ( i i i ) I f TT i s l e f t e x a c t t h e n p r o j . dim C o r o l l a r y 3.20 (Remark o f F r e y d [ 8 ] , page 109) Z i s an e x a c t f u n c t o r *£/sd. P r o o f Z : t/sO-* i s the unique/\exact e x t e n s i o n o f Z : /(P £fj/(P , hence i t s u f f i c e s t o prove Z p r e s e r v e s monies. L e t f : A—*»B be monic i n f O •> A" * A -> A' —>0 H i i 0—y B" ? B =? B'-^O U s i n g the c a n o n i c a l f a c t o r i z a t i o n o f f g i v e n i n Theorem 2 . 7 , one can assume A" = B" and f" i s the i d e n -t i t y . But then 0 » z'(A) > TTU") — » 77/(A) f )l 4, 0 > 2 ( B ) • — » T ( A " ) — * -TT(B) i m p l i e s Z(A) — ^ Z ( B ) i s monic. // C o r o l l a r y 3.21 F o l l o w i n g i s a p r o j e c t i v e r e s o l u t i o n o f A : Remarks (a) The e x t e n s i o n o f the Syzygy f u n c t o r ^/P^f/P i s not the Syzygy a s s o c i a t e d w i t h ^5 but the 3rd Syzygy f u n c t o r , (b) S t a r t i n g w i t h the diagram Z(A' ) >P* >A' w / 1 H 0 * A" • ,A => A' * 0 , l e t f x=A -»A* , f 2=A"->A, f 3=-w:Z(A r )-»A" . T h i s g i v e s r i s e t o an i n f i n i t e sequence C o r o l l a r y 3.22 ( i ) I f f f a c t o r s o v e r a p r o j e c t i v e t h e n p.d. A £ m-1. ( i i ) I n p a r t i c u l a r , i f f ^ f a c t o r s o v er a p r o -j e c t i v e t h e n A = T r ( A ' ) , and i s p r o j e c t i v e . P r o o f I n the e x t e n s i o n o f (#) i n Thm. 3.16 t o the p r o j e c -t i v e r e s o l u t i o n g i v e n l n Cor. 3.21, the sequence o f maps f i s formed from the bottom row. I f f f a c t o r s r m m o v e r a p r o j e c t i v e t h e n the c o r r e s p o n d i n g map between e x a c t sequences i s z e r o . // (c) I f f : A-—»B i n of , l e t g : Q-*>B , Q p r o j e c t i v e . Then / f | . A <£) Q-^B, and 25 o n e c a n d e f i n e t h e p r o j e c t i v e d i m e n s i o n o f f as t h e p r o j e c t i v e d i m e n s i o n o f 0 — * K —•» A<3> Q — * B — * • 0 i n £/j*3 . C o r o l l a r y 3.23 ( a ) I f p . d . A " n t h e n p . d . A < 3n + 1, n ^ 0 , : p . d . A -6 n t h e n p . d . A 3n, rv 0 p . d . A ' ^ n t h e n p . d . A 3n-l, n ^ 1 p . d . A 1 = 0 t h e n A = 0. ( b ) I f p . d . f l / ^ n t h e n p . d . £ / J ^ 3n-l , n ^ l p . d . af m o i f f S/J = o. P r o o f TV* k i l l s p r o j e c t i v e s , a p p l y r e s o l u t i o n o f C o r . 3.21.// R e m a r k A l l r e s u l t s o f t h i s s e c t i o n d u a l i z e f o r i n j e c t i v e s , r e s u l t i n g i n a f u n c t o r u s i n g i n j e c t i v e c o - p r e s e n t a t i o n s . We a r e now r e a d y t o p r o v e T h m . 3.16. T h e p r o o f commences e x a c t l y a s t h e c o n s t r u c t i o n o f t h e l o n g E x t h o m o l o g y s e -q u e n c e , a n d i n f a c t T h m . 3.16 c o u l d b e p r o v e d u s i n g t h e l o n g E x t s e q u e n c e , b u t we p r e f e r t o w o r k w i t h i n t h e c a t e g o r y P r o o f o f T h e o r e m 3.16 L e t P ' — > A» a n d P " - ? > A " , P ' , P " p r o j e c t i v e . S e t P = P ' ( ± ) P " , t h e n f o r m ( + ) 0 0 0 I I I 0 0 0 T h e maps r , s , u , v r e s u l t f r o m t h e s p l i t t i n g o f P » — * p _ 5 , pt ^ v i a c a n o n i c a l p r o j e c t i o n s a n d i n j e c t i o n s a n d g i v e t h e p r o p e r t i e s o f P r o p . 2.4, s o t h a t s t a n d § a s s e q u e n c e maps a r e h o m o t o p i c t o z e r o . ( I ) Making use o f Thm. 2.7, one s t a r t s a p r o j e c t i v e r e s o l u t i o n o f A. K» < W »P'> > A" © P» I *J » A 1) * . h * K K » p ? p t _ _ _ > A i A « . L_ „ A 1 , A , w the i n d u c e d map on k e r n e l s , s i n c e t h e l o w e r r i g h t square commutes by (+). Top sequence i s t h e k e r n e l ; t o c o n t i n u e the r e s o l u t i o n , use Thm. 2.7 a g a i n t o f i n d ar* e p i from a p r o j e c t i v e sequence t o t h e k e r n e l . ( I I ) K * K ' ( ? ) P » A " ® P 1 K » P — ? A k ' i (-r,p ) l K" - 7 — A " © P » I 3 / •* A One need v e r i f y the bottom squares commute. F o r lo w e r l e f t , one needs k' fi' = /3p which i s c l e a r from ( + ) , and k' UJ = - / 3 r . F o r t h e second e q u a l i t y a p p l y the monic f . (k'w + / 3 r ) f = k'wf + /3 r f = k ' / 3 ' s + /3( ps) = k' /6 's - /3ps = (k« /3 • -/3p;)s • 0 A s i m i l a r c a l c u l a t i o n f o r l o w e r r i g h t s q u are. ( I l l ) C o n t i n u i n g t o p r o j e c t on the k e r n e l , the obvious c h o i c e i s (0,^»") ( p , r ) K „ ____ s, p _ . ? p , 0 A „ I II K 5> K» © P ? P ' 0 A " 27 H o w e v e r ( p , r ) : )?' ( £ ) ? " P ' G D A " ( P > r ) = (o = ( t h i s f o l l o w s b y d e f i n i t i o n o f r , a n d s f r o m p r o j e c -t i o n s a n d i n j e c t i o n s , a n d d i a g r a m c h a s e ) . S o t h e t o p row i s i s o m o r p h i c i n £/d t o 0 — 9 K " —9 P" — » A " — * 0 . W e u s e t h i s r e p r e s e n t a t i o n , a n d t h e n c o m p u t e t h e k e r n e l ( k , / 3 » ) K" K" I * K (+) P " » K ' © P » A " ( 0 , 1 ) | ( 0 , - 1 ) p n K ( k ' , / 3 ) K 1 P * P f (±)A" L o w e r l e f t commutes b y ( + ) . F o r l o w e r r i g h t , ( i p , - * r ) = ( 0 , - * " ) ( b y ( + ) ) . ( 0 , i ) \ p - r , F o r u p p e r r i g h t , C .1) C) w h e r e k ' w = - ^ S r i s p r o v e d i n ( I I ) . ( I V ) T h e k e r n e l o f ( I I I ) i s i s o m o r p h i c t o K " - * K - > K » v i a » ( - k » w - / 3 r , i r ) ( 0 , o i " ) K 0 I K° Y I K K" -> K -» K' K " (k,{3 ») ( l , u ) ( l , v ) K ' @ P K K' 2 8 A l l squares commute, the only n o n - t r i v i a l one being the upper ri g h t (l,u) / J. - (k ' , fi + ui) = (k ' ,v) = .<k'(i,v). That (1 -dip ) ~ 0 and (1-<^0)~O i s clear because l e f t sides for both are.the zero map and so are t r i v i a l l y homotopic to zero. I,II,III,IV est a b l i s h (#) of the theorem. // 29 CHAPTER *\. • THE FUNCTORIAL APPROACH Another method o f s t u d y i n g &j i s t o study the a s s o c i a t e d c a t e g o r y o f a d d i t i v e c o v a r i a n t f u n c t o r s from of t o Ab ( A b e l i a n groups) ( c o n t r a v a r i a n t ) . These f u n c t o r c a t e g o r i e s i n h e r i t most o f the p r o p e r t i e s p o i n t w i s e from Of, and u s i n g the Yoneda lemma, the assignment A—>(-,A) [ A — > ( A , - ) ] i s a f u l l - e m b e d d i n g o f &t as a c l a s s o f r e s o l v i n g p r o j e c t i v e s . However i f of i s not s m a l l , t h e s e f u n c t o r c a t e g o r i e s are to o l a r g e t o m a n i p u l a t e . To make the embedding ' t i g h t e r ' , one can c o n s i d e r the s u b - c a t e g o r y o f coherent f u n c t o r s . F i s coherent i f i t I s the c o k e r n e l o f a t r a n s f o r m a t i o n between r e p r e s e n t a b l e f u n c t o r s (which are s m a l l p r o j e c t i v e s i n the f u n c t o r c a t e g o r y , so coherent f u n c t o r s are analogues o f f i n i t e l y p r e s e n t e d modules i n the module c a t e g o r y ) . The f u l l s u b c a t e g o r y o f coherent f u n c t o r s i s a b e l i a n and has p r o j e c t i v e d i m e n s i o n at most two ( a q u i c k p r o o f : i f F i s c o k e r n e l o f ( - , B ) — * ( - , C ) , by Yoneda t h i s a r i s e s from a morphism B—>C, i f 0—> A—=>B—>C i s ex a c t t h e n 0—?(-,A)—?(-,B)—»(-,C) — 0 F —?>0 i s a p r o j e c t i v e r e s o l u t i o n o f F ) . Now the n o t i o n o f k i l l i n g p r o j e c t i v e s , by t h e passage of—* of'/(P , can be combined w i t h the study o f coh e r e n t f u n c t o r s by c o n s i d e r i n g the f u l l s u b c a t e g o r y o f coherent f u n c t o r s t h a t f a c t o r t h r o u g h of/(P ; t h a t i s , those coherent f u n c t o r s t h a t v a n i s h on p r o j e c t i v e s . With each coherent f u n c t o r , one can a s s o c i a t e a l e f t e x a c t sequence 0-^A—^B-*C where F = c o k e r (-,B)—=»(-,C). P r o p o s i t i o n 4.1 I f F i s c o n t r a v a r i a n t c o h e r e n t , and F = c o k e r (-,B) — > ( - , C ) , t h e n F f a c t o r s t h r o u g h Of/(p i f f 0 — ? A —*B —^C — ? 0 i s e x a c t . P r o o f ( ^=T> ) L e t B —?C —•> D —->0 be e x a c t . C o n s i d e r U/P ^ )fj , P p r o j e c t i v e (we assume B—5>C—>D —-? 0 s u f f i c i e n t p r o j e c t i v e s ) . 30 There i s an Induced s i n c e P i s p r o j e c t i v e . But 0 — > ( P , A ) — ? ( P , B ) — ^ ( P . C ) — > F ( P ) — 5 > 0 i s e x a c t =^ 0-^(P,A)—»(P,B)—(P,C)—» 0 i s e x a c t , s i n c e F ( P ) = 0 * f can be f a c t o r e d over B ^ " ^ v but the n v = B — * C — » D ~ * 0 D =7> P = 0 . ^ G i v e n 0 —» A —»B —* C—9 0 t h e n 0 — * ( - , A ) — = > ( - , B ) — ? ( - , C ) — ^ F — ? 0 i s e x a c t , so f o r any P s 0—» (P,A) — » (P,B) —» (P,C)-»F(P)—3»0 i s e x a c t . But any P—5>C can be f a c t o r e d t h r o u g h the e p i B-*>C, so (P,B)-3>>(P,C) , i m p l y i n g P(P) = 0. // Theorem 4.2 The assignment F » > 0 — > A - » B — * C — > 0 e s t a b l i s h e s an e q u i v a l e n c e between <y ( t h e ca t e g o r y o f coherent f u n c t o r s v a n i s h i n g on p r o j e c t i v e s ) , and . P r o o f To make t h i s a f u n c t o r we must f i r s t d e f i n e i t i n morphisms. Suppose F —>F' i s a n a t u r a l t r a n s f o r m a t i o n ' 0 - * ( - , A ) —»(-,B) -> (-,6) —» F — ? 0 ^ si' i * 0 — * ( - ,A' )—>(-,B» )~^(-,C« )-» F'—> 0 T h i s i n d u c e s a commutative diagram on t h e p r o j e c t i v e r e s o l u t i o n s o f F and F', and by Yoneda t h i s a r i s e s from a commutative diagram i n <V 0 — » A — * B -> C —> 0 j, X I 0—=>A'-*B«—-*C'—>0 and t h i s can be c o n s i d e r e d as a morphism i n To check t h a t t h i s i s w e l l - d e f i n e d , suppose 0 — * A —f B ? C - — * 0 f i l s i l h i ^ b o t h l n d u c e p~-? p'» 0 ?A'—»B'—?C<-—»0 i = 1,2. Then the d i f f e r e n c e w i l l i n d u c e t h e z e r o t r a n s f o r m a t i o n F — 7 F 1 : e v a l u a t e a t C : 31 (C,C) >F(C) > 0 I h r h 2 i 0 (C,B' ) — * (C,C' ) ? F ' ( C ) — > 0 F o l l o w 1 : C—-?C, i m p l i e s ^2~h2 i s i n the k e r n e l o f ( C , C * ) — > P ' ( C ) , ^ by e x a c t n e s s t h a t t h e r e e x i s t s V i n (C,B') such t h a t * Oh-^-hg) w h i c h means t h a t h-j-h,, f a c t o r s o ver . B'—>C*. So a p p l y i n g Prop. 2.4 t h e d i f f e r e n c e map on sequences i s homotopic t o z e r o , hence i s z e r o i n The f a c t t h a t t h e assignment i s w e l l - d e f i n e d e a s i l y y i e l d s t h a t i t i s a l s o f u n c t o r i a l . P o r the i n v e r s e , g i v e n an e x a c t sequence 0 — » A — » B — * C —> > 0 , a s s i g n the c o k e r n e l , and any morphism o f sequences i n d u c e s a unique t r a n s f o r m a t i o n on the c o k e r n e l s : 0 — * A -—»B—•> C 5>0 \ I j i n Cy«D l e a d s t o 0 3 A 1—=>B« —9 C — » 0 0 — ^ ( - , A ) — > (-,B) > (-,C) > F — 0 if i/ -V \i-0 — » ( - , A » )—>(-,B» )—->(-,C ) — > F ' ^ 0 T h i s g i v e s r i s e t o a f u n c t o r £~m> °f , and s p l i t sequences are a s s i g n e d the z e r o f u n c t o r , so i t y i e l d s a f u n c t o r . Now of —s> —> <y i s the i d e n t i t y . C o n s i d e r Z/A —> —* f/J, say 0 — ? A — 9 B —^ C — ^ 0 '—* P H> 0 - ^ A ' - > B ' - ^ C 1 - ? 0 , Then 0 -—» (-,A*)—=» (-,B' ) — » ( - , c ' ) — > F 0. 0 — 3 ( - * A ) » (-,B) >(-W,C) — > F - ^ 0 0 — , (-^j A' ) ( B ' ) —=> (-"^  C 1 )—-> F — a 0 T a k i n g the d i f f e r e n c e o f the ' . i d e n t i t y map and the c o m p o s i t i o n o f t h e s e maps, r e s u l t s i n the z e r o t r a n s -f o r m a t i o n n F — > F , so the i n d u c e d map o f d i f f e r e n c e s between sequences i s z e r o i n ~i[J) , and hence t h e map i n d u c e d from 0 *-> A* —^ B' —~> C* — 0 t o i t s e l f i s the 32 i d e n t i t y i n ^IM , hence of £/J>-^> i s naturally equivalent to the i d e n t i t y transformation.// The following hold by duality. Proposition 4.3 I f F i s a covariant coherent and F = coker : ( B , - ) ( A , - ) , then F factors through °f/*J I f f 0-5 A—* B-^ > C—=>0 i s exact. Theorem.4.4 The assignment F<—* 0—=? A — B 5 C — 0 establishes a contravariant equivalence between of (the category of covariant coherent functors vanishing on i n j e c t i v e s ) and . Corollary 4.5 There i s a contravariant equivalence — — — — — — — — — A w' between °f and of . Remarks Auslander proves that °7 i s abelian,so Thm. 4.1 would establish that i s abelian (Thm. 2.7).. The proof i s easy once i t has been established that the subcate-gory of coherent functors i s abelian, but this i s non-t r i v i a l (Auslander [2.] ). . We now examine the equivalences Of, c./tQ s <y and interpret r e s u l t s of Chapter 3 i n terms of functors. is 4.6 (a) Injectives i n °? The sequences 0—z> K —* P —=> A —>0 are projective i n "t/^d » for P p r o j e c t i v e , and i f Of i s co-complete a l l such projectives are of t h i s form (by 3.2 and 3.13). Under the contravariant equivalence <V , the r e s u l t i n g functor i s coker : (P,-)—*(K,-) = Ext • (A,-) , and so these are i n j e c t i v e i n O? . The projective objects 7fz^(A) correspond to the functors Ext n(A,-) and the projective resolution of Cor. 3.21 i s the standard long Ext homology sequence, truncated of the f i r s t three terms 0-»A->Ext' (A' ,-)->Ext' (A,-)->Exf (A" ,-)-?• Ext 2(A' , - ) - * . . . This i s an i n j e c t i v e co-resolution i n erf . 33 A (b) Projectives In Of The functors corresponding to sequences 0 —> K — P A —;> 0 under the equivalence => ^ 3 are coker : (-,P)—.;>(-,A). Now for fixed X, images of (X,P)—>(X,A) are those morphisms which factor through P and hence by Prop. 3.4 those morphisms factoring through any projective and so coker (X,P) —>(X,A) = (X,A) / <^(X,A) = Hony//? (X,A) (Prop. 3-5) Thus projectives are 'representable' functors Hom^,(-,A) , following H i l t o n [ ], we denote these as 7T(-,A) ( i t i s for t h i s reason we chose If: tf/f—> ^ /^> as embedding functor). Note that since 7T : °?/<P i s . f u l l ^ H o m w ( X , A ) ^ Homjr^ (7T(X), 7 T ( A ) ) . To carry the correspondance further, set y T N ( A ) = ^ ( Z N ( A ) ) , and ^ N ( - , A ) = ^ ( - , Z N ( A ) ) . Then the projective resolution of Cor. 3.21 i s - T r ^ A ' ) - * 7 ^ ( A ) - ^ ^T^(A' ) 71 ( A " ) - ^ 7 r ( A ) - ^ 7 T ( A ' ) - ^ A}0 and correspondingly a long homology sequence (Hilton [ ]) . . r ? 7 r 1 ( - , A ) - ^ 1 ( - , A ' ) ^7n;-,A " ) -> ; r(-,A) A« ) ^ > A-?O A which i s a projective resolution i n Cf . 4.7 (a) Injectives i n The sequence 0 — > k — > I — » N — * 0 Is i n j e c t i v e i n f o r I i n j e c t i v e , and i f of i s complete a l l i n j e c t i v e s are of this form (dual of 3.2 and 3.13). The co-Syzygy functor W can be defined on by 0 —^ k-^> I—? W( A)—?> 0 and W N(A) =W n_ 1(W(A)). F i n a l l y set Y (k) = 0 —"> A — 9 I — ^ W(A) — ? 0, and V N ( A ) = y^(W N(A)) , so that i s a full-embedding of as a co-resolving class of i n j e c t i v e s . 34 A Now examine the equivalence > °7 , P(A) ' * coker : (-,1) —>(-,N) = Ext«(-,A) and V n ( A ) '—>Ext n(-,A). The dual of Cor. 3.21 i s an i n j e c t i v e co-resolution of A =0 —* A"-—» A —» A' —* 0 , 0 —oA—? WA , f)-» V(A)~> V (A' )->y / 1(A")'^ ^ ( A ^ V ^ A * ) ~ > . . . and i n ty t h i s i s the truncated long Ext sequence 0 ^ A Ext * (- ,A" ) —* Ext * (-,A)-* Ext 1 ( - , A ' ) E x t 2 ( - , A " ) . . . (b) Projectives i n °7 With notation as above, these are of the form coker : (I ,-)"*> (A,.-) = (A,-) /<J(A,-) = *f (A,-) . In analogy with the 7T(-,A) functors, where aJiAfX) - maps A-^X which factor through an i n j e c t i v e , and s e t t i n g ^ n(A,-) = '•fCW (A),-) , one gets another homology sequence ... (A",-)-^> V(A',-)-» V'CA,-)-* Y ( A " , - ) - * A-^0... which i s a projective resolution i n NOTE V(A ,3E) = Hom^/(J (A,X) = H o m ^ ( <ft A), ^ ( X )) and ^(A,-) = Hom^/^ (A,-) i s representable. We would now l i k e to transfer some homological algebra into the category 4.8 Example The functor Ext'(-, ZL) i n Ab corresponds to the sequence 0 2 —> CBl/ 2 —•* O In £/i > which then corresponds to 1 ^ ( 2 , - ) i n Ab. The Whitehead conjecture i s Ext'(A,Z) = 0 ^ A i s projective A natural dual would then be V( 2",A) = 0 implies that A i s i n j e c t i v e . This holds. Proof We show A i s d i v i s i b l e . Let a^A and n an integer. We need to solve nx = a I Complete this diagram i n &//Q : a A 35 0 — T 0 = W ( Z ) A I A 0 —» A —» I — ^ W(A) ^ 0 = . V (A) . By assumption L>( *,A) « ; H o m 2 / ^ ( ^ ( £ ), ty-(A)) - 0, so there exists a f a c t o r i z a t i o n over ©(Prop. 2.4). To solve nx=a, follow 1 from Cfy—^k. // Proposition 4.9 (Hilton and Rees [ K ] , Cor. to Thm. 1.3) Every natural transformation O: Ext' (B,-) — ? E x t ' (A,-) i s induced by a map f : A—>B. Proof Using the contravariant equivalence of "2/<£> and &f , Q can be regarded as a morphism from IT (A) to TT (B) , but IT : &fI<P —^^/^ i s f u l l so t h i s morphism i s induced from a morphism A—>B i n tf/tP, and hence represents a morphism f i n <rf. // Proposition 4.10 A map f : A — B induces the zero map Ext'(B,-) —9 Ext 1(A,-) i f f f factors over a projective. Proof The map f : A—*B regarded i n #/<f i s zero i f f f factors over a projective. Since —=> i s f u l l , the contravariant equivalence of and <Y gives the r e s u l t . // Proposition 4.11 (Auslander and Bridger [ + ], Thm. 1.40) 0-* P ( A , B ) — 9 (A,B) — = 5 [Ext'(B,-),Ext'(A,-)] -9 0 i s exact. C , ] - natural transformations = Horn set i n functor category Proof 0-»P(A,B>—» (A,B) —9 TT (A,B) —^ 0 i s exact by defin-i t i o n . IT (A,B) = Hom«y/(p (A,B) = ttome/£ (TT (A),TT (B)) > 9Tl<P^>£l& i s f u l l = Hom^ - (Ext'(B,-),Ext•(A,-)) contravariant eq. of and &t = [ ( E x f ( B , - ) , E x t 1 ( A , - ) ] since the subcate-gory G7" i s f u l l i n the functor category.// 36 We also extend a resu l t of Hilton and Rees [ 14 ], . Thm. 2.1. Theorem 4.12 For f : A—=?B, the following are equivalent : (I) Ext'(B,-) >—9 Ext'(A,—) i s monic. ( i i ) Ext ,(B,-)>—»Ext'(A,-) s p l i t s , ( i i i ) There exists B' with Ext' (B ,-)>—»Ext 1 (A,-) • ^ E x t ' (B f ,-) ( s p l i t ) exact, (iv) IT (-,A)-^7 ir(-,B) e pi. (V) i r ( - , A ) - ^ 7 1T(-,B) s p l i t epi. (vi) There exists B» with 0 —=> IT (-,B') ">—=> IT (-,A) ~^ir (-,B) —>0 (s p l i t ) e x a c t . ( v i i ) /iT> ( A ) — ^ 7 l f ( B ) epi (in Z/A). ( v i i i ) 1T(A)-^7 <TT(B) s p l i t epi. (ix) There exists B 1 with 0 —^TT (B' ) —9 TT (A) —^TT (B) — ^ 0 s p l i t exact. (x) Given Q-^^B, Q projective, A © Q - J > > B s p l i t s i n . (xi) A-^>B i s s p l i t epi i n Proof I f ( v i i ) , ( v i i i ) and (ix) are equivalent, then category equivalences handle ( i ) through ( v i ) . (ix) =^(viii)=s> ( v i i ) t r i v i a l ( v i i ) ~ > ( v i i i ) since T[(B) i s projective i n . ( v i i i ) =$?(ix) by lemma 3.12. (ix) <~=^ (x) i s done i n proof of lemma 3.12. (xi) <5=> ( v i i ) since c//<p-s> i s f u n . // Theorem 4.14; I f *f i s co-complete, then every direct summand of Ext'(A,-) i s also of the form Ext'(B,-) for some object B. (Auslander [3]). Proof This i s a restatement of Prop. 3.13 using contra-variant equivalence of "—5> ^ .// 37 Corollary 4.13 For f : A — > B The following are equivalent : (i ) Ext'(B,-)—»Ext'(A,-) i s an isomorphism, ( i i ) I t (-,A)—>ir(-,B) i s an isomorphism, ( i i i ) If (A)—» i r(B) i s an isomorphism i n . (iv) Given Q-^> B , Q projective, then Q (£) A — » B s p l i t s and has a projective kernel, (v) There exist projectives P,Q with an isomorphism A © Q — * • B <£> P , where f i s the component A—>B. (vi) A—*-B i s an isomorphism i n &f]<P Proof Again ( i ) , ( i i ) and ( i i i ) are equivalent by category equivalences, and ( i i i ) <5=?> (vi) since % :°f/P >—* i s a f u l l y f a i t h f u l embed-ding, ( i i i ) =4>(iv) :By Thm. 4.12 B 1 >-> A©2->> B s p l i t s ; then f f ( B ' £ > B ) ^ 7T(B' ) © f (B) ~ 1T(A©Q) =T(A) implying ^ ( B ' ) = 0 so B' i s projective, (iv) (v) «=^> (vi) t r i v i a l . // Remark Condition (v) i s the d e f i n i t i o n of stable isomorphism. CHAPTER 5 PURE AND COPURE SUBCATEGORIES 38 We return to the i n t e r n a l structure of by consider-ing the subcategory of pure and copure sequences. Before doing so, we establish a few lemmas. An object i n an abelian category i s cal l e d small I f any map into an arbitrary sum - ' factors through a f i n i t e sum v i a the canonical map of the f i n i t e sum into the t o t a l sum. An object i s f i n i t e l y generated i f an epimorphism to i t from an arbitrary sum can be reduced to some f i n i t e sum and can remain an epi. Lemma 5.1 For any abelian category, a quotient of a small object i s small. Proof Let B be small, and C —*<JK^ Where C i s a quotient of B ® j x l • A f l n l t e sumfi? JX 1 X B » c —*<2>x. D exists which factors the composite map, since B i s small. Taking coker D , then B - » C -+®X±-i>?D i s zero, and B — C i s e p i , so can be cancelled. Hence C—*@X.± factors through ker : (£> X^  —}>D , which i s (J^X // Lemma 5.2 In any abelian category, a small projective Is f i n i t e l y generated. Proof IfOK^-^? P , P small projective, i t s p l i t s so P — — t h e id e n t i t y for some P—> © X ± . But this factors through a f i n i t e subsum P-^ © J x i ® x i —>P , and hence <3),X. —><$X..—^P i s epic. // 39 Lemma 5.3 I f C i s small i n of , then 0->A—>B—*C—>0 i s small i n 2//^ . Proof 0—*A -*B—* C ~*0 i s a quotient of 1T(C), so by lemma 5.1 i t s u f f i c e s to show TT(C) i s small. Given 0 —=> K P =* C » 0 = TT (C) I I ± 0 —><3lA.—> ®B.—> (££. *0 i i i then C — !»(3C i factors through a f i n i t e subsum and 0 — > K ? P * C * 0 0 — 9 ® T A —*<3TB —=»®,C 9 0 i ) i 0 0 A i —=»®B i *®C± * 0 is; a f a c t o r i z a t i o n of the sequence morphism through a f i n i t e subsum.