UBC Theses and Dissertations

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UBC Theses and Dissertations

Finitely presented modules and stable theory Gentle, Ronald Stanley 1976

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FINITELY PRESENTED MODULES AND  •STABLE THEORY  Ronald B.Sc,  Stanley Gentle  U n i v e r s i t y o f Toronto, 1974  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  i n the Department of Mathematics  We a c c e p t t h i s t h e s i s as conforming required  t o the  standard  THE UNIVERSITY OF BRITISH COLUMBIA August, 1976 (c)  Ronald Stanley Gentle, 1976  In p r e s e n t i n g t h i s thes i s in p a r t i al fu 1 f i Invent o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and I f u r t h e r agree t h a t p e r m i s s i o n  f o r e x t e n s i v e copying o f t h i s  that  study. thesis  f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h i s  representatives.  It  i s understood that copying o r p u b l i c a t i o n  o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my written  permission.  /V/l~TH^^4T^  Department of  S  The U n i v e r s i t y o f B r i t i s h Columbia 2075 W e s b r o o k P l a c e V a n c o u v e r , Canada V6T 1WS  Date  ^  /~?  6  (ii)  ABSTRACT  T h i s t h e s i s i s a two pronged a f f a i r .  P a r t one  i s a study o f f i n i t e l y p r e s e n t e d modules u s i n g the techniques o f homological a l g e b r a .  We e s t a b l i s h a  theorem i n v o l v i n g c e r t a i n exact sequences, which  proves  to he h i g h l y e f f i c i e n t i n d e a l i n g w i t h the t h e o r y o f f i n i t e l y presented modules.  An attempt  to u n i f y many o f the r e s u l t s found  has heen made  i n the l i t e r a t u r e ,  ( w i t h the i n c l u s i o n o f some o r i g i n a l  results).  P a r t two, which can he read i n d e p e n d e n t l y o f p a r t one,  i s a study o f t h e c a t e g o r y o f s h o r t e x a c t sequences  modulo s p l i t  sequences.  S p e c i a l a t t e n t i o n i s paid to  p r o j e c t i v e s i n t h i s c a t e g o r y ; an e x p l i c i t c o n s t r u c t i o n of a p r o j e c t i v e r e s o l u t i o n , with i t s ' an a r b i t r a r y o b j e c t i s g i v e n .  consequences, f o r  P a r t two i s r e l a t e d t o  p a r t one i n p r o v i d i n g a c a t e g o r i c a l bedding, e n r i c h i n g the t h e o r y o f f i n i t e l y  presented  thereby  modules.  (iii)  TABLE OP CONTENTS  Introduction  P a r t One:  1  F i n i t e l y P r e s e n t e d Modules  1/  Preliminaries  Ie  2/  D u a l i t y and F i n i t e l y Presented Modules  3  3/  Absolute P u r i t y ( f . g . I n j e c t i v i t y )  17  4/  Coherence and f .p. . I n a c t i v i t y  22  5/  G e n e r a t o r s and R e l a t i o n s  P a r t Two:  Stable  '  27  Theory  6/  The Category  7/  P r o j e c t i v e Homotopy  39  8/  K i l l i n g Projectives  44  9/  Syzygy F u n c t o r  51  10/  Auslander's f . p . D u a l i t y Functor  57  11/  Odds and Ends  60  Bibliography  o f E x a c t / S p l i t Sequences  32  63  (iv)  Acknowledgements  Dr. S. Page r e c e i v e s my acknowledgement advice,  for his  guidance, encouragement and f o r p l a n t i n g a  seed o f an i d e a . f o r h i s reading  I a l s o wish to thank Dr. J.L.MaeDonald of t h i s t h e s i s .  R e a l i z a t i o n o f the  t h e s i s would not o f been p o s s i h l e w i t h o u t the f i n a n c i a l support of the N a t i o n a l Research  Council.  My  appreciation  to the a u t h o r s , M. A u s l a n d e r , P. Preyd and P. H i l t o n f o r f i l l i n g my head w i t h i d e a s . to B a r b a r a f o r i n s p i r a t i o n and  F i n a l l y my typing.  gratitude  (1),  Introduction For  a r i n g R, one has the a b e l i a n c a t e g o r i e s o f  r i g h t and l e f t R-modules.  