// Lemma 5.4 I f C i s f i n i t e l y generated i n °j then 0 — A —» B — * C - * 0 i s f i n i t e l y generated i n ZJ& . Proof 0-H»A—s.B-^C-^0 i s a quotient of IT (C) , which i s small and protective, hence f i n i t e l y generated by lemma 5.2. Hence also 0 —» A—"» B -» C ~~> 0 i s f i n i t e l y generated.// Remark I t i s not true f o r abelian categories i n general that f i n i t e l y generated implies small or vice versa, and a direc t proof of 5.4 avoiding smallness Is n o n - t r i v i a l . Assume °f has a generating set of small projectives ( i n p a r t i c u l a r of w i l l be l o c a l l y small, i . e . every object has a set of subobjects). So one can consider the set of f i n i t e presentations 0—=>K—^P—=>A—^0 , P f i n i t e l y gener-ated p r o j e c t i v e , K f i n i t e l y generated. Let be the f u l l subcategory generated by thi s set. Objects of CT are quotients of di r e c t sums of f i n i t e presentations. Define *S such that Horn (Of 3&) = o, i . e . a sequence i s i n the only morphism from a sequence i n >J i s zero. Por &f = MOD R, / i * i s the class of pure sequences (in the sense of Cohn, remaining exact under tensoringj for a 40 proof, see my Masters Thesis, Gentle ['2.], or Fieldhouse[6]). So we adopt th i s terminology and c a l l & the category of pure sequences and CT the category of copure sequences. Clearly and -J are additive. Much of what follows i s standard 'torsion theory' simply applied to the p a i r (3*tjSf) but w i l l be included f o r completeness, and f o r ease of reference. Proposition 5.5 ( i ) ^ i s closed under quotients (taken i n */4 ). ( i i ) I f T^-^T* i s epi i n Cf', then i t i s epi i n £/4 . ( i i i ) Cjf i s closed under colimits (which are taken i n £/4 )• Proof (I) T r i v i a l by d e f i n i t i o n of & (this i s n o n - t r i v i a l i f one f i r s t defines purity i n Cohn's sense). ( i i ) Suppose T 1 -» T 2 i s epi i n J " , and l e t T x -> T 2 —» X be zero. Then T^ ~* J_2 -» T >-*X i s also zero (factoring T_2 —?X into epi-monic). This implies that ^ - ^ T g - ^ T i s zero. But T i s In CT by (I) so T_2 —=>T i s zero since T_2 —>T 1 i s epi i n -J . Hence T_2 —»X i s zero. ( i i i ) XT i s closed under dire c t sums, so combined with ( i ) gives r e s u l t . // Proposition 5.6 T^—» T_2 i s monic i n C i f f i t s kernel Is pure i n . Proof <^= C T i s additive, so we need only show T-dT^—>T 2 zero implies T-*T, zero. But T — * T, —f T~ factors ••• y through the pure kernel K. .:. r But then T—>K i s zero implying that T—^T.^ i s ajso zero. =^ Form the kernel K. I f T-*K with T i n 3 then T -^ K >-> T ^ T _ 2 i s zero, so T-»K i s zero. 41 Cancel the monic to get T-*K i s zero. Then by d e f i -n i t i o n , K i s pure. // Proposition 5.7 0*ls closed under extensions. Proof Suppose 0 -» T^ X -* T_2 0 i s exact. By d e f i n i t i o n of C, one can choose projectives (also copure) with P 1-^ >>T 1 7P_2 T_2 . Since X-»>T_2 the map P_2—>T2 factors over X-?>T2 . Then the sum map P1<3)P2'*J»X . Hence X Is a quotient of ? 1 © P 2 and i s i n 3*. // Theorem 5.8 For any sequence E there exists a subobject T i n T with 0 - > T - » E - » S - * 0 , S i n / S 1 ' , and T i s unique with t h i s property. (Character-ized as being the largest copure subobject of E . ) Proof Let E • 0 — A ~ » B ~ » C - * 0 , and { x J be the set of f i n i t e l y present objects of °l. For each X^ , l e t Y. = (±) X. where X. i s a copy of X. 1 g6(X 1,C; l j g 1 , s 1 for each g of Horn ^  (X^,C) . Then there i s a canonical map Y. = (£) X. — C , the image being the trace of 1 ( x i , c ) 1 X± i n C. Set Y =(±)Y 1 , and Y—*C the sum map. Now form the pull-back T from th i s map 0 —=> A — * E — » Y 9 o II i V 0 A — * B — C —> 0 T i s a copure subobject of E ( i t i s a quotient of TT(Y) and Y i s a direc t sum of f i n i t e l y presented objects). Claim T i s the sum of a l l copure subobjects of 0 — * A — > B — » C — * 0 . In f a c t , any copure subobject i s generated by Images of morphisms *TT(X^)~-*E , which are of the form Since X i —5>C factors naturally through X^ — ? Y — » C , there i s a f a c t o r i z a t i o n If (X^) —* 7T ( Y ) — * E . But the map 1T (Y)-* E factors as ir(Y)-=» TV* E . Hence a l l copure subobjects are contained i n T. To prove S i s pure, i t s u f f i c e s by Prop. 5.5 to snow i t has no copure subobjects. Suppose X>-*S_, X copure., In , form the pull-back 0 —> T —> E 1 —> X — * 0 li I I 0 —> T —^ > E — S _ —» .0 E' >—E i s monic (pull-back of monic i s monic). E' i s copure by Prop. 5 .7. But Tyls sum of a l l copure sub-objects so E'>—>E factors over T. Then T —> E* T —* E' T I - I I T —+ E E T —> E Cancel monic to get T —> E' . Hence E 1 >-» T i s 1 also epi and so an isomorphism, which then Implies T—>E' i s an isomorphism and X i t s cokernel i s zero. For uniqueness, I f 0 T' -» E—» S 1 -* 0 with T' copure, S' pure, then the maps induced on kernels and cokernels T — * E —=» S * It v T' — » E * S' T 9 E ? S Show that T = T 1 as subobjects of E. // Corollary 5.9 E Is copure i f f Horn (E,/SO = 0. Proof =^ By d e f i n i t i o n of A S * <#= Form 0-*T —>E—»S 0 as i n theorem. S=0 so T = E i s copure. // An object X i n 0 / i s c a l l e d pure projective i f given X 0 9 A —>B > C-^O = E with E pure, there i s a f a c t o r i z a t i o n over B. This i s equivalent to If(X) being i n J . By construction of $ , the set of f i n i t e l y presented objects are pure-projective,: and c l e a r l y direct sums of pure projectives are pure projective. Suppose now X i s pure pro j e c t i v e , so that 0—>L—9 p — — > 0 , P projective i s i n 3* . Then using the construction of the largest copure subobject, Y = © Y ± , Y = © X 1 1 g€(X i,X) 1' g the subobject 0 — ^ L — ? E — » Y — * 0 • n i l 0 — » L — » P »X — » 0 i s actually the sequence i t s e l f , i . e . the cokernel i s zero, i . e . i t s p l i t s . The cokernel i s 0 — > E — * P © Y — 9 X — = ? 0 . Hence X i s a direct summand of P © Y . Now Of has a generating set of f i n i t e l y generated projectives (by assumption) , so P can be taken as a direct sum of f i n i t e l y generated projectives, thus establishing Corollary 5.10 ( i ) An object X i s pure projective i f f i t i s a dir e c t summand of a di r e c t sum of f i n i t e l y presented objects. // For any C there i s a pure sequence 0 — * N — » X — » C —»»0 with X pure projective. In f a c t , take X = P © Y of the theorem, with P taken as a dire c t sum of f i n i t e l y generated projectives. This property i s usually stated as the property of ' s u f f i c i e n t pure projectives' i n the l i t e r a t u r e . This i s w e l l - j u s t i f i e d i n t u i t i v e l y , but also i n the following sense : that the sequence 0 —? N —3X—=> C — ? 0 i s projective i n $ . In f a c t , we have an exact sequence 0—=>T—> < TT(C)—*S—>0 where S = 0—>N—?X—*C—>0 i s pure and T i s the maximal copure subobject of TT(C). Suppose 0 T —»-7T(C) — * S — * 0 There i s an induced map from 'IC(C) by p r o j e c t i v i t y , but this w i l l factor through the cokernel of 0 — » T—*TT (C) since T i s copure. Then IXiC)—*S -JT(C) 1f(C)--**1 *1 Cancel epi to get S—>S_^ factoring over S_2-> .S^  . So S i s projective i n Now suppose E = 0—»A - ^ » B —> C — * 0 i s pure. Then T - — ^ TC(C) — * S shows S-»> E , v ' E ^ ' thus esta b l i s h i n g Corollary 5.10 ( i i ) Given C, there i s a sequence 0 — > N — — > C — » 0 In ^  which i s projective as an object of and X can be taken to be pure projective. Every pure sequence 0—* A—^>B —»C —*>0 i s a quotient of t h i s sequence. Hence as an abelian category,$ has s u f f i c i e n t projectives.// Lemma 5.11 ( ± ) jS^ i s closed under subobjects (taken i n ( i i ) "Jf i s pure i f f a l l S± are pure, ( i i i ) /S' i s closed under l i m i t s (taken i n ~£jh ). Proof ( i ) S i s pure i f f S has no copure subobjects, so (I) i s t r i v i a l . ( i i ) X—*W S ± i s zero i f f X - ^ T T S j — i s zero for a l l i , so ( i i ) follows from Horn (SyS) = 0. ( i i i ) follows from (I) and ( i i ) . // Theorem 5.12 $>-> ~^/jS ^ s a f u l l exact embedding. (l.e.S i s an abelian f u l l subcategory of and the incl u s i o n i s exact). Proof It w i l l s u f f i c e to prove $ i s closed under quotients taken i n . Let S-*> S». We need to show that 'ft'(C)—>S* i s zero f o r C f i n i t e l y presented; but ft(C)—3>S f factors over S by p r o j e c t i v i t y , and th i s must be zero since S i s pure. // 45 Proposition 5.13 x ^ i s closed under co - l i m i t s . Proof By Thm. 5.12, i t w i l l s u f f i c e to show A S * i s closed under dire c t sums. I f VT(C)—*<S>Si , then since 7 C ( C ) i s small by lemma 5.3 ( i f has a generating set of small projectives then a f i n i t e l y generated object i s a quotient of a small object, hence small; so C i s small), we have 1f(C)-»(i)j 2> —^GlS^ for some f i n i t e subsum, but a f i n i t e sum i s also a f i n i t e product, so ©jS^ i s pure by 5.11 and 1t(C)—> 0 j S± i s then zero. // Proposition 5.14 $ i s dense i n ^ 5 (Closed under sub-objects, quotients and extensions). Proof A l l that Is needed i s extensions : Consider 0 —> S 1 — » X —> S 2—•» 0 , with T copure, Then T —>X factors through the kernel of X—»S_ 2 since T—»X—>S_ 2 i s zero. But then also T —PS^ i s zero, r e s u l t -ing i n T-—>X zero. // We now r e c a l l the d e f i n i t i o n of a torsion theory (Dickson [ 5 ], pages 223-235) for an abelian category & : i s a couple (S^fi) of classes of objects of C s a t i s f y i n g (±)jn ? = lo} ( i i ) I f T - » A — ^ 0 i s exact with T t J, then A6 S . ( i i i ) I f 0—5»A—»F i s exact with P e # , then A e $ . ( i v ) For each object X of C , there i s an exact sequence 0 —*T —*X —»F —»0 with T t J , Fer ft , The p a i r ( J * i s thus a torsion theory for , and by theorem 5.12, ^ i s closed under quotients, so i t i s cohereditary. We w i l l return to the study of these subcategories as a torsion theory i n a l a t e r chapter. Proposition 5.15 ( i ) The inclus i o n has a right adjoint. 46 ( i i ) The i n c l u s i o n JU. : $ —> £ ^ has a l e f t adjoint. Proof Define t(E) and r(E) by 0 —> t(E) —•> E —»r(E)-—» 0 with t(E) copure and r(E) pure (using Thm. 5.8) (for torsion theories t the r a d i c a l , r the c o r a d i c a l ) . Then the uniqueness and Horn = 0 easily shows t and r are functors Hom^ (T,t(E)) ~ Homfe/^j (\J (T), E) i s simply the statement that CT i s closed under quotients. Hom^ (r(E),S) * H o m ^ ( E ^ S ) ) assigns E — 5 u(S) the induced map E — * S r - i * r(E) out of the cokernel r ( E ) , since t(E) I E » S i s zero. // Remarks J i s generated by f i n i t e l y presented, a set of small projectives. This would y i e l d an abundance of results i f S were abelian because then $ would be equivalent to a functor category. Unfortunately T w i l l rarely be abelian as i s suggested by prop. 5.6. On the other hand, jSf i s abelian, however i t i s doubtful that i t w i l l have a generating set of small projectives. Indeed, i t w i l l even l i k e l y not be l o c a l l y small. That i s , subobjects of a given object may not form a set. As to throwing the pair (CT,^) into the machinery of torsion theories and l o c a l i z a t i o n , the major obstruction i s that a l l such l i t e r a t u r e on the subject imposes a minimum condition that the underlying category be l o c a l l y small, and more usually that i t i s Grothendieck. Just when i s Grothendieck, or even just l o c a l l y small ? The next section w i l l take t h i s subject up. H 7 5.16 F i n i t e l y Presented Objects (a) We assume &f i s equivalent to a functor category (co-complete with a set of generating small p r o j e c t i v e s ) . In p a r t i c u l a r , Of i s Grothendieck and every object i s a d i r e c t l i m i t of i t s f i n i t e l y generated subobjects. Suppose X i s f i n i t e l y generated. Then X i s a quotient of a f i n i t e l y generated pr o j e c t i v e . Form 0 — * K — * P — » X —*0, then K = 11m K± , f i n i t e l y generated subobjects of K. Then 0 — * K, —=> P — * X. —=> 0 I1 II i1 0—=» K — » P —-» X —=» 0 The X± are f i n i t e l y presented and X = l i ^ X ± . Every object w i l l be a d i r e c t l i m i t of i t s f i n i t e l y generated subobjects, which i n turn are dire c t l i m i t s of f i n i t e l y presented objects. Combining these l i m i t s gives that every object i s a d i r e c t l i m i t of f i n i t e l y presented objects. (b) Suppose P i s a small projective. Consider a map * l i m Y, (d i r e c t l i m i t over a directed s e t ) . Now we * i have an epi © Y i — » lim Y^ , so by p r o j e c t i v i t y P © Y± ?> lim Y ± But P i s small so t h i s can be reduced to a f i n i t e subsum. Then since t h i s i s over a directed s et, there exists a Y, with P * lim Y., . This establishes V -v-lim Horn (P,Y 1) ~ Horn (P,lim Y ±) , where lim Horn (PjY^) —> Horn (£,liin Y.^ ) i s the unique map out of the direct l i m i t Induced by the compatible maps Horn (PjY^)—=»Hom (P,liin Y^) which arose from 48 Proposition 5.17 (stated without proof by Stenstrom [24], page 323) I f A i s f i n i t e l y presented, then for any dir e c t system (Y i) , 11m Hom (A,Y ±) ~ Hom (A,lim Y±) . Conversely, i f lim Hom (A i Y ± ) - — * > Hom (A, limM^) for any directed system, then A i s f i n i t e l y presented. Proof Let P• —» P —> A —* 0 be exact, P, P* small projectives. 0—* lim Hom (A,Y,) — * lim Hom (P,Y,) >lim Horn (P' ,Y,) 0 -->Hom (A,lim Y±) — 9 Hom (P,lim Y±) — » Hom (P 1 ,11m Y±) implies that the l e f t side i s also an isomorphism. Conversely, i f A = 11m A i f o r some directed system (A^) of f i n i t e l y presented objects, and i f lim Hom ( A , A 1 ) — » H o m (A,lim A^) = Hom (A,A) then the i d e n t i t y factors over some A^ : A. 1 v . Hence A i s a dir e c t summand of a f i n i t e l y presented object and i s also f i n i t e l y presented, / / 5.18 Construction of Pure Sequences Suppose 0—*A—*B—>C—"?0 i s exact, and C = lim C^ , f i n i t e l y presented. Form the pull-backs 0 — * A # B, -—» C,—•> 0 li I 1 1 0^—* A * B »C *0 Then i n the category <5 , 0 — ^ A — » B — » C — * 0 i s the direct l i m i t of 0 — * A —>B^—* C±—=>0 . • Now i f 0 — * A — » B — - > C — » 0 i s also pure, then each 0 — ? > A — > B ^ — — 9 0 s p l i t s since i t i s a copure subobject of a pure object, hence zero. So a pure sequence i s a direct l i m i t of s p l i t sequences, i n . • Conversely, given such a di r e c t l i m i t and a map from a f i n i t e l y presented object X factors over C. for some j , so X 0 •> A, B, Z * C. > 0 V I 1 0 —>llm A±—z lim B i — » lim C± shows X — » l i m factors over lim B^ ^ . This shows the l i m i t sequence i s pure, establishing Proposition 5.19 0 — * A — » B — » C — > 0 i s pure i f f i t i s a dir e c t l i m i t , i n <E , of s p l i t sequences.// Corollary 5.20 0 —=><£) A ± — — > E — » 0 i s pure. Proof This i s the dire c t l i m i t of sequences 0 — J » © j A 1 — > T A ± — E j — > 0 , J f i n i t e . // Corollary 5.21 (of Prop. 5.17) The sequence 0 » K —»(?) A ± — » lim A± > 0 used i n the construction of dire c t l i m i t s from the sum i s pure. Proof I f X Is f i n i t e l y presented, then any X — » lim A i factors through some A^, hence through © A^ 50 CHAPTER 6 PURE SEMISTMPLE CATEGORIES Rather than impose that be Grothendieck, we w i l l f i n d s u f f i c i e n t conditions on of to force ^-/^d to become Grothendieck. Suppose ~£/A has a generator 0 — = » A — » B — > C — * 0 , so 7t(C) also i s a generator. C i s an object of and we can assume i t generates ty . ( I f necessary, replace C by C(±)U, with U generating ty , and 0 —> A—» B —• C —*0 with 0 —? A —*B (£)U —>C © U —*0.) Given X in of , form E = 0—> K —»0jC —=>X — 0 for some direct sum of C. Now Jt(C) generates t h i s sequence, so Gfc) 7f(C)—» E f o r some index set K. Then the cokernel must s p l i t , and th i s i s the sequence 0 - » L —•» ( ( 2 ) K 0 ) 0 ( 0 3 - C) — ^ X — > 0 , L the kernel of the sum map. This gives the n o n t r i v i a l part of Proposition 6.1 has a generator i f f there exists an object C i n such that every object of of i s a dir e c t summand of a di r e c t sum of copies of C. // By Prop. 2.8, i f °f i s (co-)complete, then so i s EL^ . (*) , This i s condition Ab 3 . Condition Ab 4 i s : given a family of monies •^ Ai—=> B ^ , then (±) h^-^G) B ± i s monic (M i t c h e l l c a l l s t h i s condition C 1 ) . Proposition 6.2 Of, Ab 4 M , Ab 4. Proof I f A^ — * B^ are monic, r e a l i z e these as 0 —> A i" —> k± — * 1 —=> 0 0 — » B " > B ± Bj^' 9 0 Then the kernel 0 - * A ± " - * A ± ( ± ) B ^ — > E ± — * 0 s p l i t s . * 0 The sum map i s 0 —5>(±)A. » — » ( 3 A . — ^ S A , ' -0 ~ D < 2 B 1 ' , - ^ 3 B 1 — = 9 ^ ' -with kernel 0 -XSA^' * (®B^") 0 ( @ A ± ) — » E — ^ 0 . 0 But (®B±") ® & ± ) ~ © (B±" <DA±) , so the sum of the s p l i t t i n g maps s p l i t s the kernel sequence. Hence © A ^ — ^ © ! ^ i s monic. // A category i s C 2 i f for any di r e c t s u m © X ^ , the natural map i s monic. Module categories are t r i v i a l l y C 2, i n fact most reasonable categories are C 2. However thi s i s a very deep imposition on. ~£/^ > . Proposition 6.3 i s C 2 i f f given any set of monies C j — i n <y, the map <£)C± ![ C ^ i ^ ) s p l i t s . Proof This i s simply a restatement of the d e f i n i t i o n from tl& to Cff , l.e.®C±>-* ( l e ^ i e ^ H E - ^ O i s the kernel of the map from sum to product. // Lemma 6. 4 Given A—=>B monic,and A—*>I, I i n j e c t i v e . I f A ?—> B <£)I s p l i t s , then A >-» B s p l i t s . Proof L i f t A — * I to B - ^ - » I . Then " ( i ) ~ A >-» B > B © I = A>—> B © I s p l i t s and hence also A >—>B s p l i t s . (Note In jargon, the kernel of 0 —•* A — > B — » N — > 0 r° i i i 0 » I > E •» N —^0 s p l i t s hence i s zero, so this i s an isomorphism, but the bottom row s p l i t s because I i s i n j e c t i v e , hence top row also s p l i t s . ) // A l o c a l l y Noetherian category i s a Grothendieck category having a set of Noetherian generators ( i . e . Mod R i s l o c a l l y Noetherian i f f R i s Noetherian). When OJ has a generating set of f i n i t e l y generated objects, t h i s i s the equivalent to the condition that the di r e c t sum of in j e c t i v e s i s i n j e c t i v e . Corollary 6.5 I f $1 i s l o c a l l y Noetherian, ^/>^ i s C 2 i f f © A ^ *"TVA^ s p l i t s for any direc t sum. Proof Take A±—» I ± , I i i n j e c t i v e , apply Prop. 6.3 and Lemma 6.4. // 52 At t h i s point Of w i l l become a module category over a r i n g R, although much of what follows probably generalizes to functor categories. D e f i n i t i o n (a) M i s Pure - i n j e c t i v e i f given any map 0 _ * A — » B - » C — » 0 i M with 0—>A-^»B—»C—*0 pure, A—>M factors through B. This Is more frequently c a l l e d a l g e b r a i c a l l y compact, (b) M i s 21 - a l g e b r a i c a l l y compact ( pure-i n j e c t i v e ) i f any di r e c t sum of copies of M i s al g e b r a i c a l l y compact. By a Theorem of Wolfgang Zimmermann [ 2 7 ] , this i s equivalent to 0 j M—*~TT I M s p l i t t i n g f or arbit r a r y sums of copies of M. (Note - Cor. 5.20 gives the implication one way.) Theorem 6.6 The following are equivalent : (i ) A l l modules are al g e b r a i c a l l y compact (pure i n j e c t i v e ) . ( i i ) A l l modules are pure projective, ( i i i ) A l l pure sequences s p l i t , (iv) A l l sequences are copure. (v) i s C 2 and &7 i s l o c a l l y Noetherian. (vi) S/^i> i s C 2 and has a generator, ( v i i ) i s Grothendieck. ( v i i i ) -£\j£> i s equivalent to a functor category. Proof Equivalence of ( i ) , ( i i ) , ( i i i ) , a n d (iv) i s playing with language. ( i v ) ^ ( v i i i ) The ( K - ^ P — » A ^ = of of f i n i t e projective presentations, P f i n i t e l y generated p r o j e c t i v e , K f i n i t e l y generated, i s a generating set of small projectives i n the co-complete abelian category Hence "E/^ i s equivalent to (<£ , Ab). ( v i i i ) ^ ( v i i ) -=f> (vi) t r i v i a l . 53 (vi) ^ ( v ) By Prop. 6 . 1 , every object of CTj- Mod R i s a d i r e c t summand of a di r e c t sum of copies of some fixed module (subobject would s u f f i c e ) . This implies that R i s Noetherian (e.g. F u l l e r & Anderson [ 1 ] , page 297 Cor. 2 6 . 3 ) . (v) -=^(i) By Cor. 6 . 5 , © M - ^ T T M s p l i t s for arbit r a r y sums. Applying the Zimmermann re s u l t gives that every M i s /C a l g e b r a i c a l l y compact.// Corollary 6 . 7 A rin g R s a t i s f y i n g the condition of the theorem i s a r t i n i a n . Proof Since a l l pure sequences s p l i t , f l a t projective (M i s f l a t i f f 0 —^A —^ B —» M—» 0 pure for a l l such sequences). Hence R i s both perfect and Noetherian, which implies A r t i n i a n . // Corollary 6 . 8 The conditions of Thm. 6 . 6 are also equivalent to : every module i s a d i r e c t sum of f i n i t e l y generated modules. Proof A module i s pure-projective i f f i t i s a d i r e c t summand of a direc t sum of f i n i t e l y presented modules by Cor, 5 . 1 0 ( i ) . Now each f i n i t e l y presented module i s a direc t sum of indecomposables (necessarily f i n i t e l y presented) with l o c a l endo morphism rings since R i s Art i n i a n . By a theorem of Crawley-Jonsson-Warfield and i t s corollary ([ ], pages 2 9 9 - 3 0 0 ) , any d i r e c t summand of a dire c t sum of f i n i t e l y presented modules with l o c a l endomorphism rings i s again of thi s form. Conversely, R Is Noetherian (again see reference i n (vi) =^>(v) of theorem). Hence we can assume f i n i t e l y presented instead of f i n i t e l y generated, and so Cor. 5 . 1 0 implies that a l l modules are pure-projective. // 54 6.9 Remarks Can the condition of Cor. 6.8 be weakened to 'every module i s a dir e c t sum of indecomposables' ? For there i s a s t r i k i n g s i m i l a r i t y with the rings s a t i s f y i n g Thm.6.6 and semi-simple rings. For semi-simple rings, one has that ( i ) a l l sequences s p l i t ( Z<1& = 0) ; and ( i i ) a l l modules are direct sums of simples are equivalent statements. Replacing a l l sequences by pure sequences and simples by f i n i t e l y generated indecomposables, the equivalenbe remains i n t a c t . Daniel Simson has coined (or at least promotes) the name pure semi-simple rings for the rings s a t i s f y i n g pure =^  s p l i t . For semi-simple rings, one has the Wedderburn structure theorem, which uses matrix rings as b u i l d i n g blocks. Is there a structure theorem for pure-semi-simple rin g s , and what i s the suitable replacement for matrix rings (simple rings) ? The Wedderburn theorem yields two important results : That r i g h t semi-simple l e f t semi-simple, and a quick proof that there are only f i n i t e l y many non-isomorphic simples. Even i f there i s no structure theorem akin to the Wedderburn, i s i t true x^ -hat right pure semi-simple =^ l e f t pure semi-simple , and i s there only a f i n i t e number of non-isomorphic ( f i n i t e l y generated) indecomposables ? Towards a solution of these problems, M. Auslander has shown that a r i n g i s both r i g h t and l e f t pure semi-simple i f f i t i s of f i n i t e representation type ( l e f t A r t i n i a n , with a f i n i t e number of non-isomorphic f i n i t e l y generated l e f t decompo-sables). So the problem becomes one of showing that l e f t pure semi-simple rings are of f i n i t e representation type. A great deal of e f f o r t has been put into t h i s . I had the opportunity to t a l k to M.Auslander (and others at the Canadian Mathematical Conference, December, 1980) concern-ing t h i s problem. M.Auslander at f i r s t thought the proof would be straight-forward, and i n fact thought he had solved i t , but caught his own error when writing i t up. Kent F u l l e r was l ^ s s fortunate and Auslander caught his 55 mistake during Puller's presentation at a r i n g theory conference.D.Simson even less fortunate, published an incorrect proof, again error was pointed put by Auslander, (see Simson [23]). L.Gruson also believed he had solved the problem but f e l l short of completion. As of December 1980, Auslander s t i l l f e l t that l e f t pure semi-simple f i n i t e representation type, but had stopped working on the problem. Kent F u l l e r also had given up, commenting that he f e l t i t was *undecidable', and that a solution would involve set theoretic considerations (akin to Martin's axiom f o r the solution of the Whitehead conjecture that Ext'(A ,"2.) =0 T> A i s f r e e ) , and D.Simson now believes the conjecture i s false.(see Simson [23]). It i s unfortunate that at present I cannot conquer the dragon. However I hope that the previous discussion indicates that this i s an important area of i n v e s t i g a t i o n . So the following results may seem lacking i n content standing on t h e i r own, but the hope i s that they can be used as b u i l d i n g blocks towards a solution. The f i r s t move towards a solution w i l l be to express the condition of f i n i t e representation type into a statement concerning (a more categorical condition). Proposition 6.10 A r i n g R i s of f i n i t e representation type i f f i s equivalent to a module category. Proof ^ Rings of f i n i t e represented type s a t i s f y the conditions of Cor. 6.8 (see f o r instance F u l l e r & Reiten [»°]). So S/^> i s a functor category with the set of ^ K—? P—* A^ of f i n i t e projective presentations as a set of generators. But i f there i s only a f i n i t e set of f i n i t e l y generated indecomposables thi s can further be reduced to a f i n i t e set. So has a small projective generator and i s co-com-pl e t e , implying that i t i s equivalent to a module category (the r i n g being the endomorphism r i n g of the small projective generator). 56 By Thm. 6 .6 , S/^S i s a functor category with £/7T (A)J = [ K—>P —-»A} , with A indecomposable f i n i t e l y presented, as a set of small projective generators. Lemma 6.11 ( i ) I f A has a l o c a l endomorphism r i n g , then ft (A) i s Indecomposable projective (also with l o c a l endo^morphism) ( i i ) 0 y TC (A) = 7T(A') i f f : A » A F (where A' has l o c a l endomorphism). Proof, ( i ) T r i v i a l since End ( i f (A)) ^ End A/P(A) ( P(A) = endomorphisms factoring over a projective) ( i i ) I f 7r (A) ^ TT (A') t h i s isomorphism must arise from a map A—>A! inducing the isomorphism 0 —> K- A - ^ * 0 \t 1 >l 0—> K' —>P' —» A' —=» 0 But then the cokernel i s zero, i . e . the sequence 0 — » E — » P ' ( ± ) A — = > A ' — 0 s p l i t s . But since A' has a l o c a l endomorphism r i n g , P 1 ©A—=»A'—=>0 s p l i t s , forcing P 1—^A'— * 0 or A - * A ' — > 0 to s p l i t (see Lemma 6.14 ahead). The f i r s t would imply Tr(A') = 0 ( i . e . A' p r o j e c t i v e ) . Hence A —» A' —» 0 s p l i t s , and so must be an isomorphism.// Proof of Proposition 6.10 continued So {lC{h)} , A indecomposable f i n i t e l y generated, i s a generating set of small indecomposable projectives with l o c a l endomorphism rings. Then U = (±)7T(A.) , l e i taken over the set { A ^ of nonisomorphic indecomposable f i n i t e l y generated modules, i s a .generator. Now, by assumption, has a small projective generator, so there i s an epi ® U — » V . But then the s p l i t t i n g monic V > ~ * © U can be reduced to a f i n i t e subsum of copies of U, and then further reduced to a f i n i t e subsum of ( i f (A )] . That i s , there i s a s p l i t t i n g (±)7T(A,); 1 J f i n i t e 1 this Implies that ® 7 T ( A 1 ) i s a generator. Now f i n i t e 57 given any A^ , TC(Aj) i s the ( s p l i t ) epi image of ( <j>> X(A,))n—->?r(A.) f i n i t e 1 J f o r some n. But "7C ( ) has a l o c a l endomorphism r i n g so.'TTCA^)—»7T(Aj) s p l i t s f o r some i , by the Lemma 6.11, A^ ^  Aj . Hence there are only f i n i t e l y many nonisomorphic A^ . Hence R i s of f i n i t e representa-t i o n type (R i s Artinian by Cor. 6.7). // 6.12 Remarks Thm. 6.6 was proved for of a module category, the c r u c i a l step being the use of the Zimmerman resu l t that M i s 21 - a l g e b r a i c a l l y compact i f f (S^ . M—»"TFjM s p l i t s f or arbitrary index sets I. A l l the rest of the theorem i s v a l i d providing that Cfj has a generating set of small projectives and i s co-complete ( i . e . a functor category). The Zimmerman resu l t probably holds for functor categories, but I have not proved i t yet. This raises the question of generalization simply for the sake of getting new r e s u l t s . The theory surrounding Thm. 6.6 arises from pure-semi-simple rings which are important enough to neglect the added constraint of proving the re s u l t for functor categories. However one i s faced with problems which can not be dismissed as just 'generalizing'. For the conjecture i s now: functor category =^ ^/^S module category . I f the problem i s to be solved, a f i r s t b attle plan would be to study as a functor category. To a great extent this w i l l require v e r i f y i n g that certain r i n g theory results hold i n *-• functor categories, so we w i l l assume for the next section that of i s a functor category with a generating set of small projectives. For the results concerning semi-perfect objects, the proofs i n the r i n g theory case (module categories) can be found i n Mares [ IB] . 58 6 , 1 3 Baer-Injective Test X i s i n j e c t i v e i f f X i s i n j e c t i v e r e l a t i v e to [p„^ . Proof Consider A>—>B . Let A be a maximal extension X i n B. (Zorn's lemma can be. used since there i s only a f i n i t e set of non-isomorphic monies into B). Assume A B. Form K — > P,—> C It (C) A —•> E — * C E' •ii i r x A —>B — * C E \> \> v i X >I >X' (X) where C i s a non-zero f i n i t e l y generated subobject of C , by assumption K —>A—>X can be l i f t e d to P^  , i . e . f t ( C ) — » E ' >—> E — » C i 9 ( X ) = 0 where O (X) i s the Injective sequence. Hence E ' ^ E — ^ ( X ) = 0 which means A — » X can be extended to E (Prop. 2.4). But E >—» B since i t i s a pull-back of the monic C W C . Then the maximality of A implies that A>—» E i s an Isomorphism =^  C = 0 a contradiction. // Lemma 6.14 I f C has a l o c a l endomorphism r i n g and N C£ n © C ± ' — > 7 C s p l i t then C ± ><±)C±—> C s p l i t s for sp^ me i . Proof Let 01 ± « C 1>^(£)C 1 —>> C and n \f± = C>—><DC±—»C± where ^ i s the s p l i t . Then V ± U?± = l c i n End C. So at least one of the i s a unit, implying that C? 1 i s a s p l i t e p i . // Lemma 6 . 1 5 I f P i s indecomposable projective with a l o c a l endomorphism rin g , then P — » X i s a projective cover. (That i s , a l l subobjects are superfluous.) 59 Proof Let 0 — > K — * P — 2 > X —^Q. Assume K—*P i s not superfluous, then there exists 1C—* P with K + Y = P. Form 0 — * K — » P » X — » 0 1t(X) i i ll ^ \^  II 0 — * K'—» Y — * X — 0 . I r II 0 — * K — » P — * X — = » 0 It (X) Let 1? be P-*Y-*P. Consider the difference 1-0 —> K — » P »>X ^ o 0 » K ? P — » X » o Commutativity implies 1— U? i s not a unit i n End P, hence # must be a unit since End P i s l o c a l . Hence Y<=->P i s also e p i , implying that Y = P. // Remark I f C i s f i n i t e l y generated with l o c a l endomorphism r i n g , then 1t(C) i s a small projective with .local endo-morphism r i n g i n , and a l l Its subobjects are super-fluous. I f 7T(C) had only a set of subobjects, the t o t a l sum would be the unique maximal subobject of "7T(C) (since "7t(C) i s a small object, a direct sum of proper subobjects i s proper ) , and so the quotient would be simple. Un-fortunately, this quick way of producing simples w i l l f a i l i f there i s more than a set of subobjects. Lemma 6 . 1 6 Let 0 / X be a small subobject of A, such that X $= N, N A. Then there i s a subobject "Sf containing N maximal with respect to not contain-ing X. Proof Order the subobjects of A containing N but not X. I f c—» Ng^—* N^'^-^Njj ... i s an ascending chain, l e t N = U N i . Since X i s small, X £ N. This i s the essence of smallness, so we give a proof : l e t P be a small projective with P—»X. Form p I. V X 60 •^ UNj^  = N assuming X N P i s small so there i s a factor-i z a t i o n through a f i n i t e subsum P N / N, By diagram chase, P I v X .N N / N k = 0 • N/N and the epi P X can be cancelled, so X—* N i s zero and X—>N can be factored through N k c—> N , that i s X <^-? N^ contradiction. Hence Zorn's lemma can be applied to achieve maximal elements. // Corollary 6 . 1 7 I f X + N = A and N i s maximal with respect to containing N but not X, then N i s a maximal subobject of A. // Corollary 6 . 1 8 Rad A = 2L" superfluous subobjects of A. Proof D e f i n i t i o n of Rad A - f\ maximal subobjects. I f X i s superfluous, and M i s maximal then X + M j ^ A ^ X M. Since any object i s the directed l i m i t of i t s f i n i t e l y generated subobjects, and f i n i t e l y generated implies small ( i f there i s a set of small projective generators, see Lemma 5 . 2 ) , i t su f f i c e s to show that small subobjects X of Rad A are superfluous. Suppose X + N = A. I f X ^ H , there exists a maximal subobject not containing X, by Cor. 6 . 1 7 , which contradicts X z~r Rad A. // G.M.Kelly [ i?] defined the r a d i c a l of a category by QL(Hom^(A,B)) » [f£Hom(A ;B) | 1-gf i s a unit i n End A 0 for a l l g£Hom(B,A) } Equivalently, Equivalently, Equivalently, = £f£Hom(A,B) | gf£Rad(End A) YgeHom(B,A)} « £f£Hom(A,B)( 1-fg i s a unit i n End B for a l l g£Hom(B,A) j" = £f£Hom(A,B) \ fg€Rad(End B) tfgeHom(B,A)} Ther next proposition i s a simple extension of the fact that Rad (End P) Is the set of morphisms with superfluous images. In f a c t , the proof i s almost i d e n t i c a l , but i s included f o r completeness. Proposition 6.19 ^ If:> P ,,is projective, then -(Q,P) i s the subgroup of morphisms with superfluous Images. Proof Suppose Im (^  i s superfluous, l(J: Q—>P. Given any y : P—->Q, P = Im l p = Im ( d p - *fi? ) + Wtf) S- Im ( l p - ^  ) + Im ( 4 ^ ) so P • Im ( l p - ytC) + Im ( ^ ). But i f Im l{ i s superfluous then so i s Im , hence Im ( l p - V ^ ) = p » lp - Y ^ i s an e p i , P—>P hence s p l i t s implying i t i s a unit i n End (P,P). So by d e f i n i t i o n U i s i n Rad (Q,P). Conversely, i f U i s i n ^ _(Q,P) suppose K + Im U = P, then Q —* P—">P/K i s epi , and - -P lk gives (1 - s ^ )h - 0. But 1$ e V(Q,P) , l - s ( ^ l s i n v e r t i b l e hence h i s 62 zero implying K = P , so Im (Ji i s superfluous. // Corollary 6.20 Rad P = Im ( @ ( (±) P^ a ) where P^  ^ i s an isomorphic copy of P^ for each U e ^ P ^ ,P) and P ^ U " ^ P i s the map <J. Proof By Prop. 6.19, i f ^-(P^P) then Im i s superfluous' n e n c e t n e i m a S e o f > ) — P i s contained i n Rad P. Conversely, l e t X be any small object of Rad P. Then X i s superfluous i n P. Letting P^—-»X , by Prop. 6.19 again P(-^> X<^—*P l i e s i n ^ ( P ^ , ? ) . Since Rad P i s the sum of i t s small subobjects, t h i s implies the i n c l u s i o n the other way. // Corollary 6.21 Rad P ^ P for P projective (an extension of a theorem of Bass, see Prop. 17.14 i n F u l l e r & Anderson [ 1 ] for module case). Proof I f Rad (P) '« P , then ® ( <£> P^ ^  )—»>P i s e p i , A hence s p l i t s . For each P - — r r * P » l e t ce. be P ><S> ( ® p * ) — P ( f i r s t map the s p l i t monic), i . e . the 'dual basis'. For X a small subobject of P, the s p l i t monic factors through a f i n i t e subsum of <±) ( ) , which means that there i s a f i n i t e set u} ± with (1 - 2li± l$±)\ the zero map (on X). X But since the sum i s f i n i t e , 2_^ 1C? 1 £Rad (End P), implying that 1 i s i n v e r t i b l e , which i s impossible, hence Rad P / P. // Corollary 6.22 I f P i s a projective summand of A and P £ Rad A then P = 0. Proof Let X be f i n i t e l y generated, hence small subobject of P. Then X & P ^  Rad A » £ superfluous ^ by smallness, that X i s superfluous i n A. Since P i s a di r e c t summand, t h i s implies X i s also super-63 fluous i n P, hence X & Rad P. This implies that P = Rad P , hence P = 0. // A semi-perfect object Is defined to be a projective object such that a l l i t s quotients have projective covers. Proposition 6.2 3 P i s semi- 'perfect i f f (i ) Rad P i s superfluous i n P ( i i ) P/Rad P i s semi-simple ( i i i ) each simple component of P/Rad P has a projective cover. Proof (=^) To establish ( i ) , l e t Q-*>P/Rad P be a projective cover, claim Q P . Form Q'1 » P * Q » 0 r f ' II * 0 — » Rad P * P — » P/Rad P — » 0 The map P —> Q r e s u l t i n g from p r o j e c t i v i t y of Q must be epi since Q—•» P/Rad P i s a cover. Hence P—>> Q s p l i t s and Q' i s a projective summand of P contained i n Rad P =^ Q' =0 by Cor. 6.22, hence Q = P and Rad P i s super-fluous . To establish ( i i ) , l e t P/Rad P — . We must show that this s p l i t s . Let Q — » V be a projective cover, and consider K .Q 4 o P -» V i .•••'II P/Rad P — » V There i s an induced map from Q—-^P which i s a s p l i t monic since Q—»>V i s the unique projective cover. Then K i s superfluous i n Q hence i n P, and so K—* P/Rad P =0 inducing a map out of V = coker : K Q. This i s the required s p l i t since Q I V P * = p" >V = V i II I  P/Rad P —*7 V P/Rad P V V V Cancel the epi Q — V . ( i i i ) i s obvious. (<^=) Let A>—>P , we want to construct a projective cover for A = coker : A>—>P . W.l.o.g. , Rad P Q A. for consider A » P * P/A > 0 j : i\ i A +~*Rad P — » P > P/A + Rad P 0 P/A—»P/(A + Rad P) i s superfluous, f o r i t s kernel i s (A + Rad P)/A which i s superfluous i n P/A, since Rad P Is small i n P. Hence a projective cover of P/(A + Rad P) w i l l also be a projective cover of P/A. So assuming Rad P A, consider 0 —*Rad P->P — * P/Rad P ?0 f II A 0 * A > P * P/A * 0 Since P/Rad P i s semi-simple , so i s P/A. Let P/Rad P- = P/A (£> X with P/Rad P = (±) S, ifc I 1 and P/A ^  © S, , X ~ (J) S, , J a subset j e J J k l \ J K of I, S^ simple objects. Let P^ * be project-ive covers, and consider P ^ p/Rad P = « ± L s . ) © 0 S ) J 1 x V U exists since © i s proj e c t i v e , i t i s epi since P—»P/Rad P i s superfluous, hence i t s p l i t s because P i s projective. But then Ker i) £ (J) — ZZ (superfluous subobjects of (±)j_ P^ ) = Rad ( @ £ P^) i s a projective summand of the r a d i c a l and i s zero by Cor. 6.22. V i s thus an isomorphism, implying that ©j- K^ i s small i n ®1 P i » w h l c h l r n P l i e s © j K j i s small i n ^ Pj , so @j P j — s j = p / A l s a projective cover.// Corollary 6.24 A f i n i t e direct sum of semi-perfect objects i s semi-perfect.// We return to the category . By Lemma 6 . 1 5 , indecom-posable projectives with l o c a l endomorphism rings are semi-perfect. Suppose C & (rf has a l o c a l endomorphism r i n g . Then 'TC (C) i s projective with l o c a l endomorphism r i n g , hence semi-perfect. I f n C = © C ± o n with each v'End (C ±) l o c a l , Tt(C) ^  (D^TCC^) i s then semi-perfect by Cor. 6.24. Now i f . (S//6 i s a functor category , { ^ ( C ^ ) ! with C^ f i n i t e l y presented and End (C^) l o c a l Is a generating set of small projectives. Hence every f i n i t e l y generated object of tj^o i s an epimorphic image of © TC (C^) f i n i t e for some f i n i t e set and thus has a projective cover ; t h i s establishes Proposition 6.25 I f ^ 5 i s a functor category, i t i s semi-perfect ( f i n i t e l y generated objects have projective covers).// Remark For Grothendieck categories, the condition 'the direct sum of i n j e c t i v e s i s i n j e c t i v e ' i s equivalent to l o c a l l y Noetherian. The proof p a r a l l e l s the module proof using the Baer-Injective test as formulated i n 6 . 1 3 . 66 By Thm 6.6, a functor category implies of i s •locally Noetherian, which i n turn implies the di r e c t sum of i n j e c t i v e s i n S/^ i s i n j e c t i v e (since <jj (®A ±) ^ (±>J(A±) , where tj (A) = 0 —» A—»I—»N—=> 0 i s an i n j e c t i v e c o - p r e -sentation). So since Z/£ w i l l i n this case also be Grothendieck, this yields that T/& i s l o c a l l y Noetherian. Thus i f erf i s pure seml^simple, then 2?/<d i s a semi-perfect l o c a l l y Noetherian functor category. Next, we show Z/£ i s in fact perfect. Theorem 6.26 I f of i s pure semi-simple, then "£/<5 i s a perfect, l o c a l l y Noetherian functor category. Proof We need to show that each object has a projective cover. To do t h i s , i t s u f f i c e s to show that every projective i s semi-perfect. And since { ^ ( C ^ ) ^ , where i s f i n i t e l y presented and indecomposable, i s a generating set of projectives, we need only show that (±)K(CU) i s semi-perfect for arbit r a r y index oce i set I. Now TCCC^) i s semi-perfect by 6.25. Hence ^ (C^) s a t i s f i e s the three conditions of Prop. 6.23, and the l a s t two w i l l also hold for arbitrary sums of ^ (C^) so i t s u f f i c e s to prove that the r a d i c a l of (±) 'K(C^) i s superfluous. The following lemmas hold o t e I for a general abelain category °f. n w\. Let © A . and © B - b e f i n i t e sums i n °7. Then n m KA Hom (©A, ., © B . ) Irl m (Horn (A. ,B. )•) Cnxm matrices]. x j n * n 1 J Lemma 6.27 ^(Hom ( © A ± , © B^ )) ^  M n m (^(Hom(A±,Bj ProofThis follows e a s i l y from ^(Hom (A 1@A 2,B)) /^ J.(Hom(A1,B)) Q) ^ (Hom(A 2,B)) .^(Hom (A,B^®B 2)) "= ^(Hom(A,B 1))®^(Hom(A,B 2)).// 67 The essence of this lemma i s that n m f : ( ± ) A i — * © B j i s i n the r a d i c a l i f and only i f the component maps f i , j : A ^ O A ^ (±> B j-=» B. are i n the r a d i c a l . The extension to arbitrary sums i s the following r e s u l t . Lemma 6.28 Suppose f : (±) A^—5> © A ^ , 1 cC* I o<6l a r b i t r a r y , A^ objects of Of , has the property that each compnnent l i e s i n ^.(HomCA^jA^)). Then 1-f i s pure monic. "VA., i s the f i l t e r e d d i r e c t l i m i t of C * J a f i n i t e set. SO ( l - f ) | Proof ®-rK > ® j A J Aoc £ II 1 1-f X gives 1-f as a direc t l i m i t of the maps ( l - f ) | j . Hence i t w i l l s u f f i c e to prove that these are s p l i t monies. Now ( 1 _ f ) ( i s the matrix map (1 - (f , .)) where (^,(3) range through J ; thi s i s a unit by Lemma 6.27; hence (±)j A^ A^is a s p l i t monic. // We now return to the hypothesis that Of i s pure semi-simple. - Corollary 6.29 I f Oi i s pure semi-simple, then Rad (End (€)P^)) consists of those maps whose components l i e i n ^(A^jA^) for each ot, p . ® j A^ Z ^ A ^ - ^ ^ K 68 Proof The set of such maps i s an i d e a l containing the r a d i c a l , hence i t s u f f i c e s to show that 1-f i s a unit f o r such an f. But by Lemma 6 . 2 8 , 1-f i s pure-monic and hence s p l i t s . // For objects A,B of &1 , the natural map It : Hom^ (A,B) — H o m ^ C T T ( A ) , i r ( B ) ) sends ^_(A,B) into 'X(A) ,1t (B)). I f A ;B are indecomposable with l o c a l endomorphism rings, then ^(A,B) equals Horn (A,B) f o r A,B non-isomorphic, and equals the unique maximal i d e a l of End (A) for A B. So In these cases, It (f) l i e s i n the r a d i c a l only i f f does. (For inde-composables with l o c a l endomorphism rings, A = B i f f fiT(A) = TC (B) . ) Corollary 6 . 