How  one o f these c a t e g o r i e s a f f e c t how  does the s t r u c t u r e o f the o t h e r ?  For instance,  does the s t r u c t u r e o f the subcategory o f f i n i t e l y -  generated l e f t modules a f f e c t  r i g h t R-modules.  The  b a s i c d e v i c e , i n making the passage from l e f t t o r i g h t modules, i s the d u a l i t y however the d u a l i t y  f u n c t o r * (Hom(-,R)).  functor greatly alters  p r o p e r t i e s ; f o r example a f i n i t e l y i s not n e c e s s a r y c a r r i e d module. of  A f i r s t attempt,  the d u a l i t y  Unfortunately  structural  generated l e f t  to a f i n i t e l y  generated  t o overcome t h i s  module right  instability  f u n c t o r , i s to c o n s i d e r f o r each  finitely  generated l e f t module N, a p r o j e c t i v e p r e s e n t a t i o n P -^N and  l e t U° be coker : N* •» P*.  generated r i g h t module.  N° i s then a  finitely  However the c o r r e s p o n d i n g  p r e s e n t a t i o n P* —s*N°, has k e r n e l N* which i s a d u a l module; so t h a t the s e t o f modules [N°3 subset o f f i n i t e l y assignment relate  i s only, a  generated r i g h t modules.  i s thus d e f i c i e n t ,  special  This  and f a c t s t o e f f e c t i v e l y  the s u b c a t e g o r i e s o f f i n i t e l y  generated r i g h t and  l e f t modules. All  i s not l o s t , however i f one passes t o the more  restrictive (the  subcategory o f f i n i t e l y  importance  of f i n i t e l y  presented modules  presented  ( f . p . ) modules  (la)  cannot be under e s t i m a t e d , as every module i s the d i r e c t l i m i t o f f . p . modules). how  the s t r u c t u r e of the  T h i s t h e s i s demonstrates  category of f.p. l e f t  modules  determines the s t r u c t u r e o f f . p . r i g h t modules (modulo f i n i t e l y generated p r o j e c t i v e s ) . the  I will briefly  c o n t e n t s of t h i s t h e s i s . In s e c t i o n two,  between f . p , l e f t  Thm.  and  2.1  e s t a b l i s h e s the  f . p . r i g h t modules.  connection  This  s e t s up  the dominoes; a l l r e s u l t i n g p r o p o s i t i o n s  section  (and  indeed f o r a l l o f p a r t one)  This section incorporates Bass (3)  (who  d u a l i t y , and  McRae (14)  fall  theorem of  this  easily.  some r e s u l t s o f Jans (13)  work w i t h n o e t h e r i a n  o f f . p , modules, but are  outline  (15)  rings)  and  concerning  on p r o j e c t i v e dimensions  the use  of Thm.  substantially different.  A few  2.1  means the  proofs  o r i g i n a l r e s u l t s are  also present. • In s e c t i o n t h r e e , Fielhouse  (7).  pure theory tensor is  Use  o f Thm.  2.1  i s introduced  product).  shown to be  A b s o l u t e p u r i t y as d e f i n e d  equivalent and  following  immediately shows t h i s  c o i n c i d e s w i t h t h a t of Conn's (6)  of f . p . i n a c t i v i t y The  pure t h e o r y  (using by Maddox  to both the h o m o l o g i c a l concept  copure i n a c t i v i t y  of F i e l d h o u s e  b a s i c p r o p e r t i e s o f a b s o l u t e l y p u r i t y are  J a i n (12)  on f . p . i n j e c t i v i t y and  (7).  established.  S e c t i o n f o u r c o v e r s some r e s u l t s o f Stenstrom and  (16)  coherence, and  (18) contains  ;  (lb) • 1  the  p a r t i c u l a r l y u s e f u l Thm. 4 . 3 > f o r t e s t i n g f l a t n e s s  and  f.p. i n j e c t i v i t y .  heavily  Both o f the above a u t h o r s , r e l y  on the c h a r a c t e r  functor  ( - ) = HonigC-^/Z), 0  w i t h d u a l i t y formulas (a) Ext i(p,L°) = T o r ( F , L ) ° , J  n  (b) E x t ( F , M ) ° = T o r ( P , M ° ) , R r i g h t c o h e r e n t , P f . p . n  n  and  w i t h Tor^(M,P) =" Hom(Ext1(P,R),M), P f . p . , M  infective, R left  coherent.  (Cartan  I f e e l the i n t r o d u c t i o n o f c h a r a c t e r  (4)).  and E i l e n b e r g  modules i n t o t h i s  t h e o r y i s somewhat a r t i f i c i a l and obscures the r e s u l t s ; once a g a i n Thm. 2.1 Flatness  enables a l t e r n a t e  can be i n t e r p r e t e d  proofs.  i n terms o f r e l a t i o n s  ( l i n e a r e q u a t i o n s ! ) (Chase ( 5 ) ) .  Section  five  i s an  a n a l y s i s o f t h i s r e s u l t , ( u s i n g Thm. 4 . 3 ) , and shows how f . p . i n a c t i v i t y has a s i m i l i a r i n t e r p r e t a t i o n . Part  two o f the t h e s i s i s o f a more c a t e g o r i c a l of r e s u l t s of Auslander ( 1 )  n a t u r e , and i s a s y n t h e s i s H i l t o n (9)  ( 1 0 ) and Freyd  o f an e f f o r t  t o make Thm. 2.1  I subsequentially (2),  (8).  Part  (2),  two o r i g i n a t e d out  ( a ) (b) f u n c t o r i a l (which  found t o be h i s t o r y , S t a b l e  Module Theory  hence t i t l e o f t h e s i s ) Section  (Preyd  (8))  s i x contains gives  two major r e s u l t s .  one a home f i e l d  i n which to study the f u n c t o r s Prop. 6.4  (an a b e l i a n  of sections  category  e i g h t and t e n .  i s e s s e n t i a l f o r a l l r e s u l t s of section  concerning p u r i t y .  I t i s a l s o o f utmost  6.7  Thm.  three  importance  (ie)  because i t are  zero  a l l o w s one to determine when morphisms ( i n  maps.  S e c t i o n seven i s a study o f H i l t o n ' s homotopy ( 9 ) ( 1 1 ) . A l l r e s u l t s exception  are  known W i t h  possible  of P r o p . 7.5.  Section eight  puts some r e s u l t s  o f H i l t o n and Ree (10)  and A u s l a n d e r and B r i d g e r (2) i n t o the framework e s t a b l i s h e d  concerning projective £/A> ; i t s '  Thm. 9.3, a r e s u l t  resolutions  many c o r o l l a r i e s  ten,  o f my own,  i n the a b e l i a n c a t e g o r y  indicate  but I wish f o r a more e l e g a n t In s e c t i o n  categorical  by Preyd ( 8 ) .  Section nine contains  p a r t one,  projective  it  i s o f some v a l u e  proof.  we r e t u r n to the s u b j e c t  p u t t i n g Thm. 2.1 (a)  (b)  into  matter o f  categorical  language. F i n a l l y section and poses  eleven i s of a miscellaneous  some problems.  Through somewhat f o r t u n a t e enjoyment this  nature  on my p a r t ) ,  thesis  are  (with exception  known  circumstances  (more  a l t h o u g h most of the r e s u l t s a l l the p r o o f s g i v e n are  of  original  o f p a r t s o f Thm. 6 . 7 ) . E i t h e r I f e l t my  own p r o o f s were s i m p l i c a t i o n s ,  o r because the g i v e n p r o o f s  were bound up i n too much t h e o r y  (heavy  categorical  machinery)  and would take the r e a d e r too f a r a f i e l d o r s i m p l y because no p r o o f s were g i v e n .  (Id)  I f no r e f e r e n c e i s g i v e n i n t h i s t h e s i s , i s o r i g i n a l ; u n l e s s I overlooked o n l y a second c r e a t o r . —»  its  A f i n a l note  e p i c , (which i t i s n o t .  author, :>—*  the then  result being  denotes monic,  Good r e a d i n g ) .  (le)  1/  Preliminaries. Let R be an a s s o c i a t i v e r i n g w i t h i d e n t i t y , RM t h e  c a t e g o r y o f l e f t R-modules.  M* = HomR (M,R), then f o r  M a l e f t module, M* i s a r i g h t module by the a c t i o n (fr)  (x) = ( f ( x ) ) r  fcM*, xeM, reR.  The assignment Mi—>M* i s a c o n t r a v a r i a n t basic  f a c t s concerning t h i s functor  following  functor.  are stated  The  i n the  theorem.  Theorem 1.0. (a)  There i s a n a t u r a l  isomorphism  9:Hom(N,M*) = Hom(M,N*) where ( 6 ( f ) ( m ) ) ( n ) = ( f ( n ) ) ( m ) . Hence we have, (b)  Regarding * as a c o v a r i a n t  opposite category),  f u n c t o r RM—>MR op( t h e  then * i s i t s ' own l e f t a d j o i n t .  