3 0 Let © C ^ be a dire c t sum of inde compos ab les with l o c a l endomorphism rings i n CTj. Then Rad (End 7T( © C ^ ) ) consists of those maps whose components l i e i n ( TT ( c ^ ) , 7 T ( C^)) for each oii p . Proof The implication i s t r i v i a l one-way. So suppose 'Tf(f) i s a map i n End C7T (£> C^)) whose components l i e i n the r a d i c a l . These components can be represented as T l C f ^ ) And since CA , C^ have l o c a l endomorphism ring s , fL p l i e i n ^ ( C ^ , ^ ) . Hence by Cor. 6 . 2 9 , f l i e s In'the r a d i c a l of End ((£> C^) and so TT( f ) l i e s i n the r a d i c a l of End (IX <$> c . ) . / / Corollary 6 . 3 1 Q.(End P) • Horn (P,Rad P) for P any projective i n t-/£ . (p/ pure semi-simple) Proof Any projective i n i s isomorphic to (a d i r e c t summand of) (+)[• "X( C^) for a suitable index set I. Now i f 1 t(f) e ^ (£nd P) , i t s image i s 69 . superfluous by Prop. 6.19.. (holds i n any abelian category), hence ;7T(f) can be regarded as an element of .Horn (P,Rad P). Conversely, i f -IT(f) : © T ( C ^ ) - ^ © T t C ^ ) factors through Rad ®T(C^)) = (?) Rad XiC^) , then *7t C**.,.^) :^c^) "Xt^^ R a d ^ C ^ ) ^ © ^ ^ ) - ^ * ^ ) factors as ITCC^) —» Rad-7r(C/3 )c-*1C(Cp ) and hence has a superfluous image since Rad TttC^) i s superfluous i n "^(C^ ) (It i s a small pro-j e c t i v e ) . So by Prop. 6.19 each7Ttf. .») l i e s i n J Horn ( 7T ( C j , ?T (C^ )) and by Cor. 6.30 'fl'(f) i s i n the r a d i c a l . / / Corollary 6.32 Rad P i s superfluous i n P, for P any projective i n 2/4 (G/, pure semi-simple). Proof Again w.l.o.g. P = © ^ ( C ^ ) . Suppose N +CD(Rad TC(C^)) = S>^(Coc) . Then the epi N <±> ( ($ R a d ^ C C ^ ) ) — » <©7f (C^) s p l i t s , so the Identity can be written as © ^ ( C ^ ) ^ N © ( ©Rad7r(Cet))-^> (BTCiCj , that i s , i t i s the sum of QTiO^)-^ © R a d T C t C ^ ) ^ © ^ ( C ^ ) and © ^ ( C ^ ) — * N ^ ©-Tir (C^_) ; but the f i r s t l i e s i n ^End (QKiC^)) by Cor. 6.31, so t h i s forces the second to be a unit, and so N = ©7F( C,^).// This also completes the proof of Theorem 6.26.// One now has Corollary 6.33 of i s of f i n i t e representation type i _ €fj£ i s equivalent to a module category over an Artinian r i n g . Proof Prop. 6.10 and Thm. 6.26 y i e l d r e s u l t . // 71 CHAPTER 7 THE REPURE' SUBCATEGORY In Chapter 6 we concerned ourselves with the s i t u a t i o n i n which every sequence was copure (pure semi-simplicity). The other extreme i s more f a m i l i a r : " a l l sequences pure" i s Von Neumann re g u l a r i t y . We return now to the general case and investigate the relationship between the pure and copure sequences. The category /S* of pure sequences i s a f u l l exact abelian subcategory and i s dense i n (Thm. 5.12, Prop. 5.14). For the moment, we turn our attention to density. Let & be any class of sequences i n <E , the sequence category. Denote by C m (respectively, CQ ) the corresponding class of monies (epis ) . C i s c a l l e d a proper class i f (Maclane [ ] page 367) : P. l . Every s p l i t sequence i s i n ^ . P. 2. I f * , j 3 e C m , then y6W e <^ m i f defined. P. 2 i f oc^^ e Ce > t n e n fi* e C-e i f defined. P.3 I f |Sot e C m , then °c e C m . P. 3* I f j k e C e then /3 £ C g . Denote by C . the class of representatives of C . i n £/«^> . Since we w i l l always regard s p l i t sequences as zero, we t a c i t l y assume P . l i s s a t i s f i e d . Proposition 7.1 C - i s proper i f f O i s dense i n /^>& Proof Remark This i s f a i r l y routine , just a matter of reformulating the concept of proper i n i t s 'proper' s e t t i n g . That i s to say, the axioms of properness seem more awkward than the concept of density. The de t a i l s are as follows : (4=-) P. 2 Let oc be A >—»B, p> be B>-»C. Let ct be the corresponding sequence A >—> B—*>A' and ft be B >—* C--» B'. 72 Consider the morphism i n "£-/*£> : A > * C *» C' * l H i -B > » C =» B» Taking the kernel and image i n (using Thm. 2.5) leads to A » B <£> C * D K |\ i t X A => C * C /&t B i. E » C L I  I A I B => C -» B Now K i s also a subobject of oc since A » B €> C ?»D K H * * i A » B * A' o c ^represents a 'subobject i n Hence K i s i n C and L i s a subobject of j3 hence also i n C , but 0 — » K —» j&c —=•> L —=> 0 i s exact i n , so ^S*t i s i n C . This gives e C and P. 2 i s established, * / m P.2 i s established i n a completely 'dual' fashion. P. 3 Again l e t <± = A ^ B ^ > A' and £ » B A C ^ B ' with &c i n C . 1 m Forming A -=-» B » B 1 °t A » C * C _ shows that °i i s a subobject of , so «^  i s i n Thus <*: Is i n C m , establishing P . 3. Again P. 3 i s a dual argument. ( ) Let C be a proper c l a s s . F i r s t we show O i s closed under isomorphisms i n "&/J> . Let E ~ E 1 with E i n C . Factoring the Iso-morphism i n canonical method (Thm. 2.5) : 7 3 0 — » A — > A' (D B — ? D — 3 - 0 K ( = 0 ) l l i l X 0 — » A » B » C — * 0 E . 0 — » A' » D ? C —9 0 D 11 I 1 * s 0 — 9 A1 => B* » C ' — * 0 E' I A II ir O - ^ D - ^ B ' 0 C » C ' ^ > 0 N ( = 0 ) Kernel and cokernel are both s p l i t sequences and D i s also isomorphic to E. By P . 3 , D-^7C i s i n £ c , so D i s i n C , hence A' >—> D i s i n C , . N s p l i t s so l i e s i n . Then A'>—>D>—»B' © C , which equals A'>—> B f>—»B' <£> C, i s i n C ^ , by P. 2 . Then P. 3 gives A'>—» B f i n so E' i s i n C . Supposes now 0 —> E_1 —> E_2 —=> E^—> 0 i s exact i n . Since by the above C Is closed under isomorphisms, we can represent this exact sequence i n as a quotient with corresponding kernel : 0 E 1 I 0 — * A * B * C ? 0 E V 0 => A' * B' » C 0 E_3 (a) I f E 0 i s i n C , p . 3 gives A>->A' © B i n C so — ^ in i s i n C , and P . 3 gives B'—>C i n C g so E^ i s i n C . (b) I f E± and E_3 are i n C then A' © B - ^ B ' — * > 0 and B»—=>C—^ 0 are i n C and by Pi2 ' e A' ® B —*B' —» C —» 0 = A' © B—>B — » C — * 0 i s i n . Applying P . 3 gives B —*C i n C g , hence E_2 i s i n C . This establishes the density of C . / / » A — * l\ A ' © B -I —* B' 1 11 > B — •is C 1 I 1  * A ' —> B' — — • c 7M Remark A dense subcategory of an abelian category w i l l be a f u l l abelian subcategory with exact i n c l u s i o n . Returning to the category of pure sequences, by Cor. 5 . 1 0 ( i i i ) , has s u f f i c i e n t projectives. This r e s u l t rests upoa the fact that given an object C of <Vy there i s a pure epi X — » C —=>0 with X pure projective. I f < V i s a module category, then given C there i s a pure embedding C—-»Y with Y pure i n j e c t i v e ( = alg e b r a i c a l l y compact, Warfield). Stenstrom has extended th i s to functor categories.For the remainder of this chapter, &/ i s a functor category. Lemma 7 . 2 I f 0 —» C—*'Y -* N —* 0 i s pure exact and Y i s pure i n j e c t i v e , then i t i s the unique maximal pure subobject of the i n j e c t i v e ^ ( C ) i n &/J> (contains a l l pure subobjects of cJ(C) ). Proof Suppose 0 —> C —> E — » Z — » 0 l\ I i 0 — • » C — » I — » c» — * 0 = <J(C) i s a pure subobject of \ J(C). Then 0 — * C — = > E Y l i ? t s to Y.Since Y i s pure i n j e c t i v e , t h i s y i e l d s 0 — » C — » E — » Z * 0 )\ A • » showing that 0 — ^ C — ^ E — » Z 0 — ? c —=> Y —=>N— * 0 i s a subobject of 0- » C — * Y * N —= > 0 . / / Corollary 7.3 ( 1 ) E v e r y o b j e c t E of e/^ has a unique maximal pure subobject t ' ( E ) . ( i i ) t' i s an additive functor. Proof Let E = 0 — — » D — » C—90. Then E <=-s>t-5-(C) and, by Lemma 7 . 2 and Stenstrom's r e s u l t , <iJKc) has a unique maximal pure subobject S. The required maximal pure subobject of E Is the interse c t i o n of E and S. The uniqueness of the maximal pure subobject follows from the uniqueness of S and ea s i l y establishes that t' i s an additive subfunctor of the i d e n t i t y . // 75 At t h i s stage of the development, i t w i l l be useful to Introduce a new subcategory : for want of a better term, K. w i l l be c a l l e d the repure subcategory. Define, ; R £ <£L i f f Hom^ ( - c T , * ) = O. Since i s closed under epis, this i s equivalent to having no pure subobjects. (/S*,^ ) i s a torsion theory since /S* i s closed under quotients and sums and ^ i s Its 'complement'. Proposition 7.4 ( i ) K, i s closed under subobjects. ( i i ) I f R 1—>R 2 i s monic i n ^  , then i t i s monic i n £/s§ . ( i i i ) ^ i s closed under l i m i t s (which are taken i n ). Proof Dual to Prop. 5.5. // Define r'(E) by exactness of 0 —» t' (E) —» E — r ' (E)—s 0 Proposition 7.5 r 1 i s an additive functor and r'(E) l i e s In . Proof This holds i n a general torsion theory s e t t i n g , but we supply proof i n our setting. Punctorially of r' follows from that of t 1 . Form the pull-back i n tj& 0 — * t 1 (E) —» E1 * S l\ I z 0 —> t' (E> —» E *> r» (E)—> 0 where S i s a pure subobject of r'(E). Since $ i s dense, E' i s pure, so E' t1(E) since t'(E) contains a l l pure subobjects. This forces t'(E) = E' y i e l d i n g S - 0 hence r'(E) a . // Corollary 7.6 S i s pure i f f Hom^y^ (S ) = 0. Proof ( ^ ) by d e f i n i t i o n of . (<£=-) Consider S-*>r'(S). Since this must be zero, t'(S) = S. // Proposition 7.7 R x — * R 2 i s e p i i n ^ i f f i t s cokernel i s pure In £(M . 76 Proof the proof i s almost 'dual' to Prop. 5.6 but using Cor. 7.6 i n place of actual d e f i n i t i o n of p u r i t y . / / 7.8~ Characterization of t'(E) The e x p l i c i t construction of t'(E) i s as follows : given E = 0 -^C —»D —*C • — » 0 , l e t S = 0 —* C —=» Y — » N — > 0 be pure exact with Y pure i n j e c t i v e . t r ( E ) i s the in t e r s e c t i o n of S and E i n (C) which i s 0 —*=»C —* D © Y —> M —=»0 (2.11) . Proposition 7.9 has s u f f i c i e n t i n j e c t i v e s . Proof Pure sequences of the form 0 — ^ C — * Y — » N —^0 , with Y pure i n j e c t i v e , w i l l be ..injective i n and any pure sequence 0 —* C —-* B —» A —-"> 0 w i l l be embedded i n 0 —» C —^> Y—* N—» 0 . Proof i s e n t i r e l y dual to proof of Cor. 5.10 ( i i . ) which shows jS has enough projectives. // Proposition 7.10 ( i ) ju : $ ^—» £/jS has a right adjoint t ' . ( i i ) /| : ^ ^ - ^ Sfj£> has a l e f t adjoint r ' . Proof Dual to Prop. 5.15, but we s h a l l supply i t : ( i ) Horn^ (S,t'(E)) ~ Hom^( JLL(S) , E) follows from the fact that the image of S—*E l i e s i n /J* , and so can be regarded as a map to f ( E ) . ( i i ) Hom^(r'(E), R) ~ Horn ^  (E, ^ (R)) assigns E —»/\jR) the unique induced map E » R 4, 71 out of the cokernel. // V •(E) ' Remarks The existence of the unique maximal pure subobject could have been established d i r e c t l y as the t o t a l sum of a l l pure subobjects, provided there were a set of pure subobjects. However thi s i s doubtful. On the other hand, the maximal copure object can be e x p l i c i t l y calcu-lated as the pull-back of B — C and<3X^—=?C, the d i r e c t sum 77 of f i n i t e l y presented objects, the sum taken over the set of morphisms from f i n i t e l y presented objects to C. (see 5.10) 0 *A =» E * < © X f ' t(E) ll i X £ 0 •» .A ?• B * C — * 0 E The key ingredient i s that J has as generator the set of f i n i t e presentations. Indeed, t h i s was how 3* was defined for us. An alternative approach i s to f i r s t define $ , purity i n Cohn's sense, by means of the tensor product. Then £f i s defined by T £ -O' i f f Hom (T,/^) = 0. One must then show that 3 has f i n i t e presentations as generator which would then y i e l d the maximal copure sub-object as above. The advantage of this approach i s that J and can simultaneously be defined. However one needs to v e r i f y that the class /S1 can act as both the torsion free part of a torsion theory, y i e l d i n g 0" as torsion, and the torsion part of a torsion theory, y i e l d i n g ^ as torsion free. On the whole I f e e l the approach i s easier to define 3 f i r s t then $ and f i n a l l y . But now does duality r e a l l y preside for 3* and Most of Chapter 7 was a dualization of Chapter 5. However does the key fact concerning dualize? Does 6(. have a set of cogenerators? We investigate t h i s topic further i n the next chapter. 78 CHAPTER 8 • LOCALIZATION AND COLOCALIZATION IN tj£ The t r i p l e (3 ,S,ft) i s a T.T.F. theory for (torsion-torsion f r e e ) . That i s , and are torsion theories. We have an exact sequence 0 — * t ( E ) — * • E —=>r(E)-*0 where t(E) i s the unique maximal copure subobject of E, and r(E) i s pure, t i s the right adjoint of the i n c l u s i o n J ' e — 5 » Z/£ . r i s the l e f t adjoint of the i n c l u s i o n tf^ZfJ,. We also have an exact sequence 0 * t' (E) E — ^ r'(E) => 0 where t'(E) i s the unique maximal pure subobject of E and r'(E) i s repure. t' i s the right adjoint of the i n c l u -sion AS'C—=> tl/^ . r' i s the l e f t adjoint of the i n c l u -sion * . This holds f o r general T.T.F. theories i n abelian categories, and most of what follows could be formulated as results for T.T.F. theory; at times one must impose the existence of enough projectives and i n j e c t i v e s (which has) but reasonable abelian categories have these properties. However we s t i c k to the notation of rather than attempt complete generality. Unfortunately a few d e f i n i t i o n s must follow ; we s h a l l dispose of them immediately before applying them. 8.1 Category of (Additive) Fractions Let Of be an (additive) category, and Z l a class of morphisms. The couple (T, Crfz ) i s a category of (additive) fractions for Of and 21 , i f ^ i s an (additive) category, T an (additive) functor of—^O/r , such that T(s) i s an isomorphism for any s e l l ; and universal with t h i s property. That i s , i f T 1 i s an (additive) functor of-=? ^ 5, with T'(s) isomorphism for s e "EL , there i s a unique (additive) functor T such that TT i s naturally equivalent 79 commutes i n the category of (additive) categories and T i s unique. 8.2 D i v i s i b l e and Codivisible Objects Let (y,W) be a torsion theory, i n an abelian category d$ . B e ^ i s c a l l e d d i v i s i b l e ( c o d i v i s i b l e ) i f (Hom^tB,-) ) i s exact on a l l short exact X » _ * o with X<£ Honig (-,B) sequences 0—»X'—=>X i . e . 0 — » X' — * X — * X " — > 0 B ^ •B l i f t s to B provided X" i s i n V . / i . e . 0 -> X — » X " ->0 \B—>X" factors over X provided X' i s XnW 8.3 L o c a l i z a t i o n and Colocalization notation as i n 8.2 g : A —*B i s a l o c a l i z a t i o n ( f : B—>A i s a c o l o c a l i z a t i o n ) I f ker g and cok g 6 \J , B e W and B d i v i s i b l e ( i f ker f and cok f £ W , B 6 \J and B c o d i v i s i b l e ) . (Co)localizations are unique and I f every object has a ( c o ) l o c a l i z a t i o n , then ( c o ) l o c a l i z a t i o n becomes an additive functor (Tachikawa & Ohtake [£fo]). 8.4 Category of Fractions Relative to a Dense Subcategory Let C be an abelian category and«£) a dense subcategory Define a class £ to be those morphisms whose kernel and cokernel l i e i n X> . I f the additive category of 80 fractions e x i s t s , i t i s usually denoted Popescu [2-2.]) 8.5 (Co)Section Functor Let T : C — * be the functor associated with the category of f r a c t i o n s . I f T has a l e f t adjoint S,,£) i s c a l l e d a l o c a l i z i n g subcategory and S i s c a l l e d a section functor ( Popescu [22], page 174). ( i f T has a right adjoint R,S w i l l be ca l l e d c o l o c a l i z i n g and R the cosection functor). 8 . 6 Outline of Remainder of Chapter 8 ( i ) For the torsion theory (Cf»S^)» l o c a l i z a t i o n e x i s t s , which w i l l be denoted S(E). ( i i ) For the tors i o n theory ( t¥ tfij) , c o l o c a l i z a t i o n e x i s t s , denoted R(E). ( i i i ) i s a dense subcategory of ; the category of fractions (£/•£> )/j? exists and i s abelian, and in fact equals Cf f) ^K. (iv) S i s both l o c a l i z i n g and c o l o c a l i z i n g . (v) ^ (vi) commutes; t i s right adjoint to ff C—J> r 1 i s right adjoint to c o l o c a l i z a t i o n T = r ' t i s r i g h t adjoint to c o l o c a l i z a t i o n so R i s l e f t adjoint to T and i s the cosection functor. f' also commutes : r' i s l e f t adjoint to t i s l e f t adjoint to l o c a l i z a t i o n T = t r ' i s l e f t adjoint to l o c a l i z a t i o n so S i s ri g h t adjoint to T, and i s the section functor. 81 Note We w i l l need to establish T = t r ' = r ' t . ( v i i ) Cjn<?\ i s a functor category. ( v i i i ) Consequences of ( v i i ) Remarks Popescu handles the general theory of l o c a l i z a t i o n i n h is book, but we cannot appeal to the results (which would y i e l d ( i i i ) the existence) because of his t a c i t assumption that the underlying category be l o c a l l y small, (sets of subobjects) . We have the advantage of dealing with an abelian cateoory with enough projectives and i n j e c t i v e s , but the disadvantage of being unable to assume l o c a l l y small. Lemma 8 . 7 ( i ) t preserves monies and epics. ( i i ) r' preserves monies and epics. Proof ( i ) 0 — > t ( E ) - — » E — > r(E) -—> 0 \j s i ^ 0 •—«"t(F)-—F — > r ( F ) — > 0 and map E-^ =>F induces unique maps, t(E)—»t(F) (into kernel), and r(E)—»r(F) (out of cokernel) making diagram commute. These are the maps t(g) and r ( g ) . I f g i s monic, c l e a r l y t(g) i s monic. I f g i s epi then the connecting morphism from ker r(g) to coker t(g) i s e p i , but ker r(g) i s a subobject of r(E) which i s pure hence ker r(g) i s also pure and then coker t(g) i s an epimorph of ker r ( g ) , and i s also pure. But i t i s also an epimorph of t(E) hence copure. Thus coker t(g) = 0 . ( i i ) Argument i s dual (epi easy part, monic using connecting morphism). // Lemma 8 . 8 ( i ) I f A—>B has pure kernel, then so does t ( A ) ^ t ( B ) . (II) I f A—> B has-pure cokernel, then so does r 1 (A) — > . r ' (B). 82 Proof (1) t(A)—=>A—»r'(A) t ( B ) — * B — > r ( B ) ker ( t ( A ) — > t ( B ) ) c — 5 > ker ( A — > B) by ker-coker sequence.// ( i i ) Dually. Proposition 8.9 r ' t ^ t r ' [hereafter define T to be r ' t . ] Proof Apply r' to the inclu s i o n tE^~> E which has a pure cokernel. Note: r'tE a quotient of tE i s copure. 0—^r'tEv^r'E—»N -»0 , N i s pure jby 8.8 . Then 0 — » r ' t E — » r ' E — N —> 0 0 —=> tr'E — » r'E—> rr'E — 0 There i s a unique map to the kernel t r ' E (and out of cokernel N). Uniqueness implies that i t i s natural and ker-coker sequence yie l d s i t i s monic. The connecting map from ker (N—>rr'E) to coker (r'tE—»tr'E) i s an isomorphism but the former i s pure and the l a t t e r copure hence both are zero. So r ' t E — * tr'E i s a natural isomorphism. // Proposition 8.10 Lo c a l i z a t i o n exists for the torsion theory ( x ^ , ^ ) . Proof Sublemma 8.10(a) Any repure sequence embeds i n ^>(D) f o r a suitable pure i n j e c t i v e D. (cj> (D) i s then i n j e c t i v e and repure.) Proof of Sublemma I f E i s repure, E = r' ( E ) . To commute r'(E) form the pushout of E = 0 —> A — * B » C > 0 X i II 0 — ? D => B*—» C -—> 0 where D i s pure i n j e c t i v e and A)—> D i s pure monic. So E = r * ( E ) c — > Q J ) ( D ) and ^ ( D ) i s repure by def i h t i o n of repure sequences and pure i n j e c t i v e s . 83 Proof of Proposition Given M any object of ^£}s£> , embed r'(M) i n an i n j e c t i v e and repure sequence' I (by sublemma). Form r' (M) >—^'M —*t''(N) 11 t**- r V ( M ) > — * M — » r ' - ( M ) c — » I — » N • — * 0 by taking the pull-back of the cokernel map iE-^N and i n c l u s i o n of the maximal pure subobject of N. Claim 0-*t'('M) —> M—> 'M t» (N) » 0 i s the l o c a l i z a t i o n of M. (i ) By dropping the monic r'(M)>—> wf, ker (M—* M ) = ker ( M - - » r F ( M ) ) = t*(M) , which i s also pure. ( i i ) By dropping the epic M — » r , ( M ) , cok (M — > *M) = cok (r' (M)<=—» M) = t'(N) , which i s also pure. ( i i i ) —>I , I_ i s repure so M i s repure also. (iv) coker ( M c - * l ) = coker (t'(N ) c —* N) = r'(N) i s repure. So given 0 ^ X ' — » X — > X " — T O with X" pure, form 0 => X' =» X » X" . > 0 0 ? M > I — » r' (N) > 0 X — I by i n j e c t i v i t y , and X11—> |r'(N) out of cokernel. But i f X" i s pure, X"—> r 1 (N) i s the zero map. By Prop.2. H t h i s yields a map X—>M such that A/ 0 —^ x' »X commutes. Hence M i s d i v i s i b l e . // I / ' M ^ Proposition 8.11 Col o c a l i z a t i o n for the tors i o n theory ( $ ,$) e x i s t s . 84 Proof Sublemma 8.12 v Any copure sequence i s the epimorphic image of 'IT'(D) f o r a suitable pure projective (D) Oir'(D) i s then projec-t i v e and copure). Proof Dual to 8.10(a) The proof of 8.11 i s dual to 8.10 but we s h a l l give the construction as reference. Given M, t'(M) i s the epimorphic image P — » t ( M ) for some projective and copure P. Form K i—> P -^>tM M — » r(M) i A II r(K) —> M —=> tM then 0 —> r ( K ) — M — » M -—> r ( M ) — > 0 gives c o l o c a l i z a t i o n . / / For each M, set S(M) so that M —»S(M) i s a l o c a l i -zation. Then S Is an additive functor (Tachikawa & Ohtake [2fe], Cor, 1.6) , and also R(M) by R(M)—>M a c o l o c a l i z a t i o n . , r _ Proposition 8.13 <f n ^ r' i s l e f t adjoint to n t i s l e f t adjoint to S ( r e s t r i c t e d to ( ^ n j ~ ) Hence T=r't i s l e f t adjoint to S. Proof The objects S(M) are repure and T i s naturally equivalent to t r ' , so diagram commutes. By Prop. 7.10 r' i s l e f t adjoint to n. To estab-l i s h Hom^ ^ (t(R),E) = Hom^ (R,S(E)) since E i s i n ^ , i t s l o c a l i z a t i o n i s 0 > E —> S(E) X -~> 0 where X i s pure ( i . e . ( i . e . ker ( E — * S(E) = t'(E) = 0). Given R—*SE. , form t R C 9 R 85 This assignment i s the unique map into the kernel (where tR—*X = 0 since i t i s copure to pure). This i s natural by uniqueness. For the inverse statement : given t R — > E , 0 VtR — * R > rR -—* 0 0 * E » SE —* X * o By d i v i s i b i l i t y of SE , there exists a map R—*SE But one must v e r i f y that i t i s unique. I t s u f f i c e s to show that the zero map tR —» E induces only the zero map R — * SE. 0 p tR » R > rR * 0 0 » E > SE —* X - * 0 By Prop. 2.4 there exists a map rR—=>SE such that rR ^ commutes. But rR i s pure and SE i s repure, so t h i s i s the zero map. Hence rR—*X i s also the zero map. The l e f t and right sides are both zero mappings, forcing the middle to be the zero map also. The assignments are c l e a r l y inverse to each other which establishes the adjoint r e l a t i o n s h i p . // -t Proposition 8.14 r' i s r i g h t adjoint- to R t i s right adjoint to \> Hence T=r»t i s r i g h t adjoint to R. Proof Dual to 8.13. // Theorem 8 . 1 5 tU-^RnT i s the additive category of fractions for the dense subcategory Proof Let 0 -—> K>—> A — » B — * > N - — * 0 be exact with K and N pure. We must show T( A)- g > T(B). Consider K" —+ K > K' I 4 t ( A ) — * i A I —»r(A) 1 A f ( B ) — * 4> B 1 — » r ( B ) i J r N"—* N —=»? N' The ker-coker sequence i s K " — K —^ K' —» N"—-> N - » N ' - » 0 . Now N i s pure, and K1 i s pure, so K' • > N" * N , X and Y are pure. Then by density of pure sequences, N" i s also pure. ,But N" i s also the epimorphic image of t(B) hence copure. Therefore N" = 0 . Also K"*=—*K i s pure. Now form o * L' » K" =? L i I v 0 j t ' t ( A ) . => t(A) *>r ft(A) - TA-^O . I t 1 0 > t't(B) » t(B) ? r' t (B) = TB - % 0 r ,L" The ker-coker sequence immediately gives TA—>?TB. And then 0 — » L ' —>>K"'—»L—*»L"—? 0 i s exact with K" and L" pure, which again yields L pure by density of iS1 . But Lc—>r't(A) so L i s also repure hence L = 0 . ^ This establishes TA >TB. Suppose now T' i s an additive functor with T'(g) an / isomorphism i f ker g and coker g are pure. Now i f T exists to make TT naturally equivalent to T;, then i t i s obviously unique because T r e s t r i c t e d to SOif( i s the i d e n t i t y , so T(E) = TT(E) = T* (E) , E e 3 n <R. We need only v e r i f y t h i s works: that i s , for any M we need TT(M) = T'(M) naturally. Now tM >—> M and tM -»*> r 1 tM have pure kernels and cokernels, so T'(M) = T'(tM) = T'(r'tM) = T(r'tM) = TT(M) .// Lemma 8 . 1 6 I f A >—* B [A• •—*> B] then TA >—> TB [TA—*?TB] in S / * 5 • Proof T = r ' t so t h i s follows from Lemma 8.7. // 8.17 Remark Any functor having a l e f t adjoint preserves l i m i t s , and any functor having a right adjoint preserves collmits (Maclane [2.o], page 114 Thm. 1). We note that kernels are a sp e c i a l l i m i t and also cokernels (Maclane [20], page 64) are a colimit. Proposition 8 . 1 8 T : ?,/,£>—*£inT i s exact (Note Regarded as a functor *tf£—^ T need not be exact.) Proof T has a right and l e f t adjoint,:, .so i t preserves kernels and cokernels. So given 0 — » A — » B — » C — » 0 , T(A)—=>T(B) i s the kernel of T(B ) — ^ T ( C ) , and T(A) =>T(B) i s monic and T(B)—=> T(C) epic i n ~*f& by Lemma 8 . 1 6 , hence also i n (R^>X, . So 0 -*T(A)—?T(B)--*T(C) —* 0 i s exact i n Gin 7 .// 88 Proposition 8.19 (R/lCj i s abelian. Proof (%.n 7 i s cl e a r l y additive, so i t su f f i c e s to prove that any map f A. >B has a f a c t o r i z a t i o n f = JjK where h i s a cokernel and g a kernel (Stenstrom [25], page 87). Let A . v B be a f a c t o r i z a t i o n i n Claim that this same fa c t o r i z a t i o n works i n O T : D c — * B hence D i s repure A-—^">D hence D i s copure so D l i e s i n &n 1 . Let 0—? K >-*A—»B-»>N—»0 be exact i n . Then Prop. 8.18 and the fact that T(A) = A , T(B) = B , T(D) = D y i e l d that 0 =» T(K) » A =? B >T(N)- *• 0 i s exact i n (£n T , so A-?D i s coker of T(K)-»A and D-^B i s kernel of B —»T(N). // 8.20 Remarks <%-rt ? i s not an exact subcategory of " 2 . / ^ . For example, to compute the kernel of a map A—=?>B, with A, B e c7? n J , one applies the exact functor T to 0—=» K —=a A —*B giving T(K) » A »B . Now T(K) = r't(K) and t(K) i s i n <R since K i s i n <R , so T(K) = t(K). So the kernel i s the maximal copure subobject of the kernel i n . S i m i l a r l y , c o k e r ^ ^ (A—»B) = r' ( c o k e r ^ A—*B) Corollary 8.21 n i s (co) complete . Proof Since i s abelian, to show cocompleteness, i t suf f i c e s to show that arbitrary sums ex i s t . Given 89 {E^} take In Z/J , then apply T. Since T preserves sums ( 8 . 1 7 ) T((±)E o i) i s the direct sum i n (%,/*) 3*. Completeness i n dual manner. // The subcategory / S * i s dense i n t / A and the incl u s i o n £ J M i s exact. For H - £ > l s & we have : Proposition 8.22 The following are equivalent : ( i ) 3 i s an abelian category, ( i i ) C C-<Z. ( i i i ) C7" i s hereditary, (iv) y i s a dense f u l l exact subcategory of . Proof Assuming ( i ) , the inc l u s i o n functor J<^> preserves isomorphisms. Suppose E i s copure. The epimorphism E — w r ' ( E ) i s also an epi i n 5", i t s kernel i s t»(E) which i s pure, so by Prop. 5.6 i t i s monic i n CT .• So i f 3 i s abelian, E —?» r'(E) i s an isomorphism i n 5 hence also i n . But t h i s forces t'(E) =0. So E i s repure. ( i i ) = ^ ( i i i ) Let X be a subobject i n of T J Forming t(X) >X >r(X) * 0 w c y t(X) —•> T 9 T 1 » o T' a quotient of T i s copure, hence by assumption repure. But then r ( X ) c — > T ' i s the zero map, hence r(X) = 0 and X i s copure. ( i i i ) =i>(iv) S i s always closed under extensions (Prop. 5.7) and quotients (Prop. 5 . 5 ) . So S hereditary implies T i s dense, and then c l e a r l y J"«=—? ^/^> i s an exact embedding. (iv) =^ ( i ) T r i v i a l . // 90 Dually, Proposition 8 . 2 3 The following are equivalent : ( i ) i s an abelian category, ( i i ) (XC J * ( i i i ) (X, i s cohereditary. (iv) (ft. i s a dense.full exact subcategory of . // The adjoint pairs (T,S) and (R,T) have units and counits. 8 . 2 4 Unit for (T,S) : 1 — ^ ST Given E, TE i s repure, so l o c a l i z a t i o n y i e l d s an exact sequence TE >—>S(TE)—» X —* 0 with X pure. Form tE 5> E •> rE A V xi' N/ . TE y—> S T E — » X By d i v i s i b i l i t y of STE a map i s induced E — ^ STE . It i s unique. (Proof as i n Prop. 8 . 1 3 , the zero map t E — > T E would induce a map r E — > X out of the cokernel, which would factor over ;? "STE by Prop. 2 . 4 . This would necessarily be the zero map since rE 6- A? and STE €(R. . This would then force E — * S T E to be zero also.) This i s )J E : E—-> STE . Note also that from the diagram t'tE > > K — » K 1 l~ I I tE ? E »rE TE - — * STE — 5 » X 1 1 ^ 1 0 » L » L » L = L ' = coker (rE —>X) i s pure and K i s an extension of t'tE and K ' i s also pure. Then Ve 0 » K - — * E — » » S T E — L *0 i s a morphism of , the class of morphisms associated 91 with the category of fractions with respect to ft*. Also since STE i s d i v i s i b l e and repure, ??E i s the l o c a l i z a t i o n of E. 8.25 Remark We defined l o c a l i z a t i o n as the functor S (Prop. 8.10). Here, S i s an . additive functor ^/J ? . The functor S was shown to be the right adjoint of T but to be p r e c i s e , since T : Z/J)—*-<%nT , the actual right adjoint i s S r e s t r i c t e d to ^ . However we wish to reserve S as the right adjoint rather than change notation, but the difference i n usage should be noticed, ( s i m i l a r l y for c o l o c a l -i z a t i o n R). So i f the insistence Is that S be the right adjoint to T, then the actual l o c a l i z a t i o n functor i s ST, not S as before. 8.26 Counlt for ( T , S ) : £ : T S — ^ 1 For E i n tf.nO~ we have E >—> SE •—* Y—i>0 where Y i s pure. Since E l i e s In 3 , this sequence yields E t(SE), hence E = r'E ~ r'(tSE) = TSE so TS and 1 are naturally equivalent functors v i a the counit (that the above isomorphism i s the counit map i s 'easy' to e s t a b l i s h ) . Corollary 8.27 S i s f u l l and f a i t h f u l . Proof This i s a general categorical theorem concerning counits : (F,G) adjoint p a i r , £ : FG — ^ l i s equivalence i f f G i s f u l l and f a i t h f u l . // IN an adjoint s i t u a t i o n , (F,G) with counit an equiv-alence F Is said to be the ' l e f t - a d j o i n t - l e f t - i n v e r s e ' to G (Maclane [20], page 92). Then i f F : % ^ , G i s an isomorphism of ^ to a r e f l e c t i v e subcategory } ( of X ( r e f l e c t i v e means an inclusion has a l e f t a djoint). Applying t h i s to (T,S) gives 92 Corollary 8.28 S factors as &.n T * -X^ 3 c — * with Im S the class of d i v i s i b l e repure sequences (denote t h i s as e^nC^). So JbD(R. i s a r e f l e c t i v e abelian subcategory of Proof SE Is d i v i s i b l e and repure by d e f i n i t i o n of c o l o c a l -i z a t i o n . And i f E i s d i v i s i b l e and repure i t i s Isomorphic to i t s l o c a l i z a t i o n . // Turning now to the adjoint pair (R,T) : 8.29 Counit 6 : R T — 1 Procedure i s dual to 8.24. 0 —•» X — * RTE—=» TE v X 0 —* t'E —> E —»> r'E £ E w i l l be the (unique) mapiinduced out of RTE using c o d i v i -s i b i l i t y . Again kernel and cokernel are pure (using ker-coker sequence) so that £ £ : R T E — E i s c o l o c a l i z a t i o n . 8.30 Unit A: 1 ^TR Dual to 8.26. For E i n (ZnT we have 0 —5> Y —s> RE~3» E —9 0 with Y pure. Since E l i e s i n , th i s sequence yields E ^ r'(RE). E = tE ^ tr'RE ~ TRE and th i s naturial equivalence i s the unit transformation. Corollary 8.31 R i s f u l l and f a i t h f u l . Proof Again general category theory, but the idea i s simple so we sketch the proof : ( i ) faithfulness i s immediate from TR ^  1. ( i i ) f ullness : Any map i n Hom (RE, RE 1) i s naturally assigned-by:^M5&intn^B's<:to.-ia^nap-.--ln ' \ Horn (E,TRE') ^ Horn (E,E«) [ v i a Z T * , ] and then apply R to the map i n Hom (E,E') y i e l d o r i g i n a l map. // 93 For the p a i r (R,T) , T i s the • r i g h t - a d j o i n t - l e f t -inverse' to R, and this yields : Corollary 8 . 3 2 Factoring R as <%,/)J"Im R <=—> Im R Is the class of c o d i v i s i b l e copure sequences (denote t h i s as C ^ T * ) . G^tfZ i s a cor e f l e c t i v e abelian subcategory of . Proof Dual to 8.2B. // Corollary 8 . 3 3 The categories &n6{ and C n - y are equivalent abelian. Proof Both are equivalent to (kZ^ ? ,// Proposition 8 . 3 4 ( i ) S preserves i n j e c t i v e s . ( i i ) R preserves projectives. Proof ( i ) Again general theory : S i s the right adjoint of the exact functor T. [Details. Let I be i n j e c t i v e in^CT , with 0 — ^ E' —=> E — E " —•> 0 exact i n . Apply T to get 0 -—=> T E ' — * T E — 5 T E " — * 0 exact, I i n -j e c t i v e gives 0 Hom (TE",I)—» Hom (TE,I)—jHom ( T E ' , I ) - * 0 exact and adjointness gives 0 (E" ,SI) —•> ( E , S I ) — * (E' , S I ) — * 0 exact which gives SI i n j e c t i v e . ] ( i i ) Dually. //' Lemma 8 . 3 5 ( i ) I f 0 — ^ X — , R — ? S — > 0 i s exact with R fc<^ and S e /S , then X—=>R i s an essentu'o.1 monic. ( i i ) I f 0 - » S — i T - ^ X ^ 0 i s exact with S f r / S ' and T e S , then T —»X i s a superfluous epic. Proof ( i ) Suppose X'OX = 0 . Then X' R —-*> S i s monic, so X' V-^S gives X' eS^CR^ = 0 . ( i i ) Dually. // 94 The i n c l u s i o n not exact; however we have the following results : Proposition 8 . 3 6 Given A JL»B , A,B £ <%,n , f has property fP i n O l n (J* i f f i t has IP i n , where IP i s any one of : (i) monic ( i i ) epic ( i i i ) isomorphism (iv) e s s e n t i a l monic (v) superfluous epic. Proof (<£=-) T r i v i a l i n a l l cases since ( £ n J ' i s f u l l . ( = 7 0 ( i ) I f kernel of A-^B equals K i n tyx2>, kernel of T(A) —=> T(B) (= A-^B) in <Rn7 i s T(K). But T(K) = 0 . Now Kc> A ^(R. hence Kefc and so T(K) = t(K). t(K)= 0 means K i s pure, but also repure.Hence K = 0 . ( i i ) Dually. ( i i i ) Follows from ' ( i ) and ( i i ) and (Zn 7 abelian. (iv) Suppose X H A = 0 i n , then tXflA = 0 . But t X f c ^ O j ' hence tX = 0 and so X i s pure but also repure implying X = 0 . (v) Dual to (iv) .// Proposition 8 . 3 7 (i)'S preserves e s s e n t i a l monies. ( i i ) R preserves superfluous epics. Proof ( i ) Por E 0 -*E —=»SE —>X—^ 0 with X pure. Hence E—=»SE i s ess e n t i a l by Lemma 8 . 3 5 . The commutative diagram E >—» SE i I E ' >—-> SE • yields the re s u l t , ( i i ) Dually. // 95 Corollary 8 . 3 8 ( i ) S preserves i n j e c t i v e h u l l s . ( i i ) R preserves projective covers. Proof By 8 . 3 7 and 8 . 3 1 * . / / We turn our attention now to the exact functor T : * cltn -T Proposition 8 . 3 9 ( i ) I f P i s projective in £/J and P e J" then T(P) i s projective i n </lrl J ' . ( i i ) I f I i s i n j e c t i v e i n ~ZJA and I 6? then T(I) i s i n j e c t i v e i n (ZnT . Proof Since P ej , T(P) = r'(P). So given T(P) i n J7)^ , by Prop. 8 . 3 6 A-*>B \» A — = » B i s epic i n T . / ' S o P /A / T(P)=r'(P) / r A » B there i s an induced map out of P. But ker (P-»7r'(P)) = t'(P) i s pure, and A i s repure hence P—=> A factors through the coker , T(P). This i s the required map showing T(P) i s projective i n d . ( i i ) Dual. // Lemma 8.40 I f P i s a small projective i n with P f ^ then T(P) "is a small projective intfg/Oj' . Proof By 8 . 3 9 , T(P) i s projective. Given T(P) * 0 A, i e i 1 the d i r e c t sum i s taken i n (KO 3" where A^ (ztfZnCf , But t h i s equals T ( © A . ) where sum i s taken i n i e i 1 (Cor. 8.21). Now © A. i s copure, hence i e i _ 1 T ( ® A ± ) = r ' t g j A ^ ) . This leads to 96 t«(P) r • p — T(P) = r'(P) * r « ( © A ± ) -> © A X 3 where P A ^ i s induced because P i s projective. Since P i s small, t h i s factors through a f i n i t e sum ®j &± . But a f i n i t e sum of repure objects i s repure so Qj A 1 e Otr> <T • r*(P) ' > r'\<£>A±) and the map P —•> (§)j A^ factors over the coker of (t'(P)—»P) = r'(P) . A diagram chase, cancelling the epi P—y>r'(P) shows the map r'(P)—r> © j factors r'(P)—^r'(<£>A i) through.ia f i n i t e subsum. So r»(P) « T(P) i s small. // Theorem 8.41 (fcl Cj i s a functor category. Proof Sf has where P^ i s a f i n i t e presentation, as a set of generators. P^ are small and projective i n & , hence T(P^) are small and projective i n (RSI'S by Lemma 8.40, and c l e a r l y generate (k^ J . By •'. .1 Cor. 8.21 fen 3 i s complete and cocomplete. Hence re s u l t follows by characterization of functor categories.// Since T(P^) i s a small projective i n an abelian cate-gory, i t i s also f i n i t e l y generated and w i l l have maximal subobjects. Hence there w i l l be epimorphisms T C P ^ ^ S i n (Kr\3 hence i n %jh (by Prop. 8.36) with S simple i n Otnr. Lemma 8.42 I f S i s simple i n (R.of , then i t i s simple as an object of Proof I f X S then tX f S ' , But tX 6 <£^7 J"-=^ > tX = 0 . 4 X i s pure, but also repure X = 0 . / / Theorem 8 . 4 3 Given C f i n i t e l y presented, there i s an exact sequence 0--£,A r ^ J 3 —^ >C —-> 0 which i s simple as an object of . Proof Follows from TCft'(C)) being a small projective i n Jnfi. // We w i l l return to the nature of simple sequences sh o r t l y , but f i r s t derive some further consequences of Thm. 8.41. Let •£ S_^| be the set of simples for (Zo JT . Then since Gins' has i n j e c t i v e h u l l s ( i t i s functor hence Grothendieck), I(S^) J i s a cogenerating set of i n j e c t i v e s ( i n j e c t i v e h u l l s of simples). By Prop. 8 . 3 4 S(I(S^g)) are i n j e c t i v e i n Z/£ and also l i e i n . Theorem 8 . 4 4 ^SdCS^))} are a set of (Indecomposable) i n j e c t i v e cogenerators for (%, . Proof KS^) i s repure hence . KSg) >—> S K S ^ ) — > X^ * ° with X^ pure. Given E € 6{ , TE = tE l i e s i n fan? so there exists an embedding TE >—^ T ^ V for some product. (taken i n <R.nT ) of the ^I(S^)^ . This remains Nan embedding i n by 8 . 3 6 . Now —n— T T K S O ) = tc T T KSJ)) ( i . e . the product of copure objects need not be copure). This v leads to 0 — > TE = tE > E > r(E) ? 0 - t d T K S p ) ) 0 — ^ T T K S ^ ) » > T\si(Sp) ^ T ^ f 98 |\ SKSp) i s i n f e c t i v e i n t/& so the induced map E —»TTSI(S^3) e x i s t s . The kernel-cokernel sequence gives ker (E -VTTsi(Sp)) ) — ^ ker (r(E) —> T x,tf) hence i s pure, but also a subobject of E hence repure, thus zero. So E embeds i n a product of TTsKS )^. To show SKS^) are indecomposable, suppose - l ® - 2 = S I ^ ^ • Apply functor T to get T ( X 1 ) © T ( X 2 ) = TSKS^) ~ KS^) by 8 . 2 6 . But KS^) i s indecomposable i n Oil CT(hence also i n being the i n j e c t i v e h u l l of a simple . Now say T U ^ = 0, since SKS^) ^ X± e d{ so T(X±) = ttX^) = 0 implies X_1 i s pure also ^ X± = 0 . Sim i l a r l y T(X_2) = 0 =^ X 2 = 0. // The objects SKS^) are i n j e c t i v e i n S/^ . Hence by the dual of Prop. 3.13 are of the form J U ^ ) = — * 1 ^ — * Z^—-*0 . But this i s also an element .of (R. . Hence A^ i s pure-i n j e c t i v e . Theorem 8 . 4 5 Given Ctf a functor category, there exists a set of alge b r a i c a l l y compact (pure i n j e c t i v e ) objects f A f l t ^ e x s u c n that X i s alge b r a i c a l l y compact i f f X i s a direct summand of a di r e c t product of copies of . Proof (<^ = ) Since products of pure i n j e c t i v e s and dire c t summands of pure i n j e c t i v e s are pure In j e c t i v e . (=^) Given X al g e b r a i c a l l y compact, then *J(X) &(R hence embeds i n a suitable product 1T(J?(A^), where {Aysj as above. So there i s a monic 0 •—> X * I =3> Xf => 0 (I i n j e c t -_1 1 ive) -*Tl 1^ » li X^ 0 99 Since Of i s a functor category, one can assume I i s the product of i n j e c t i v e h u l l s of simple :. objects of Of', and i n j e c t i v e objects are alge-b r a i c a l l y compact. But being a monic simply means X >I(?>(TrA / g) s p l i t s . This gives the r e s u l t . (Note that i n j e c t i v e h u l l s of simples i n Of have been thrown into the o r i g i n a l set of .) // This completes the duality of J with (K , i n the sense that (K\ can be described as the set of objects cogenerated by the class of oJ(X) with X alge b r a i c a l l y compact (pure i n j e c t i v e ) , 7 the set of objects generated by the class of (X) with X pure projective. The structure theorem for pure projectives 5.10 ( i ) reduces t h i s to the set {fiTCCLj} over the { c ^ of f i n i t e l y presented objects. Thm. 8.44 allows the reduction to a set {cJKA^)} for some (undetermined as yet) set {A^j of alge b r a i c a l l y compact objects. The category may not have i n j e c t i v e h u l l s . However we have the following. Proposition 8.46 (1) Objects of (R have Injective h u l l s which are objects of (R. i n the category . ( i i ) I f <£n7 i s perfect, objects of 7 have projective covers ( which are objects of, 7 ) i n the category >>6 . Proof ( i ) Given X fc^ , tX & (Hn 7 has an i n j e c t i v e h u l l i n the category (Rn 7 which i s Grothendieck. I f this i s t X < ^ E then by Cor. 8.38 StX >—>SE i s an i n j e c t i v e h u l l of St(X) i n . But St(X) = ST(X) and since X ^ t f ? , 0 -* X —> STX —»X 0 i s l o c a l i z a t i o n where X* i s pure. By Lemma 8.35 • X >-> ST(X) i s e s s e n t i a l . Thus X >-» ST(X) >—^SE i s 100 the i n j e c t i v e h u l l of X, and SE e<%-( i i ) Dual. // Lemma 8.47 ( i ) I f E i s i n j e c t i v e and E£(K then l o c a l i z a t i o n E — * S T E i s an isomorphism, ( i i ) I f P i s projective and P <r S then the col o c a l i z a t i o n RTJP—> P i s an isomorphism. Proof ( i ) Since E e-^ , E >—> STE —» X' >° with X1 pure, but E i n j e c t i v e implies this s p l i t s , so X' i s a direc t summand of STE which i s repure, hence X' = 0. ( i i ) Dual. // Lemma 8.48 ( i ) Eetf{ i s i n j e c t i v e i n <R I f f i t i s i n j e c t i v e i n Z/A . ( i i ) P €j i s projective i n S I f f i t i s projective i n t}& . Proof ( i i ) By.Prop. 5.5(H) T^ — > T_2 i s epi i n CT i f f i t i s epi i n ?JA . So projective i n implies projective i n f . Conversely, suppose P, then P i s \ A >y B copure so P—> B factors as P—»tB<^-»B; and by Lemma 8.7(i) t preserves e p i , so P gives the required f a c t o r i z a t i o n through A. ProposItion 8.49 ( i ) There i s an equivalence of categories between the i n j e c t i v e subcategory of 0£ and the i n j e c t i v e subcategory o f ^ - o j * . ( i i ) There i s an equivalence of categories between the projective subcategory of CT and the projective subcategory of O^/lT . 101 Proof (1) Regarding T as a functor 3~^^"n^(the r e s t r i c t i o n of T to 7 ) , then by Cor. 8.27 the counlt TS •-" >1 i s an equivalence. But also, by Lemma 8.47 ( i ) on Injectives, the unit 1 ST Is an equivalence. Hence J *- * JoR, r e s t r i c t s to the required equivalence (lemma 8.48 t a c i t l y used i n applying Lemma 8 .47) . ( i i ) Dual. // Now S has a canonical set of generating small pro j e c t i v e s , the set of f i n i t e presentations •{V(Cot)} , C,^  f i n i t e l y presented. SO v i a the equivalence above, C^nl has { n 7f (C ^ J = {r* KiC^J as a set of generating small projectives. Now (ZnJ' i s a functor category, hence by the fundamental characterization of functor categories & n 7 = ( {Tft-CCLJ j * , Ab ) (contravariant functors on the set of T fl'tC^) ), hence this y i e l d s Proposition 8.50 <Rn J = ( { 7T(CA) j * , Ab ) where {^ViC^j i s the set of f i n i t e present-ations (of f i n i t e l y presented objects C* ) i n l/A .// Remark Let & be the small additive category of f i n i t e l y presented objects i n the underlying category Cf/ upon which i s established. And 7f(^3) i t s image under crf-^—^tJ^ . p rop. 8.50 gives <Zn7 = (TC(<3)* , Ab). .Now ,5T(03) can be described without r e f e r r a l to £/^ > In the following fashion ( r e c a l l of — 3 . factors as erf — = > <rf/(p — * ^/^S and °1/(P > — w a s a f u l l embedding) : objects of X((73) are f i n i t e l y presented objects of Crf and Horn., (X,Y) = Hom^(X,Y) / morphisms factoring through projectives = Hom^(X,Y) /<P(X,Y) . CHAPTER 9 FUNCTOR CATEGORY TECHNIQUES 102 In t his chapter, we w i l l be concerned with two functor categories and t h e i r relationship v i a adjunctions. One functor category w i l l be y and the other w i l l be the module category where R i s the functor r i n g for the set of f i n i t e l y presented objects of Of ( d e f i n i t i o n s h o r t l y ) . Muchr-^of th i s chapter w i l l be of a peripheral nature to the theory of the sequence category £fj£ , however functor rings are a useful and important t o o l , so an examiniation of how t h i s concept f i t s into the frame-work of the sequence category should be of some value, i f not immediately then at least as the groundwork for future research. Let (Q be the small additive category of f i n i t e l y presented objects of We use 'Tf to also denote the r e s t r i c t i o n of the f u l l embedding Tt: of/(p to the image of $ i n . By Prop. 8 . 5 0 , ^ 3 " « (7Z-(<53)*, Ab). Considering the functor category (Co ,Ab) , one can form The v e r t i c a l inclusions are the Yoneda embeddings X H>(-,X). The map 1t i s the unique colimit preserving extension of the map <&^->'XU@)e-—» Cn: ( < 3 ) * , Ab) (see M i t c h e l l [ 2 , 1 ] , page 106 Thm. 5 . 2 ) . There i s also a natural functor induced by %, 7^: ( 1 t ( < 6 ) , A b ) — * (<& ,Ab) where IC*(F) = > F fir. Proposition 9 . 1 "Tt* i s the right adjoint of 7t . Remark T h i s , i s a standard reu s l t about functor categories i n a more general s i t u a t i o n , but we remain with our speci-f i c framework. (<3 , Ab) J ft ->(7T < * 8 ) \ Ab) = (ZnJ-S 103 Proof F i r s t to est a b l i s h ((-,X), Tt^G) = (7T(-,X),G) : By Yoneda, ((-,4 7Tj.G) = ^ G ( X ) = (G^)(X) - G(TTX). By Yoneda again, G(7TX) = ((-,7TX),G) = ( (- ,X), G) . Then for arbitrary sums ®z (->XOC),^ *G) = TTj ((-,xj,7r*G) = Tfj <7f (-.X J^.G) - (<£>T (-,X^ )),G) ^ (7f (®j.(-,X r f),a) , since ^ i s colimit preserving. And f i n a l l y , i f F i s i n ((2 , Ab) since sums of representables are resolving, F i s the cokernel P 2 — ^ P 1 — > F — » 0 with P 1,P 2 'free*. Then 0 — ^ (F, 7t*G) =» ( P ^ ^ G ) *(P2,5f*G) 0 —> (TT F,G) 9 (Tt P1,G) =» (7tP2,G) The bottom row i s exact because Tt i s cokernel preserving, then apply (-,G). The induced map i s the adjunction isomorphism. // Proposition 9.2 7t* i s (I) exact ( i i ) f a i t h f u l ( i i i ) l i m i t and colimit preserving. Proof (1) ft* i s l e f t exact since i t i s a right adjoint, so i t s u f f i c e s to show that preserves epis. I f F — G then F( 7C(X)) — G ( Tt (X)) for a l l X which becomes (7%F)(X)—=»(7t SG)(X). ( i i ) Suppose "u i s a natural transformation F—*G such that TTjfV = 0. Then ( T t # V ) x : (Tt*F)(X) »(TC.G)(X) i s the map V l r x : F(ftX) —-> G(XX) and since 104 every object of f({(p>) i s of the form 7TX , V must be the zero transformation, ( i i i ) 'Tt* preserves l i m i t s because i t i s a right adjoint; and since i s exact, for colimit pre-serving one need only v e r i f y that sums are pre-served : ( 7l*( (%Foc)M x> = ( ( ® T P o c ) ^ ) U ) = (©j-F^K-TTX) = ® T ( F J 7 T X ) ) = ^ ( F ^ T M X ) = <3>rCr*FO6)(X) = ( ( J ) 7 L « F ^ ) ( X ) . / / 9.3 The Unit For (71 ,71* ) : 7[: 1 =»7T^7r For (-,X) e( (3*, Ab) 7t*^(-,X) = (-,7TX) = (7T-,7CX) . This w i l l y i e l d the exact sequence 0 —=>P(-,X) > ( - , X ) (7f -, 7TX) > 0 where P ( Y , X ) = maps Y—^>X factoring over projectives. Since 7[^_ x ^ i s e p i , and representations generate, the fact that both Tt and 7T# are colimit preserving implies 4 p i s epi f o r any F ( i f P 2 —* P^—^> F -*>0, P 2 » P l 'free' then P ? P, — — y F -=? o 7Ts7r P 2 * ^ ^ P - L - + - K i K F -> 0 implies >\.F epi ) . G 9.4 Counit f o r ( f t ,TC«) : £ : 'X .TC , The functor TT Is onto, for i f G l i e s i n (7C(<8)*,Ab) and ( ± i ( - . T T X ^ ) >®j; 2(-,'^X / 3 ) -i s exact then G =1tG where G = coker (©j- _^ x (- X )) By the theory of adjoints, i s the i d e n t i t y , so using the above notation 105 - - _ _ _ -7C<^ ^ 5 > ? r K^TTCGr) * TV <i i s the i d e n t i t y , giving - K — the Identity. By 9.3, XIQ i s e p i , and 7T preserves colimits so K \^ i s an isomorphism, hence also £ Q . Proposition 9.5 The counit £: % 1^ V i i s an equiva-lence.^^, Corollary 9.6 ^ * i s f u l l y f a i t h f u l . Proof F u l l y f a i t h f u l i s equivalent ( i n adjoint situation) to components of counit being isomorphisms (Maclane page 88 Thm. 1) . // 9.7 The Right Adjoint of "7t« Suppose a r i g h t adjoint (unique up to equivalence) Tt e x i s t s , then ( 7x*(-,-rcx),F) ((-,itx),-7r*p) (7C*F)C7tx) . Yoneda Since a l l objects of 7C(^3) are of the form 7CX, t h i s isomorphism can actually serve as a d e f i n i t i o n : 7T*F(XX) = (7r»(-,"ltX),P) . Using the fact that TC- i s colimit preserving yi e l d s (proof s i m i l a r to 9.1) that (7Z%GtF) ^ (G, 7C F) . 9.8 Counit for ( 7T», 7C*): £: X. 7C* » 1 To the exact sequence 0 — ^ P ( - , X ) — > ( - , X ) — ( 7T-,?tX)—=>0, apply (-,F) to obtain 0 —•> ((7C-,7CX),F) »((-,X),P) which i s 0 > (7t sTC*F)(X) => F(X) , this w i l l be the counit. Note that each component of £ i s monic. Lemma 9.9 For (Q: F-*G, 'JC* <Q i s an isomorphism i f f c£ i s an isomorphism. 106 Proof Let 0 —*K—*F- —>G —*N —> 0 be exact. Then since 7^ i s exact, the sequence 0—=> 7 L K K — ^ 7 t # F — — » 7 C s N - ^ > 0 i s exact. I f 'TtT^F—»7L#G i s an isomorphism, this forces TC^K = 7C,jG = 0 , but evaluating ( 7 t * K ) ( X ) = K(7TX) = 0 for a l l X , so K = 0 , s i m i l a r l y N = 0 . / / 9 . 1 0 Unit for ( TV ) : A : 1 —^>TC 7T» % ^ i r T r * i s the i d e n t i t y . By 9 . 8 , £ 7 T * i s monic, hence an i s o -morphism, which implies TT #lr\ i s also an isomorphism, so by Lemma 9 . 9 \ i s an isomorphism. Remark That the components of the unit are isomorphisms can also be established by 'dualizing' Thml and i t s lemma, page 88 of Maclane [20], y i e l d i n g for an adjoint p a i r (F,G) , F f a i t h f u l i f f unit i s monic. F f u l l i f f unit i s s p l i t epi. In our case, IT* i s f u l l y f a i t h f u l by Cor. 9 . 6 . 9 . 1 1 Tensor Product Before introducing the functor r i n g , we examine the functor 7t i n another fashion which may be more f a m i l i a r (once one swallows the elaborate d e f i n i t i o n s ) . Let C b e a small category. Then there i s a unique (up to isomorphism) functor - @ - : (C,Ab) x ((? ,Ab) > Ab with the following properties : (a) - (x)B and A © - are right exact (b) - <g) B and A ® - preserve arbitrary sums (c) (C,-) ® B = B(C) and A <g) (-,C) = A(C) for any C i n C . The existence of A © - and - <g) B i s established as follows : 107 For fixed A i n (C,Ab) , then A : C —9 Ab can be interpreted v i a Yoneda as mapping (-,C)J > A(C) and so has a unique right exact sum preserving extension A (x) - to ( C , Ab). S i m i l a r l y for -(5£>B. The uniqueness of extension y i e l d s that (A(g)-)(B) i s naturally isomorphic to (- <g> B)(A). 9.12 The Left Adjoint 7T of using Tensor Product (ft (P))(7CX) = CK Xs% -) (x)F Under t h i s l i g h t the computation of 7? i s hidden away by the handy * (x) 1 symbol, which avoids the actual computation of ft as the cokernel of applying 7C to a free presentation P 2 - J » P 1 - J > F - ^ 0 (see M i t c h e l l [21], page 106). One can also compute the unit (see 9.8) by applying - (3D F to the exact sequence 0 -*P(X,-)-^>(X,-)-^ X,-jr _ ) - > o to obtain P(X,-)(x)F *(X,-) © F >(TCX,7r-) <g)F-—>0 which becomes -yy P(X,-) © F =?F(X) — - * (7r*1TP)(X) 90 giving 9.13 Counit Isomorphism Revisited Consider the exact sequences 0 — * P(X,-) ( X , - ) — > (7£x,7C-)—*o and 0—?P(-,Y) =?(-,Y) — ^ (TT-,7C Y ) — ^ 0 , and form 0 P(X,-)@P(-,Y) ? (X,-XD(-,Y) — * (TC x ^ r - X p p ( - i)-*o 0 —*P(X,-)@-,Y) ;>(X,-)(&-,Y)—^> (7t xptr-XxX-,Y) -5- 0 >b •I/ p(x,-xiXit-,'}cY) — * (x,-)®e/r-;xY)-^> (7cx,r-xxXr - j rY )—> o V ^ V 0 0 C) 108 The middle row and column are exact because (-,Y) and (-,X) are projective objects. Using property 9.11 (c) 0 thi s becomes 0 P(X , - X 3 P(-,Y) P(X,Y) -P(X,-XgXTT-,1fY) V 0 A 5 P(X,Y) -(X,Y) -=> (TX,ltY) ( K X,7T-YJP(-,Y) i •* (7c X,-7r Y) — (XX ,r-)(xXx-,trY) 0 0 » 0 0 Now the map P(X,Y) •» (X,Y) (7T X,7TY) of the lower l e f t square i s zero, hence P(X,Y) P(X,-X3>(7r-,.7cY) > > ( X X.TfY) i s also zero, cancel the v e r t i c a l e p i , giving the horizontal monic the zero map. Hence P(X , - X ^ X 7T-,7rY) = 0. Sim i l a r l y using the upper right square yields P(X,Y) *> (7T X,X-)®P(-,Y) (7CX ,7TY) the zero map. Cancelling the epi gives v e r t i c a l map zero. But this map i s monic (use the fact that P(X, - X E X T T -?TY) = 0 and Snake Lemma on top two rows). Hence also (1tX,T-) © P ( - , Y ) • 0, and from this^one also derives ( 7UX,T-) (5p (7T-,7t Y) = (2TX,7tY). Now P(X,-) (3p ( 7T-,7fY) = 0 becomes P(X,-) ® X # ( - , X Y ) = 0 and 7T* and P(X,-) @ are colimit preserving (for fixed object P(X,-)). Then since P(X,-) ®X*- k i l l s the projective (-,7CY) (7C(d3)*, Ab). i t must be the zero functor on 109 Applying - ^ T T ^ G to the exact sequence 0 — 3 PCX,-) —MX,-) *C#"X,r-)—5*0 then yie l d s ^ (X,-) ® K^G - ^ ( 7 t X , T - ) ® 7r*G or ^ X*G(X) = ? X ( 7C#G) (ftX) or _ _ G(7CX) — 9 ^G(7C X) This i s the counit isomorphism, 1 — > ft"7** 9.14 The Functor Ring Let be an arb i t r a r y abelian category and 1l = - f l J ^ a set of small objects of 0 /. Then the functor category ,Ab) can be interpreted as a module category i n the following manner (Gabriel [11]) : Let R - { (Q e H o r n e t ©U^, ©U^) j (Jij^ - 0 a.e. } R usually does not have a unity (unless the set £ l u j i s f i n i t e ) , but thi s i s replaced by a 'complete' set of idempotents {e^} where e., '• <$> U, —^ > U, c—» © n . By a l e f t R-module M, one has the usual meaning but with the added property that RM=M, so that M =(J)e^M. The category of l e f t R-modules i s then a cocomplete abelian category with ^ RE<J a s a generating set of small projectives. Setting U « (±> , l e t Horn (U,M) = ^({6 Horn (U,M)| My = 0 a.e.j then Hom^(U,-) i s a functor ,8 which assigns U^i—» Re^ and U«—»R. Then the unique colimit preserving extension to ( ^ U^] » Ab) i s an equivalence of categories : 110 c Ab ) - — - > R 9 * i Yoneda J J embedding r •> r 7 9.15 The F i n i t e l y Present Functor Ring(s) For our purposes, l e t & = / c ^ J , the set of f i n i t e l y presented objects of Ofand 7X{&) - / ^ ( C ^ ) J the set of f i n i t e presentations i n . Let R be the functor r i n g HOm^^*, $0^) and 7C(R) = Hom^ (£>7T(C^), <&X{CU)). Then R ^ = (c%* > Ab) and ^ ( R ) ^ 7 ~ (W(<3Y , Ab). The natural functor % \ of/tf> > induces a r i n g homomorphism TC: R—^7t"(R) sending the complete set of idempotents {e^ j for R to a complete set f^i^)^ f o r 7T(R). This r i n g homomorphism induces a change of r i n g functor 7C(R 5 7fl\ where each l e f t 7T(R)-module i s naturally a l e f t R-module v i a 7T . 7 T V i s t n e functor defined i n 9 . 1 . Just as for rings with unity, IX. ^  has a l e f t and r i g h t adjoint. 9 . 1 6 Left Adjoint of 7T» Revisited The l e f t adjoint i s 7T(M) - 9T (R) ^ M . Only a few minor changes must be incorporated due to the lack of unity element. R(M,7iy*) = ^ ^ ( ^ ( R ) ® R M , N ) sends $ Ir—>tQ, where T£,(xc]Dm) = x d2.(m) and conversely, "tjj> I—=> Q, where i f m - 2Z e^m^ (the sum i s d i r e c t MNBe^M) J f i n i t e then &(m) =U( le^®^ )• 9.17 The Right Adjoint of 7t» (Once Again) Just as for rings with u n i t , T* (M) » Hom R(«(R),M), those l£: 1TR—>M with ^(Tf e^) = 0 a.e., and HomR Cr#N,M) = Hom / 7 r ( R ) (N,7T*M) * * sends I ? U2 where di (n)(x) = 4>(xn) and conversely i—»c£ where i f n = 2ZjCr( ) n^ J a f i n i t e set, J J * * 9.18 Unit f o r (7T(R) ® - , ^ ) This i s the map M T # (7|T(R) (x)RM) sending m = X e^ rn^  ? ^ -"TCCe^) (x) m^ . 9.19 Counit for ( 7T(R) (£) - ,7C») The counit for ( 71 (R) Ci) -, %*) i s the isomorphism 7T(R)(x^ r*M — * M sending Z/TT(eJ ® I * 27 ^(e^Jm^ (using the fact that ^ ( R ) M « <S) TVie^M ). 9.20 Unit for ( TC*,7T*) T\ M : M ^7r*7T»(M) - HomR ( 7C(R) ,TT,(M) ) where ( fl (tn))(x) = xm for x £ 7C (R) with inverse ^ 1 ^ J - ^ * ^ 6 ^ where sum i s actually f i n i t e since (SICK e ) = 0 a.e.. 9.21 Counit for ( 7T» , 7T*) <?H : 7r*7T*M ^ M : T# (HomR C7C(R),M)) >M sends ^ * 2Zu2(7Te) «* * Note that #(7r e^) 6 e<M since (£( 7Te^) = ($. (e/( TCe*)) 112 so the above sum i s d i r e c t , implying i s monic. We w i l l drop the topic of functor r i n g for the moment since I t i s not the t o o l that we wish to u t i l i z e i n t h i s thesis. However I t w i l l be convenient to c a l l upon this theory when necessary, Since IT* i s f u l l y f a i t h f u l , any map 7 t #F — a r i s e s uniquely from a map F—»G. Lemma 9.22 ( i ) F —5G i s epi i n (7t(<8)*, Ab) i f f f T » P — ^ ^ G i s epi i n Ab). ( i i ) F—>G i s superfluous i n (7r(£B)*,Ab) i f f 7C#F—»Tr*G i s superfluous i n (<6 , Ab). Proof ( i ) Proof same as Lemma 9.9. ( i i ) ( ^ > ) I f X — ^ 7 T # F — ^ ^ * G i s epi , f i r s t note X > X *F 7T*7rx — : —5> TCJZ so we only need to v e r i f y that the lower map i s ep i . Applying IT , Tt X > 7t7t#F ^ 7C#G P _ y G Hence Tt X if TCKF i s epi since F—?G i s superfluous. Now apply 7CS to conclude ^7CX 5T*7r7C #F i s e p i . ) I f Y —>F—3>G i s epi , apply : 7TjfY ? 7 C # F - — ^ ^ G i s also e p i , so ^T^Y— 9 7C XF i s epi , implying that T r ^ Y —^TTTC^F i s epi Y > F gives Y —? F epi. // * 113 Proposition 9.23 (63 ,Ab) perfect implies that {%{(£>) .Ab) i s perfect. Proof Let G be i n (5t(^3) ,Ab) and P — » ^ G a projective cover. 9C i s colimit preserving and takes the repre-sentable \(-,X) to (-,7rX) hence also preserves projectives. £^ Claim 7tP—-*> ft K*G ~ G i s a projective cover. Form P *>C #G p L - i **** Lower map i s superfluous since P — > f t * G i s superfluous. Hence by Lemma 9.22, TCP-—? G i s superfluous.// Remark This proposition i s the functor version of 'factor rings of a perfect r i n g are perfect'. Here the functor r i n g 5t(R) i s a factor of the functor r i n g R. Proposition 9.24 I f \ Q : Q —^> TC^TCQ, i s superfluous for a l l Q , and (7C (<*3) ,Ab) i s perfect then ,Ab) i s perfect. Proof Let M <= ((/3 ,Ab) and 7t P » ?c M a projective cover , P projective ( a l l projectives of (7r(<*3) ,Ab) are of the form 7d?, since K preserves sums and (-,X) i s the representable (-,7CX) ). Form P > > M 7TX 7T P M The induced map out of the projective P i s epi since i s a superfluous e pi, and i s superfluous since Y\ i s superfluous and 7CS7TP —»> 7ts7trM i s superfluous by Lemma 9.22. // 1 1 4 Remark For factors of rings with unity R—5> R/I , the corresponding unit map i s Q—>?R/I(x) R Q = Q/IQ , the condition of the proposition i s that IQ i s superfluous i n Q for any Q. This condition i s equivalent to I being l e f t T-nilpotent (Anderson & F u l l e r [ 1 ] , Lemma 2 8 . 3 ) . For the next proposition, l e t ty be an ar b i t r a r y abelian category and ^ . j a s e t °f small projectives. Proposition 9 . 2 5 The following are equivalent : (i ) The set of morphisms between { P p < . ^ i s l e f t T-nilpotent with respect to the r a d i c a l , ( i i ) Rad (©P^ , (+%<_) i s the set of morphisms with components ty^/i £^ ( P^  ,P^). ( i i i ) Rad © P ^ i s superfluous i n @P^ . Proof ( i ) =>(ii) The set of with 6 ? ^ = P * ^ ® ? ^ ^ ® ^ ~ ^ ?p i n ^(P^»P^) i s an i d e a l , hence i t su f f i c e s to show that 1 - i s a unit for any such (Si . The Konig Graph Theorem and T-nilpotence implies that given any small object X <^ <f) P^ (X= a f i n i t e sum of P^ would su f f i c e ) , there exists an n with $_ n (X ) = 0 . Hence the i n f i n i t e sum 1 + C $ + (& 2 + L £ 3 + . . . i s well defined on © Poc and i s the required Inverse to 1 - . ( i i ) ^ > ( i i i ) (Remark Proof same as Cor. 6 . 3 2 ) I f N + rad ( © P^  ) = © P^ then the epi NdD((±).rad P ^ ) P ^ s p l i t s so the i d e n t i t y can be written as 0 P^  —> N <£) ( © rad P*)—> © P^ the sum of \ ' ' ' ' •. -© Poi ~^<S> rad P^-^ © P«* and © P ^ —i? N — > © P^ . But the f i r s t l i e s i n rad ( © P ^ , © ? ^ ) by ( i i ) . So the l a t t e r i s a un i t , implying N = © P^ . 115 ( i i i ) "=^(i) Given a sequence of maps P^— 1 pPg-J?—^3"^ • • • with each a± € ^ ± ^ 1 + ± ) l e t 0 \ A = 0 a. 0 0 a 2 0 0 0 a 3 0 0 0 0 a^ 0 Then the image of A : © P . i s contained i n © Rad P^ which i s superfluous. Hence a A l i e s i n Rad @Pc><->©U) £ f o r abelian categories, Rad (End P) = with superfluous images j . ]. Letting B be the right inverse of (1 - A) , choose n with © P i ^<S)?± oO P i=n+l equal to zero (P-^  can be carried only so f a r by B since Is small ) . Then ( 1 - A n + 1 ) B = (1+A+...+An)(l-A)B = l+A+...+An so 1-A •£> _ .@p 5 ® p — * ® p — » p x l = P, -i>s>p n+1 1 -^P—>P n+1 But AJ' maps P ± > p i + j s o t n l s reduces to ©p -^U@p- n+l b = a i a 2 • * • a which yie l d s - a ^ ^ a ^ . . . a n + 1 b i s the (n+2,n+l)th component of B, n where 116 This gives ( a - ^ . . • a n ) ( l - a n + 1 b ) • 0 But a n + 1 6 ^ ( P n + 1 . P n + 2 ) s o a n + l b e ^ n + l ' W * lm»Wn& that 1 - a n + 1 b i s a unit, hence a 1 a 2 « « « a n = 0. // Proposition 9.26 I f OJ and (7C((&) ,Ab) are perfect, then * so i s ( Oi ,Ab).. Proof It w i l l follow from a l a t e r result (Prop. lO.tfO that (£3 ,Ab) i s semi-perfect. Hence i t s u f f i c e s to prove that arbitrary sums of (-,) are semi-perfect and, u t i l i z i n g Prop. 6.23 since properties ( i i ) and ( i i i ) are preserved i n taking sums, one needs only show that the r a d i c a l i s superfluous. By Prop. 9.24 one must show that l s T-nilpotent with respect to the r a d i c a l . We adopt the proof of Hullinger [/fe]. Let a a ? —•=—» C"2 =-> ... with a± £ ^_ ( C i , C i + 1 ) . Then VC (a^) £ ^ (TT (C i) ( C 1 + 1 ) ) , so there exists n1 with - X ( a 1 ) ^ r ( a 2 ) . .. 4C(a n ) = 0. That i s , a,a n...a factors over a ( f i n i t e l y generated) 1 2 n-^  projective P^. Repeat th i s argument s t a r t i n g with C — * C .-T^* ~ 1 1 * * then- Q 1 n 1 n.^ +1 n 2; n2 \ / \ / ' V P ; l -> P 2 > P The lower row are maps In ^ ^ ^ ^ ^ } since they factor v i a a n e ^ ( c n i » c n i + l ) ' B u t °f i s P e r f e c t h e n c e 117 { p ^ i s a l e f t T-nilpotent system i n &f . So composition of lower row becomes zero implying also upper composition becomes zero. // 9.27 Remarks Under reasonable conditions Of w i l l be perfect, i . e . i f (7f = /P^L f o r A a perfect r i n g . So that R i s perfect i f f 7C(R) i s perfect. In F u l l e r [ 9 ] , the functor r i n g i s based not on f i n i t e l y presented modules but f i n i t e l y generated modules. However th i s w i l l be the same provided one imposes the Noetherian condition, i n which case one then would l i k e a perfect Noetherian r i n g , i . e . A r t i n i a n . Theorem 9.28 (Full e r ) Let A be a r i n g with i d e n t i t y and R the functor ring from f i n i t e l y generated l e f t A -modules. Then R i s l e f t perfect I f f . c every l e f t A-module i s a dire c t sum of f i n i t e l y generated modules. // Then using Thm. 6.25, YThm. 6.6 and 9.15 , one has Proposition 9.29 I f o/ « , A - A r t i n i a n , then *>/& i s perfect i f f 7 ±s perfect i f f (rf i s pure semi-simple. // We summarize the various categories and adjoint pairings i n a quick overview of the previous sections. <23 i s the additive category of f i n i t e l y presented objects. (<a*,Ab)^==^ (7c(<6)*,Ab) = tn^or * — - tf& y > 9.30 Star t i n g with % n j- * >T gy* —ST* (R,T) i s an adjoint p a i r , which factors as r" "t again adjoint p a i r s . (T,S) i s an adjoint p a i r , which factors as 118 again adjoint pairs, ( i ) T i s exact ( i i ) — ? — * ( R . n . J i s the category of additive fractions with respect to the pure subcategory , which has ST as l o c a l i z a t i o n functor and RT as c o l o c a l i z a t i o n . ( i i i ) (a) The counit for (T,S) i s an equivalence (so S i s f u l l y f a i t h f u l ) , i . e . T i s the l e f t - a d j o i n t - l e f t -inverse to S. S then establishes an isomorphism of to the r e f l e c t i v e subcategory of d i v i s i b l e repure objects n S : <Zn T ~ > (Zn&c •> Z/J> (b) The unit for (R,T) i s an equivalence (so R i s f u l l y f a i t h f u l ) , i . e . T i s the r i g h t - a d j o i n t - l e f t -inverse to R. R then establishes an isomorphism of 7 to the co r e f l e c t i v e subcategory of c o d i v i s i b l e copure objects G f\ 3* R : (Z n 7 * e n 7 c—* £ M (iv) (a) S preserves i n j e c t l v e s and e s s e n t i a l monies, (b) R preserves projectives and superfluous epis. 9 - 3 1 K ((0*,Ab) Lg>(X(^)*;Ab) • ^ * ( TC, Ttg) and (-?»:», 7f ) are adjoint pairs. ( i ) T - i s exact and f u l l y f a i t h f u l , giving an. exact embedding of (?c(<6) ,Ab) into (<8 ,Ab). ( i i ) (a) The counit for (5r, fc^) i s an equivalence , i . e . ^ i s the l e f t - a d j o i n t - l e f t - i n v e r s e to ^ then establishes an isomorphism of CX(o3) ,Ab) to the r e f l e c t i v e subcategory of contravariant functors on & which vanish on f i n i t e l y generated projective objects of of . 119 (b) The unit f o r ( 7C*T TC ) i s an equivalence , i . e . % i s the right-:adjoint-.left inverse to ^T*. ^% then establishes an isomorphism of ( ?f (/3) ,Ab) to the co r e f l e c t i v e category of contravariant functors on which vanish on f i n i t e l y generated projective objects of . Note this subcategory i s both r e f l e c t i v e and c o r e f l e c t i v e . 9.32 Using the equivalences (^3*,Ab) =* and (?t(*3)*,Ab) - 9 C ( R ) ^ where R [respectively TC ( R ) ] i s the functor r i n g with respect to [it(^3)] , then 7C# : (X(&) , A b ) — ^ (63 ,Ab) i s the change of ring functor ^ ( R ) 7 ^ — ^ R ^ t i n d u c e d D v t n e natural ri n g homo-— # morphism R — 9 7 c ( R ) . Ttand TC are then the associated l e f t and right adjoint as i n 'standard' ri n g theory. In p a r t i -c ular, ft i s tensoring over the ground r i n g fc(R) Qg # c: 9.33 The isomorphism (?r(/3) ,Ab) — - — > 6 Z l T results from the equivalence of the subcategories >C(/3) and T7f(/3), where by X{Sd>) one means { ( - , % ( X ) ) I X 6 ^ ] , which i s a generating set of small projectives for (#(#)*,Ab) . T?e(4S) i s the set {TTC(X) ( X e (fr^f where 'Tt.(X) i s the image under 'X: &I/<P S £/*6 ; t h i s set i s a generating set of small projectives for the functor category (&r> T . 120 CHAPTER 10 SIMPLE SEQUENCES J S Q> » <B/cP/i(& We commence with a b r i e f review. *fJ0* i s the projective homotopy category (3.4). erf—°1 J(P i d e n t i f i e s objects but assigns a morphism i t s class modulo maps factoring through projectives. 7t : ofjfe—> g/£ assigns to an object X a short exact sequence terminating i n X, with middle term projective. /7T i s a f u l l y f a i t h f u l embedding of (as a resolving set of pro j e c t i v e s ) . Dually, one can consider the i n j e c t i v e homotopy category 6f/\f . i s the subcategory of f i n i t e l y presented objects. The Importance of £ i s that i t generates ( v i a % ) the copure subcategory 3" of £A6 ; every pure projective i s a di r e c t summand of a di r e c t sum of f i n i t e l y presented objects of <*/ . Note that PflOo Is the category of f i n i t e l y generated projectives. One can dualize S as follows (8.34 and 8.44) : l e t 6' be the set of pure-injectives r e s u l t i n g from taking i n j e c t i v e h u l l s of simples of T , along with i n j e c t i v e h u l l s of simples of erf (this l a t t e r set i s a suitable replacement for f i n i t e l y generated p r o j e c t i v e s ) . By Thm. 8.45, 63' cogenerates<^ . Every pure-injective (al g e b r a i c a l l y compact) i s a direct summand of a direc t product of elements from (&x . This creation of *8 ' i s not very esoteric : the duality with 3^ i s imposed rather than a r i s i n g naturally. The duality can be better illuminated i f one imposes the following conditions : (a) Every pure-projective i s a direc t summand of a dir e c t sum of pure projectives with l o c a l endomorphism rings. 121 (a 1) Every pure-injective i s a direct summand of a d i r e c t product of pure inject!ves with l o c a l endomorphism ring s . Now any simple of G{n 3" Is an epimorph of IC(C) for some C f i n i t e l y presented. However, assuming (a) , one can further assume End C l o c a l . In t h i s case, K (C) i s a projective with a l o c a l endomorphism r i n g , hence has a~» unique maximal subobject. Conversely, 7C(C) i s a small projective for C f i n i t e l y presented and w i l l thus have a maximal subobject. This yi e l d s a 1-1 correspondance between simples of fen? and non-projective f i n i t e l y presented objects with l o c a l endomorphism r i n g . Suppose S = 0 — » A ' — » B ' — * C—*0 i s the simple epimorph of OC (C). S i s copure and simple hence must be repure, so r'S = S . To compute r'S take A'>—» A" , pure monic with A" pure-injective, and form push-out 0 — > A* —> B' — * C =» 0 S i t II 1 = 0 •—> A" —» B" —*>C * 0 r'S so w.l.o.g. the f i r s t term of S i s pure-injective. Now . assuming condition (a'), one can further assume A" = T ~ A I with End A± l o c a l ? ( i . e . A" © X = IFk± for some X, but then A ' — A " CJ) X i s s t i l l pure monic ). Now form quotients by taking push-outs 0 — > ~ I T A . — B " ? C s> 0 S I II I 0 5> A, • ? B. * C * 0 S, i i — i Since S i s simple, S± = 0 or S "S" S^ . But not a l l S ± = 0 since one has an embedding 0 -? TTA, * B" » C 5> 0 S ii i j r o > TA ± — » TB1 II0, — * o T r s ± 122 Thus S can be represented as a sequence 0 — * A —» B — * C — » C with C pure-projective ( i n f a c t , f i n i t e l y presented) and A pure-injective, and both End C and End A l o c a l . Noting that \J(k) i s an indecomposable i n j e c t i v e and S t ^ t j ^ k ) i s the unique simple subobject, establishes Proposition 10.2 I f d s a t i s f i e s conditions (a) and (a 1) then any simple _S i n fcn 7" has a unique representation as 0-^A-^B—•C—*0 with End A and End C l o c a l , A pure-injective and C pure-projective. // Before proceeding further into the topic of simples, we pause to investigate i n d i v i d u a l l y the conditions (a) and (a'). Proposition TO.3 &t s a t i s f i e s (a*) i f f given C f i n i t e l y presented with l o c a l endomorphism r i n g , there exists a simple sequence 0 —» A — » B - ^ C — » 0 , with End A l o c a l . Proof (<#=•) As i n Thm. 8.45, \yJ (A)} r e s u l t i n g from the simples of <Zn 7 cogenerate d , and t h i s implies that i f D i s pure i n j e c t i v e , «J?(D) >—> TT^(A^) for some product. So D)—> I (-D ("TTA^) s p l i t s where I i s any i n j e c t i v e containing D, but w.l.o.g. I i s the product of i n j e c t i v e s with l o c a l endo-morphism rings ( i n j e c t i v e h u l l s of simples i n Of ). ( =^ ) As i n proof of Prop. 10.2. // Proposition 10.4 The following are equivalent for a functor category < V : (i ) Of s a t i s f i e s condition (a) ( i i ) 6ft i s a Krull-Schmidt category (every object i s a f i n i t e direct sum of objects with l o c a l endo-morphism rings) ( i i i ) (<3 ,Ab) i s semi-perfect, (iv) o/,and dn "f are semi-perfect. 123 Proof Note; f i r s t that a functor category C i s semi-perfect (every f i n i t e l y generated object has a pro-j e c t i v e cover) i f f C has a generating set of small projectives with l o c a l endomorphism rings (achieved by taking projective covers of simples), ( i i ) ^ ( i ) T r i v i a l ( i ) = ^ ( i i ) Every f i n i t e l y presented object w i l l be a dir e c t summand of a f i n i t e direct sum of objects with l o c a l endomorphism rings. But every dir e c t summand of such an object i s again of t h i s form (this i s a consequence of Azumaya's theorem, see AnderB'on and F u l l e r C I ] , Thm. 12.6, Cor. 12.7 and Lemma 12.3 ; the module techniques hold for functor categories). ( i i ) - ^ ( i i i ) (^*,Ab) has {(-,X)} with End X l o c a l as a set of small projective generators, hence i s semi-perfect. ( i i i ) = ^ ( i ) If (<6*,Ab) i s semi-perfect, then i t has a set of small projective generators with l o c a l endomorphism rings. But any small projective i n a functor category i s representable ( Freyd [7-], page 119). Hence there i s a set j(-,X)} with End C l o c a l generating (o$ ,Ab). And then any pure-projective w i l l be a d i r e c t summand of a dire c t sum from { x j . ( i i ) =^(iv) Every small projective i n Of i s a f i n i t e sum of objects with l o c a l endomorphism rings, hence Cff i s semi-perfect. Also (RDf^ (T(<8) ,Ab) has {(-,7tx)} with End %X l o c a l as a generating set of small projectives, hence i s semi-perfect. (iv) =^(i) For thisyimpllcation, some preparatory re s u l t s which can be found scattered throughout the l i t e r a t u r e i n various disguises : 124 Proposition 10.5 I f of i s semi-perfect and X has no projective summands, then J_(X,P) = (X,P) and ^(P,X) = (P,X) for P f i n i t e l y generated projective. Proof Since of i s semi-perfect, then P i s a dir e c t sum of l o c a l projectives. Then using f i n i t e a d d i t i v i t y of and J (X,-) (the Kelly r a d i c a l ) , one can assume P i s a l o c a l p r ojective. Then every composition B—*X—>P i s a non-unit since X has no projective summands, thus l i e s i n \(P,P). Result follows by d e f i n i t i o n of ^ . . // Corollary 10.6 I f erf i s semi-perfect, and X f i n i t e l y generated with no projective summands, then P(-,X) & Q (-,X). Proo# Suppose g £ P(Y,X), then g factors as Y —» P —•»X for some projective P which can be taken as f i n i t e l y generated since X i s f i n i t e l y generated. But-<Q e^_(P,X) by 10.5 , hence g € J_(Y,X). // Proposition 10.7 If of i s semi-perfect, every f i n i t e l y generated object X has a decomposition X = X' ® P , with P projective and X' has no projective summands. Proof Let Q — X be a projective cover. Then Q/jUQ) X/^(X) i s a f i n i t e d i r e c t sum of simples (Prop. 6.22). Now any projective summand P' of X results i n a non-zero summand P*/j)_(P f) of X / ^ ( X ) . Remove P 1 from X, and continue removing projective summands, since X/^_(X) i s the f i n i t e sum of simples, the process terminates. // Corollary 10.8 I f Of i s semi-perfect, X, Y f i n i t e l y generated with decompositions X = X' ® P, Y = YVf>Q as i n 10.7. Then di :X.-r->Y i s an isomorphism i f f X' X —»>Y—-9Y' and P->X->Y->Q are isomorphisms. 125 Proof Writing by Prop. 10.5 i s an 2 / V u ^2/ isomorphism, i f f ^ and 4?2 a r e isomorphisms. // Recall that X and Y are stably isomorphic i f there exist "projectives P and Q, and an isomorphism P © X-^Q © Y th i s i s equivalent to #(X) = ft(Y) by Cor. M.13. Corollary 10.9 For erf semi-perfect, X,Y f i n i t e l y generated, with no projective summands, then X i s stably isomorphic to Y i f f X i s isomorphic to Y. // Corollary 10.10 I f erf i s semi-perfect, X f i n i t e l y gener-ated with no projective summands, then End TT X i s l o c a l i f f End X i s l o c a l . Proof Any element of End 7T X i s of the form TCU . 9Cdl i s a unit i n End TLX i f f there i s an isomorphism (Cor. H. 13) X Q P± —> X ® ? 2 with P 1,P 2 projective and the component map X—>X the map cfl. Then by Cor. 10.8, i s a unit implies i s a unit, but converse i s t r i v i a l , and re s u l t follows re a d i l y . // We return to the proof of ( i v ) ^ ( i ) of PROP. 10.3. Just as i n ( i i i ) - ^ ( i ) , 7 = (ot (S3 )*,Ab) has {(- 3 7CX)} with End KX l o c a l as a set of generators. Now di i s semi-perfect so w.l.o.g. X has no projective summands, then by Cor. 10.10 End X Is l o c a l . Then any pure projective i s a di r e c t summand of a di r e c t sum from the { x | and the set of small projectives with l o c a l endomorphism rings. // 10.11 Remark Starting with C f i n i t e l y presented non-projective with l o c a l endomorphism rin g [ i f C i s projective i t w i l l be the 126 projective cover of a simple, associate C to the i n j e c t i v e h u l l of t h i s simple], one associates the unique simple epimorph of OC (C). This association i s 1-1 and onto the simples of (Rn 7 . The simple then determines a unique pure-injective non-injective A with l o c a l endomorphism r i n g (provided condition (a') holds). The correspondence C I—>A i s 1-1, but i s i t onto? We have the following proposition which holds i n general, and whose proof i s just a matter of d e f i n i t i o n of «J>(A) and characterization of subobjects i n ^jf6 . Proposition 10.12 Given A pure i n j e c t i v e non-injective then there exists a simple sequence 0 —!>A —?B—*C—>0 i f f cj>(k) has a minimal subobject. // Corollary 10.13 For each A pure-injective non-injective, there exists a simple sequence 0 —> A —* B — » C — * 0 i f f cZnT i s semi-Artinian (every object has a minimal subobject). Proof ( 4 = ) T r i v i a l ( ) Since { T < J ( A ) | cogenerate J (as i n Thm. 8.45) and T«/(A) has a minimal subobject i f f \J(A) has a minimal subobject. // 10.14 Remark Returning to the s i t u a t i o n for which (a) and (a') hold : i n this case {TteJ(A)} with A pure-injective and End A l o c a l cogenerates ( Z n J ' . In t h i s case, the correspondance C i—> A from the set of pure-projectives l o c a l endomorphisms to the set of pure-injectives with l o c a l endomorphisms i s 1-1 and onto i f f (Xn7 i s semi-Artinian. Now = T t(R)'^ * a n d t h e f ' u n c t o r r i n S ^ ( R ) i s semi-Art i n i a n i f f the r a d i c a l i s right T-nilpotent. But by assuming (a), 2f(R) i s semi-perfect which holds on both 127 right and l e f t . Hence 7E(R) w i l l be right perfect (the category ^ ^ ( J Q . (?C ( f o ) ,Ab) covariant functors on tf(*3)). 10.15 Remark; In attempting to analyze condition (a'), the dual of (a), f o r which every pure i n j e c t i v e i s a direc t summand of a dire c t product of pure i n j e c t i v e s with l o c a l endomorphism rings, the major stumbling block i s the mysterious structure of pure i n j e c t i v e s i n the general case. The basic tool i s Thm. 8.45, but this i s e s s e n t i a l l y an e x i s t -ence theorem. One would l i k e a closer" dual to the set i n the general case. The clue seems to be that i n working with & , projectives can be assumed to be f i n i t e l y generated. As noted above, for o~f semi-perfect i t seems natural to associate the pro-j e c t i v e cover of a simple to i t s i n j e c t i v e h u l l . We are thus led to consider f i n i t e l y cogenerated Injectives. For instance, results 10.5 and 1 0 . 1 0 have natural duals. The dual to 10.5 i s Proposition 1 0 . 1 6 I f X has no i n j e c t i v e summands and I i s f i n i t e l y cogenerated, then ft(X,I) = (X,I) and J.CI.X) = (I,X) . // ^ One need not impose r e s t r i c t i o n s on fff since every f i n i t e l y cogenerated i n j e c t i v e i s a f i n i t e d i r e c t sum of l o c a l i n j e c t i v e s (dual statement requires 0 / to be semi-perfect) . One also has that any f i n i t e l y cogenerated Y has a decomposition Y = Y 1 © I , I i n j e c t i v e , Y' no i n j e c t i v e summands.In f a c t , working with the /socle of Y, which i s a f i n i t e sum of simples rather than X/ JL(X) as i n Prop. 1 0 . 7 w i l l give existence and w i l l also y i e l d that Y i s a di r e c t sum of indecomposables. Furthermore, i f Y i s als;o pure 128 i n j e c t i v e these indecomposables have l o c a l endomorphism rings (End A i s l o c a l for A indecomposable pure i n j e c t i v e , B. ZimmermamvHuisgen [ 20 ] ) . So the machinery i s set to go, except f o r a major stumbling block, the natural dual of & . *6 has the c r u c i a l property , upon which a great deal depends : that a f i n i t e l y generated pure-projective i s f i n i t e l y presented, (since every pure-projective i s a dir e c t summand of a di r e c t sum of f i n i t e l y generated pure-pro j e c t i v e s ) . A 'natural' dual of f i n i t e l y presented could then be a f i n i t e l y c o ge n e r a t e d p ur e - i n j e c t i ve > -feather< than a c o f i n i t e l y presented object! Consider then the condi-ti o n (a") every pure-injective i s a direc t summand of a direct product of f i n i t e l y cogenerated pure-injectives. Note that (a") implies (a') (again using the Zimmermann-Huisgen r e s u l t ) , so that one can further impose that the endomorphism rings are l o c a l . I f (a") i s s a t i s f i e d , thef\ f ^ ( A ) J with A f i n i t e l y cogenerated, pure-injective and l o c a l endomorphism rin g cogenerates (*\ , hence for any given simple i n <R.nJ , there i s a monic into «J(A) f o r some A. Since subobjects of c J ( A ) are of the form 0 — * A —» X —•> Y —» 0 for some X,Y, any given simple can be represented with f i r s t term finitely^generated pure-injective with l o c a l endomorphism ring. But then again t h i s simple i s copure, hence there i s an epimorph of 7t(C) for some C f i n i t e l y presented 0 —=> K » P * C -=* 0 nt (C) I I I h 0 ». A 9 X 9 Y » 0 s But the image S' of t h i s map can be computed by forming the pull-back 0 — - * A - — * B »C ' 0 S' li I I I s 0 » A » X » Y ? 0 S 129 Hence assuming condition (a") any given simple w i l l have a representation 0—*A—*B—»C—>0 with a f i n i t e l y cogen-erated pure-injective with l o c a l endomorphism r i n g and C f i n i t e l y generated pure projective ( f i n i t e l y presented). I f one also imposes (a), one can further assume End C i s l o c a l and i n this case the representation i s unique, by Prop. 10.2. Recall from 10.1, the set <#' of pure-injectives, which was to act as the dual of &> . This duality was imposed, and I f e e l that the 'natural' dual i s the set-- of f i n i t e l y cogenerated pure i n j e c t i v e s . The above has shown that t h i s i s indeed the case i f (a") i s s a t i s f i e d . 10.17 The I n t r i n s i c Characterization of Simples An object S of tf& i s simple i f f i t has no proper subobjects i f f i t has no proper quotients, other than the zero object. Given any object S, l e t S^ , denote the subobject r e s u l t i n g from the pull-back of a morphism f : X — » C , i . e . 0 — ° > A - — 9 y -—> X => 0 S-li .1 I* I 0 » A » B -—> C 9 0 S Recall that any subobject of S can be represented as S f for some f. Also l e t Sj be the sum of subobjects S^ , f £ I for any set of morphisms I. S i s simple I f f S^ = 0 or S f = S for any morphism f. I f S^ = S the cokernel sequence s p l i t s ; i f S f = 0 , f factors over B —?C. One then has Proposition 10.18 0 —*A —»B —>C —>0 i s a simple object of 2/4> i f f given any morphism f : X—>C, either f factors over B—>C, i . e . ^'X B —>7C or the sum map X Q B—>> C s p l i t s . // 130 Proposition 10.19 . 0 —=> A — » B — * C—> 0 i s a simple object of~£/j£, i f f given any morphism g : A — » Y , either g factors through B, i . e . A * B Y or the sum map A?—* B (?) Y s p l i t s . // The resemblance to Auslander's almost s p l i t exact sequences (a.s.e.s.) i s immediate. I should mention that my i n i t i a l study of resulted from piecing together some notions of Fieldhouse [ 6 ] on p u r i t y , Freyd [ 8 ] , Maclane's [ fl] b r i e f mention of a sequence category, and Auslander [ 3 ] . Characterizing the simples as above came quite n a t u r a l l y , i t amazes me that Auslander pulled them out of p a r t i a l l y clouded mid-air, but also saddens me not to be the creator for I had never heard of an a.s.e.s., having shied away from papers dealing with 'representation theory'. We proceed then with the concept of a.s.e.s. and demonstrate that, i t i s not quite natural i n the general setting of the sequence category 10.20 Almost S p l i t Exact Sequences (a.s.e.s.) 0—»A—*B^*>C —»0 i s an a.s.e.s. i f A,B,C are f i n i t e l y generated, End A and End C are l o c a l , such that either (in which case both) (1) i f f : X—*C i s not s p l i t e p i , X f i n i t e l y gener-ated, then f factors over B — » C . (1') i f g : A — » Y i s not s p l i t monic, Y f i n i t e l y gener-ated, then g factors through A—?B. Now th i s d e f i n i t i o n was used o r i g i n a l l y i n the context of modules over an Artinian algebra, and has : been adopted for the more general case of Artinian rings (so that i n (1) and ( l f ) . X can be taken to be indecomposable). For A r t i n i a n algebras, the condition that X (Y) be f i n i t e l y generated can be removed (one of Auslander's r e s u l t s ) . 131 It follows that an a.s.e.s. Is a simple object of That i s , i f one considers the subcategory of f i n i t e l y gener-ated subobjects 0 / 1 i n «/, then forming (S/JS)1 using only objects from 0/' , and one has that ( ) 1 i s a sub-category of Z/d . Then i f °i i s a module category over an Artinian algebra ( ) 1 i s abelian and by d e f i n i t i o n an a.s.e.s. i s a simple sequence i n ( ) 1 i n which beginning and end termjhave l o c a l endomorphism rings. And i t then follows that i t i s also simple as an object of . In easing the r e s t r i c t i o n s that 0/ be a module category over an A r t i n i a n r i n g , should f i n i t e l y generated be replaced by f i n i t e l y presented? Por Artinian rings, the concepts coincide, and Thm. 8.43 suggests C should be f i n i t e l y presented. However, i t i s too stringent to impose that A and B also be f i n i t e l y pre-sented, for we have observed that the existence of simples i n leads to A being pure-injective, not f i n i t e l y pre-sented. Furthermore, i f one drops conditions that End C and End A are l o c a l , one i s dealing with the simples of cX Orf . However the complete generality achieved i n just dealing with simples does not y i e l d representation r e s u l t s . A compromise d e f i n i t i o n i s as follows : 10.21 Locally Represented Simple Sequences 0 — * A — » B — » C — 9 0 Is l o c a l l y represented simple i f (1) End A and End C a r e > l o c a l , (2) either C i s pure-projective or A i s pure-injective, and (3) eit h e r : i f f:X—*C i s not s p l i t e p i , then f factors over B —* C , or : i f ,g:A-—?Y i s not s p l i t monic, then g factors through A—?B. Eit h e r condition of (3) states that the sequence i s simple, so i f one holds then so does the other. As for the conditions of (2), suppose C i s pure-projective. Then this i s a copure sequence which i s simple, hence i t i s also repure. Consider S A ) , Since End A i s l o c a l , 1 3 2 EndcJ(A) i s l o c a l , and so <J>(A) i s indecomposable i n j e c t i v e with S as a minimal subobject, i . e . cJ)(A) i s the i n j e c t i v e h u l l of S. This implies »0(A) i s also a repure sequence, for i f *J(A) has a pure subobject, t h i s object would contain the repure object S. But <J(A) i s repure i f f A i s pure-i n j e c t i v e . S i m i l a r l y , i f A i s pure-injective, t h i s implies C i s pure-projective. 1 0 . 2 2 Remarks The condition of Prop. 1 0 . 1 8 for a simple sequence, that the sum map X(±)B-£>C s p l i t s , i s equivalent to X-^>C s p l i t s , since End C i s l o c a l . We have attempted to give a s e l f - d u a l d e f i n i t i o n , which i s why C i s assumed to be just pure-projective and not seem-ingly stronger ' f i n i t e l y presented'. However one has the following : i f End C i s l o c a l , and i f (±)X±—>C s p l i t s , then X±—^C s p l i t s for some i . I f C i s f i n i t e l y generated (which implies small) then r e s t r i c t i o n to f i n i t e sums i s e a s i l y removed. Even more remarkable i s that f i n i t e l y generated can be dropped (again a f o l k l o r e r e s u l t ) . Proposition 1 0 . 2 3 I f End C i s l o c a l , and © X ^ — > C s p l i t s , then for some oc , X ^ - > C s p l i t s . Proof Consider maps : C — © X ^ >-» <£> X^ —=> C For any f i n i t e l y generated subobject D of C, there is a f i n i t e set of ^ such that Z T i P * i s the i d e n t i t y on D. So 1 - i s not a unit, hence at least one ^ i s not i n J L ( C , C ) , and must be a unit. // One i s now faced with a problem of dualization. For pure projective + l o c a l endomorphism =^ pure projective + f i n i t e l y generated f i n i t e l y presented. But when w i l l pure i n j e c t i v e + l o c a l endomorphism =^ pure i n j e c t i v e + f i n i t e l y cogenerated ? I f this implication holds, conditions 133 (a') and (a") are equivalent. The implication does not hold i n general : for example, i f < V = Ab, <Z? i s not f i n i t e l y cogenerated, but i s pure i n j e c t i v e with l o c a l endomorphism ri n g . Hence one i s led to another generaliza-ti o n of the a.s.e.s., that of ' f i n i t e l y l o c a l l y represented simples' i n which the f i n a l term i s f i n i t e l y presented with l o c a l endomorphism r i n g and the beginning term i s f i n i t e l y cogenerated pure-injective with, l o c a l endomorphism ri n g . Note that for A r t i n i a n ^ f i n i t e l y generated w i l l imply both f i n i t e l y presented and f i n i t e l y cogenerated pure i n j e c t i v e , so agreement i s reached with a.s.e.s. 10.24 Existence Problems U t i l i z i n g the proof of Prop. 10.2, one has Proposition 10.25 Given C f i n i t e l y presented with l o c a l endomorphism r i n g , there exists a [ f i n i t e l y ] l o c a l l y represented simple with C as f i n a l term i f condition (a') [(a")] i s s a t i s f i e d . // Corollary 10.26 I f (a) holds, then condition (a') [(a")] i s equivalent to p o s i t i v e solution of existence problem. Proof Por then any simple i s the epimorph of., 7f(C) for some C f i n i t e l y presented with l o c a l endomorphism r i n g and then the corresponding set ($JKA)} cogenerate (R. (as i n Thm. 8.45) which yields (a') [ ( a " ) ] . // The reverse procedure i s to construct a [ f i n i t e l y ] l o c a l l y represented simple s t a r t i n g with a pure-injective [ c o f i n i t e l y generated] with l o c a l endomorphism ring. Por the f i r s t step, i t w i l l be necessary to assume that <aP(A) has a simple subobject, which then represents a simple sequence 0 —* A —* X — Y 0. (This problem did not arise using C, for then T9T(C) was a small projective i n CZHJ » hence has a simple epimorph.) The next step would then be to apply condition (a), to achieve the required [ f i n i t e l y ] 13^ l o c a l l y represented simple (dualizing proof of Prop. 10.2), one then has Proposition 10.27 For (Z/lJ semi-Artinian, then given A pure-injective [ f i n i t e l y cogenerated^ with l o c a l endomorphism r i n g , there exists a [ f i n i t e l y ] l o c a l l y represented simple with A as the f i r s t term i f condition (a) i s s a t i s f i e d . // Corollary 10.2 8 I f frf i s semi-perfect and 7T(B.) right perfect, then the [ f i n i t e l y ] l o c a l l y represented simple existence problem of Prop. 10.27 has a solution. Proof Condition (a) i s equivalent to bi and ?T(R) both semi-perfect; and right perfect i s equivalent to fisemi-Artinian and semi-perfect. // And a p a r t i a l converse : Proposition 10.29 I f <%<77 Is semi-Artinian, <yf semi-perfect and condition (a') [(a")] holds, then condition (a) i s equivalent to a posit i v e solution of the existence problem (of Prop. 10.27). Proof ( ^  ) 10.27. (^=) <Z n T semi-Artinian and condition (a 1) [(a")] implies that each simple w i l l be a subobject of <JI(A) for some A pure-injective £ f i n i t e l y cogen-erated] with l o c a l endomorphism ring. The associated [ f i n i t e l y ] l o c a l l y represented simples y i e l d a set [cj with C f i n i t e l y presented with l o c a l endomorphism rings. Then ^T/T(C)J are projective covers of the simples of CZnT hence generate <Kf>7, and hence {?T(C)j generate J . Then given D pure projective, there i s an epi <£> 7C(C) — f l ( D ) . So i f P-*> D i s epi , with P p r o j e c t i v e , then ( <£>C) <•£) P — » D s p l i t s . Now i f &f i s semi-perfect, then P i t s e l f i s a 135 d i r e c t sum o f f i n i t e l y g e n e r a t e d o b j e c t s w i t h l o c a l endomorphism r i n g s , which g i v e s t he r e s u l t . // T O . 3 0 C o n s t r u c t i o n o f Sim p l e s G i v e n C n o n - p r o j e c t i v e , f i n i t e l y p r e s e n t e d , by 1 0 . 2 t h e r e e x i s t s a s i m p l e 0 — » A — » B — ' C — ^ 0 . I f End C i s l o c a l , one has a pr o c e d u r e o f c o n s t r u c t i n g t h i s s i m p l e : Step 1 F o r each f i n i t e l y p r e s e n t e d X, l e t X = © X , where f o r each g Q(X,C), g e ^ ( X , C ) g * X = X ; one has a n a t u r a l map X •—>C w i t h o components g : X —» C. Note t h a t s i n c e End C i s l o c a l , J.(X,C) = f g € r ( X , C ) which are not s p l i t e p i j . As b e f o r e , l e t (75 be the s e t o f r e p r e s e n t a t i v e s o f f i n i t e l y p r e s e n t e d o b j e c t s . L e t C = (£> X th e n t h e r e i s a n a t u r a l map ty : C —^ C w i t h compo-n e n t s X — ? C as above. Step 2 Form the e x a c t sequence i n , 0 TTCO^ —> 3T(C) —» E -»0 , t h a t i s 0 — ? K * E » C 9 0 7 f(C) II I 1* t 0 — > K > P » C — * 0 T ( C ) I 4 _ II i o — = » E — > P e C -» C * 0 E C l a i m , 7C(C)^ i s t h e sum o f a l l p r o p e r copure s u b o b j e c t s o f 7T(C). I n f a c t , ( 7 C(X), X e gen e r a t e s 7 , and the image o f %(q) : 7C (X) ^ ( C ) l i e s i n -flTCC)^ i f €<J(X,C) by c o n s t r u c t i o n , i . e . I** , f a c t o r s o v e r V> i f € \(X,C) , C and i f 4 ^ ( X , C ) , a i s s p l i t e p i , and t h e n 7T(X) ft(C) i s a s p l i t e p i . 136 Furthermore, 7 t ( C ) v ¥ ft'(C), . for i f th i s were true, then E = 0, which means P <£>C—»C —»0 s p l i t s . But End C i s l o c a l , which means P—*C—^0 s p l i t s or C—)C->0 s p l i t s . The f i r s t i s not the case and i f the l a t t e r holds, then some component map X—*C—>0 would s p l i t . But a l l components are i n the (Kelly) r a d i c a l so t h i s i s not possible. [Note A non-constructive approach of achieving 7 T ( C ) y , i . e . without mention of V or C , i s to show that the set of proper copure subobjects of ?T(C) i s closed under f i n i t e union. In f a c t , i f TCiC) „ ,1=1,...n, then the sum of ^ ( C ) -* i * i i s 7T(C) f, where f^ : X I ~ ^ C and f i s the sum of the f ± , f : 0 X 1 — » C . Then 7C(C) f = vT(C) i f f f i s a s p l i t epi i f f f^ i s a s p l i t epi for some I, since End C i s l o c a l , i f f T ( C ) f . = #(C) f o r some i . Now since %{C) i s a small projective, the t o t a l sum i s a proper subobject of ^T(C).] Now every subobject of E i s pure. In f a c t , suppose E' i s a proper subobject. We show that tE' =0. Form the pull-back i n : 0 — > X(C)^ > X » tE' > 0 II £ FB. J 0 —> 7T(C V — » T ( C ) • — » E » 0 X i s an extension of 7t(C)^, and tE', so by Prop. 5.7, X i s copure. But th i s Implies X = 7C(C) or X £7C(C)^. I f the former holds, then 'tE1 = E, so E* = E contradiction. The l a t t e r implies X = 7i (C)^ forc i n g tE' =0. Step 3 Form the sequence r'E. To form r'E , f i n d a pure monic E>—>A with A pure-injective and form pushout i n OJ . 137 0 — » E —=> P <$> C — » C > 0 E X I II i 0 — 9 A > B . * C — 9 0 r'E Now, 0 — ^ t'E —> E —> r'E —» 0 i s exact, where t'E i s the unique maximal pure subobject of E (Cor. 7.3), hut since a l l subobjects of E are pure, t'E i s a maximal subobject, hence r'E i s simple. 10.31 Remarks The object C formed i n Step 1 i s quite large ( i . e . not f i n i t e l y generated). Reducing C to a f i n i t e l y generated object seems to be the crux of establishing the existence of an almost s p l i t exact sequence terminating i n C. The following technical lemmas are useful, i n c o n t r o l l i n g the size of C. Let E = 0—»A—>B—yC—>0 be an arbitrary sequence (dropping conditions on C). Lemma 10.32 I f h : X—> C factors as X —£-» Y - J L * C then the subobject E. £ E„ . -h —g Proof E_h results from the pull-back of h with B —*C. But thi s can be achieved f i r s t as pull-back with g then f. 0 —* A —> B" — » X — * 0 E h I I | 0 — ^ A »B' •> Y 3, 0 E 0 ? A ? B » C » 0 E . // Lemma 10.33 I f g : @X^—> C has components g^ then Proof Obvious from d e f i n i t i o n s . // 138 Lemma 10.34 I f S = l&±]±&i generate Hom (X,C) as an End X module, then E<, = llnom (X C) Proof I f h6Hom (X,C), there exists a f i n i t e set i n End X such that h = ^ ^ 5 ^ • Then h factors x _ f , © X l - ^ C . o ^ £ K g - E f g i ? £ ^ (equality step by 10.33). // Lemma 10.35 I f g : X — » C i s e p i , then coker 7T(g) : ?t(X) —> 7T(C) i s E = 0 — £ K —•> X — ^ C — 9 0. Proof This follows from Cor. 3.21, for . . * 7r(X) — * 7C(G) — » E —»0 i s the s t a r t of a projective resolution for E. [For an e x p l i c i t proof, 0 — = » H * P > X — * 0 7T(X) I 1 9^ i *<g) 0 —> L * Q — - * C — * 0 TC (C) I i ll 1 0 —* E — ? X © Q —=> C — ^ 0 *E Then 0 — 9 K — 9 X • =» C ^ 0 i s an Isomorphism i £ )l • o —-> E — ? X 6 > Q — * c > 0 with inverse 0 * E — * X <3) Q — > C * 0 0 •* K * X » C 9 0 where V i s the map V / Q , using p r o j e c t i v i t y i of Q . ] X—»7 C 10.36' Example Let of = Ab, the category of abelian groups. Then f i n i t e l y presented i s equivalent to f i n i t e l y generated, and C3 , the subcategory of f i n i t e l y generated abelian groups, has the property that every object i s a f i n i t e d i r e c t sum 139 of c y c l i c groups /.(pk) , k = 0,1,2,.... and p a prime. End Z(p k) i s l o c a l for k J 0, and for k = 0 one has the integers Z. For Ab, G i s pure projective i f f , G i s a d i r e c t sum of c y c l i c groups (this follows from a c l a s s i c a l theorem of Kulikov, that subgroups of dir e c t sums of c y c l i c groups are again a d i r e c t sum of c y c l i c groups, or can be derived from the decomposition theorems i n F u l l e r and Anderson [1].) I f one ignores the projective (free) summand of a pure-projective object, then condition (a) holds, ( i . e . modulo projectives (a) holds f o r Ab). *2 (Kite) ,Ab) w i l l have ^ ( - , 7 c ( 2(p k))} k=l,2,... as projective generators with l o c a l endomorphism rings, so (frOS" i s semi-perfect, but Of - Ab i s not ( r e c a l l that (a) holds i f f both Crf and (£.0 T are semi-perfect). Also for Ab, G i s pure i n j e c t i v e i f f G i s a dir e c t summand of a d i r e c t product of c o c y c l i c groups (see Fuchs [ 2 3 ] , part of Thm. 38.1), Z ( p k ) , k=l,2, and eO . Note Z(p k) c—> 2^ (p°°) i s an e s s e n t i a l monic, and XXp°°) i s the i n j e c t i v e h u l l of the simple Z(p), so a l l the cocyclic groups are f i n i t e l y cogenerated. Also a l l have l o c a l endo-morphism rings. So (a") (and hence (a')) holds for Ab. Hence every simple object of fcn3" (which i s also simple i n %/& ) has a unique representation 0 — A —*>B—»C —*>0 with C c y c l i c (and not 2 ) and A c o c y c l i c (and not Z t p 0 0 ) ) , y i e l d -ing a 1-1 correspondence C i—»A. Set C - 2(p k). We construct ^C(C)^ as i n Step 1 of 10.30. Given X—>C, X f i n i t e l y presented, since every f i n i t e l y pre-sented object i s a direct sum of c y c l i c s , we can assume by Lemma 10.38 that X i t s e l f i s c y c l i c . Then consider any map not s p l i t epi g : X—»C. I f g i s not e p i , g factors as X ? Z(p k - 1)£—^—» 2(p k) (for the moment exclude the case k =1). Hence by Lemma 10.32, 7T(C) Q 7f(C). . I f g i g i s epi (but not s p l i t ) , then X i s necessarily of the f ° r m Z / P r ) , with r >k. In t h i s 140 case, g factors as X — > Z ( p k + 1 ) 2(p k) , with V the canonical epi. Again by Lemma 10.32, 7C(C) 7C (C)^ . Hence ^ ( C ) ^ = 7C(C) ± + -X(C)p = ^ ( O i © ^ where i£>» : Z ( p k _ 1 ) C 5 Z ( p k + 1 ) - » Z p k , so by Lemma 10.37 the required copure simple of Step 2 i s coker 7t(i©^) = 0 A 2 ( p k _ 1 ) (3> 2 ( p k + 1 ) ^ > #(p k)-*0. But then A i s a f i n i t e abelian group, i n p a r t i c u l a r A i s pure-injective, so Step 3 factoring out the maximal pure subobject i s not necessary. The simple has been achieved. A simple computation shows A *= Z pk , and that the required simple i s o -+ Z ( P k ) ...(1»-P>> 2( P k- 1)©Z(p k + 1)2l2(p k)-^ o • The case k=l i s even easier. In thi s case, non-zero maps are e p i , which implies that 1V{C) . S ^ ( C ) for a l l g o not s p l i t e p i , where v : Z ( p 2 ) — * > Z ( p ) . The r e s u l t i n g simple i s then o Z ( P ) 2 ( P 2 ) — ^ Z ( p ) - ^ o . So f o r of - Ab, the correspondence C *> A achieved by constructing simple sequences 0 —» A—> B—> C —» 0 r. \. i i s just the i d e n t i t y . 10.37 Remark By Cor. 3.23, i f p.d.A = n , then p.d. ^ 3n-l. Equality holds for Ab: for consider the simple S = 0 —> Z p —•> 1^2 -—> Zp—•> 0 , we show that p.d.(S) = 2 , so p.d. tl& = 2. By Cor. 3.21, one has a projective resolution 0-^ 7C( Z ) 9t( Z 2) — * x ( Z ) — * E 0 P P • P -I f p.d.(S) < 2, then 0-? *7C( "2. ) — » 7T( Z 2) i s a s p l i t P P monic, which i s impossible since they are non-isomorphic indecomposable projectives. 10.38 The Existence of a.s.e.s. The existence of a.s.e.s. with a given f i n a l term C ( f i n i t e l y presented, l o c a l endomorphism ring) i s of a more d i f f i c u l t nature than the existence problem for l o c a l l y represented simples. The d i f f i c u l t y arises from the imposi-t i o n that the leading term be f i n i t e l y generated. This seems to be a red herring : the basic j u s t i f i c a t i o n i s that the category &JX of f i n i t e l y generated objects i s an abelian subcategory of &f. So one can form a sequence category S/^ ' , where E £ i f f E ^ E ' where E' i s an exact sequence with f i n i t e l y generated terms. Then t//6 ' i s indeed a f u l l exact abelian subcategory of , but ' Is buried within 7 and does not play the role of f i n i t e l y generated objects. I t seems more natural i n dealing with and i t s subcategories & yf^y-jT , to consider sequences E = 0 —* A — * B — » C — > 0 with A pure i n j e c t i v e and C pure protective, i . e . E (r 7" (and conversely, any object of <R.n T has such a representation). These are precisely the sequences E such that Horn (Ej/J^) = Horn (/^jE) = 0. F i n i t e r e s t r i c t i o n s can then be imposed, for instance, f i n a l term f i n i t e l y generated and/or leading term f i n i t e l y cogenerated. One should note also that i f one imposes the condition that the f i n a l term has a l o c a l endomorphism r i n g , then this i s i n fact a f i n i t e condition. For ,/ Prop. 10-23 implies 7C(C) i s f i n i t e l y generated i f End C i s l o c a l , hence any quotient of 7T(C) i s also f i n i t e l y generated. So that any sequence terminating i n C w i l l be w i l l be f i n i t e l y generated. Unfortunately, the dual does not seem to hold; that i s , End A l o c a l w i l l not imply O ( A ) f i n i t e l y cogenerated. Returning to a.s.e.s., to examine how the existence problem f i t s within the framework of the sequence category : as noted i n 10.20, an a.s.e.s. i s a simple object of 1 , the sequence category using f i n i t e l y generated objects, and every simple i s uniquely represented as an a.s.e.s. (where 142 Of , A A r t i n i a n . So.condition (a) holds, so any simple can be represented with leading and f i n a l terms f i n i t e l y generated with l o c a l endomorphism r i n g ) . Now given C f i n i t e l y presented with l o c a l endomorphism r i n g , one can proceed as i n 10.30 to construct the unique simple ( i n ) epimorph S of 7T(C). The object E a r i s i n g from the exact sequence 0 — > TCiC)^ —> X(C) —> E — ? 0 i n Step 1 of 10.30 i s copure simple. That Is, E i s copure but every proper subobject i s pure. This i s cl e a r l y equivalent to every map 7T(X)—>E , with X f i n i t e l y presented, either zero or e p i . I f E = 0—* A' —* B 1 —? C —°> 0, thi s i s equiva-lent to the statement that any non s p l i t epi X—=>C, X f i n i t e l y presented, factors over B'-—*C , i . e . , " X 0 9 A' » B' • •> C • > 0 Suppose now G = 0—» A" —» B"—>C->0 i s an a.s.e.s. Then ^(C)^^—*7C(C)—x> G i s either zero or e p i , but 7r(C) i s a small l o c a l p r o j e c t i v e , so T(C)— » G i s a superfluous ep i , so the composition cannot be epi. Hence one has 0 » 7T(C) y » 7T(C) » E »0 J H ^ 0 9 L —i>ir(C) 2> G — ? 0 i . e . 7C(C).C—> L. However the epi \J : B" —>C factors through the map V since i t i s not s p l i t epi (see Step 2) so by Lemma J0.3Z , L » ft(C)^ ^ T ( C ) y , which implies E = G. So the existence of an a.s.e.s. terminating i n C implies EC -t/A1 • Conversely, i f E 6 £-^5' , i t i s a simple object of 1 . Also HI (C)—» E, so w.l.o.g. E terminates i n C (this i s only a t e c h n i c a l i t y but i n detail,represent E = 0—^ A' —;> B 1—?C—*> 0 and %{C)-^y E as 0 —=9 K =>P =>C—>0 *7r(C) i l l i 0 — A 1 —•> B ' — C 1 — 9 0 E then E ^ Im (7C(C)—=> E) which i s computed as the pushout of K—3>P and K > A' , i.e.' E ^  0 —«»K —*A» © P - ^ C - » 0 143 and a l l terms are f i n i t e l y generated.) Set E = 0-*A'^»B'—»C-> then furthermore condition (a) holds so A' = ( 5 ) A^ each with l o c a l endomorphism r i n g , which implies E — * E i s an i s o -morphism for a unique i , where 0—> A' > B' * C — » 0 E l I w I 0 — 9 k± — » B ± » C ^0 E ± (since E >-> TTE^  and E i s simple i n £,£&»). Hence for the existence of the required a.s.e.s., i t i s necessary and s u f f i c i e n t that E £ t/J> V(. To make this more t r a c t i b l e , we need the following technical lemmas : Lemma 10.39 OJ = frA , A - A r t i n i a n , i f E 1 >-> E_ 2 and E 2 & then E £ i f f E± i s copure and f i n i t e l y generated as an object of Zf^ . Proof ( ) By Prop. 5A a l l objects of 1 are f i n i t e l y generated; and i n ^ / f i n i t e l y generated implies f i n i t e l y presented so a l l objects of ' are also copure. (<£= ) Por some X f i n i t e l y presented, 7C ( X ) — » E ^ then E± ^ Im ( 7C (X) — E ^ V - > E_ 2 ) so i f 0 '—>K •—=> P =>X • — ^ 0 0C(X) 0 — , A 2 - — » . B 2 — C 2 — * 0 E 2 then E 1 ^  0 —•> E - ^ X 0 B 2 — » C 2 — > 0 which has f i n i t e l y generated terms. // Lemma 10.40 &f =flA. , A- Ar t i n i a n . I f 0 E^—=» E_2 —E^—=>0 i s exact i n and E 2 £ € ^ ' , then E^ £ ' i f f E_3 £ . 144 Proof ( I f E ^ — * E_2 i s represented as 0 — A n •—=> B , =» C. — * 0 I 1 I 0 -—^A2 — ^ B2—=> C 2 —=> 0 with a l l terms f i n i t e l y generated then coker '(Ej—*E ) = 0 - > E - ^ C 1 ( S B 2 ~ ^ C 2 — * 0 . so E_ 3 e Z/^' . (<£= ) Dual. // Returning to previous discussion, we have an exact sequence 0—»7l ( C ) y , — > T ( C )—* E —» 0 , and flf(C)y, i s copure, so E £ £ ^ 1 i f f ^ ( O ^ e (Lemma 10.40) i f f ^(COy i s f i n i t e l y generated (Lemma 10.39). This establishes part of Proposition 10.41 Assume A Art i n i a n . Then given C f i n i t e l y presented with l o c a l endomorphism ri n g f.a.e., ( i ) there exists an a.s.e.s. terminating i n C. ( i i ) the unique maximal proper copure subobject of 7C(C) i s f i n i t e l y generated, ( i i i ) the unique simple epimorph of 7T(C) i s f i n i t e l y presented i n O' . Proof ( i ) ^ 4 ( i i ) by previous discussion ( i i ) ^ ( i i i ) One has the exact sequence 0—^XtC jy .—»*a : (C )—*E~»»0 , and by assumption, #T(X)—=??#(C)^, for some X f i n i t e l y presented. Then apply the exact functor T, TE i s simple and 0-VTft(c)y-VTrt(C)^-rE^0aoA Tn Tfc(c)y„ by Lemma 8.40 T *r(X) i s a small projective i n ROT hence f i n i t e l y gener-ated so T7T(C) v f. i s also f i n i t e l y generated, ( i i i ) ^ ( - i i ) By assumption, T %(X) ->? T71 (C)^ for some X f i n i t e l y generated. Apply the right exact functor R, RTflT(X)-» RT7t(C) v. . But RT-TTU) ~ <ft(X) by Lemma 8.47, and the counit R T T C X C ) ^ — ^ K i C ) ^ i s epi since ^T(C)^ i s copure (8.29). So OC(X) * RT7T(X)-» R T ^ T ( C V ~*> ^ ( C ) ^ , shows 7C(C)y, i s f i n i t e l y generated. // Corollary 10.42 For 0/ = A ^ , A A r t i n i a n , then given any C f i n i t e l y generated with l o c a l endo-morphism rin g there exists an a.s.e.s. terminating i n C i f f the simples of <R.nT are f i n i t e l y presented. // 146 BIBLIOGRAPHY [ 1 ] Anderson, F.W. and F u l l e r , K.R. ( 1 9 7 4 ) , Rings and Categories of Modules, Springer-Verlag (New York). [ 2 ] Auslander, M. ( 1 9 6 9 ) , "Comments on the Functor Ext', Topology, 8 , pages 1 5 1 - 1 6 6 . [ 3 ] Auslander, M. ( 1 9 6 5 ) , "Coherent Functors", Proc. Conf. Categorical Algebra, La J o l l a , Springer-Verlag, pages 1 8 9 - 2 3 1 . [ 4 ] Auslander, M. and Bridger, M. (19 ), "Stable Module Theory", Memoirs of the A,M.S., #94. [ 5 ] Dickson, S.E. ( 1 9 6 6 ) , "A Torsion Theory for Abelian Categories", T.A.M.S., 1 2 1 , pages 2 2 3 - 2 3 5 . [ 6 ] Fieldhouse, D. ( 1 9 6 9 ) , "Pure Theories", Math. 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