a r e s u l t * p r e s e r v e s c o l i m i t s , but b e i n g t h i s means * t r a n s f o r m s c o l i m i t s t o l i m i t s products, cokernels to kernels, (c)  The u n i t (and c o u n i t ) njy[:M—y M'  contravariant (sums t o  pushouts t o p u l l b a c k s ) .  of t h i s adjunction i s  ( n ( m ) ) ( f ) . = f(m) M  feM*  A module M i s c a l l e d t o r s i o n l e s s ( r e f l e x i v e ) i f injective (d)  As  is  ( a n isomorphism).  The t r i a n g u l a r i d e n t i t y f o r t h i s a d j u n c t i o n i s M  i s a s p l i t monic.//  (2)  The b a s i c example 0—>• L - — » R — > R / L — > 0 , ideal, via  g i v e s 0—>• ( R / L ) * —  r  L a left  R* e x a c t , and ( R / L ) * = Ann(L)  f n ) f (1 ). A module M i s s a i d  t o be f i n i t e l y  presented  i f there  e x i s t s an exact sequence P ' — y P — j - M — y O w i t h P, P' finitely  generated  projective.  finitely  presented  ( f . p . ) modules i s c l o s e d under  cokernels  The f u l l  sub-category o f  but n o t n e c e s s a r i l y c l o s e d under k e r n e l s .  Left  N o e t h e r i a n r i n g s a r e p r e c i s e l y those r i n g s f o r which the category o f f i n i t e l y a ring i s called  left  modules i s a b e l i a n . finitely f.p.  generated  modules.  generated coherent  ( f . g . ) modules i s a b e l i a n ; i f the c a t e g o r y o f f . p .  A r i n g i s coherent  i f and o n l y i f  submodules o f f . p . modules a r e i n t u r n  (3)  2/  D u a l i t y and PiniteJLv_JPresejited_Mod^l^es F o r r e f e r e n c e the f o l l o w i n g i s a s l i g h t  of  the standard  extension  Snake-lemma.  Snake Lemma - Ker-Coker Sequence ( ( 4 ) Lemma 10.1 page 101)  0 —>B1—^2—^B3-^B4-^B5-^... commutative w i t h exact rows, then we have an exact  ..  = ker f^,  = coker f±,  sequence  * A5 -* A4 * K i * K2 * K3 * C i * 02 * C3 *B4  B5  ...//  Theorem 2.1. Given a f i n i t e  p r e s e n t a t i o n P'  there e x i s t s a f i n i t e l y  P -*• A o f a r i g h t module A,  presented  l e f t module A such t h a t  (a)  0 * 1* + P* * P -» A '+ 0 and 0 •> A* * P* -> P *  (b)  Any f i n i t e l y presented  A*  £ * 0  1  f o r some f i n i t e l y  l e f t module B, i s o f the form  presented  r i g h t module A (and vice'"  versa). (c)  0 * E x t (A,-) -y A0- + Hom(A*,-) 1  E x t 2 ( A , - ) -> .0  0 * Tor2(-,A) + -8A* * Hom(A, - ) -» Tor-} (-,A) -» 0 If0-»M-**N-?Q->0is  exact then t h e r e e x i s t s u, v  such t h a t (d)  0 -» Hom(A,M)  Hom(A.N)  Hom(A,Q) £ ASM + A0N * A0Q v 0  (e)  ...  (f)  F o l l o w i n g commutes i n every p o s s i b l e way  T o r ! (A,N) -> Tor-j (A,Q) ^ E x t ( A , M ) 1  Hom(l,Q) * Ext (A,M) 1  T o r i ( A , Q ) 9 AOM  E x t (A>N) 1  ..  (g)  a l l o f t h i s i s expressed  i n f o l l o w i n g commutative diagram o f exact sequences  0 V Tor3(A,Q) - ^ T o r 2 ( A , M ) ^ T o r 2 ( A , N ) - ^ T o r 2 0  i  Tor* (A?Q) *  >A*8M  >A*8N t  i  > A*8Q  0  G  >0  I  [  w 0 w  >Hom(A,M)^Hom(A,N)^Hom(A,Q)-^Ext U,M)^Ext1U,N)-^Ext1(A,Q) 1  -» T o r ! (A,M)->Tor (A,N)->Tor-| (A,Q) 1  y  0  0  0  • ASM •  ->A8N-  V  -*A8Q  ->0  *Hom(AtM)—»Hom(A*N)^Hom(A*Q)-*Ext (AtM)  >  -^Ext (A,M)^Ext (A,N)->Ext2(A,Q)->Ext3(A,M)  >  1  2  2  v 0  o  o  (5)  Remarks;  ( a ) i s p a r t i a l l y used by everyone,  m o s t l y the A t — ? A correspondence,, t h a t  but is  Part  generally neglected.  (1):  M. A u s l a n d e r :  Part  ( c ) can be found i n  Coherent F u n c t o r s ,  i s d i f f e r e n t , more c a t e g o r i c a l . theorem w i l l difficulty  (b) A 4 - ) i  but the .approach  The p r o o f  o f the  be d i v i d e d i n t o s e c t i o n s ; the o n l y  great  i s part ( f ) .  Proof:  ( i ) Let P'-^-P—-»A—>0,  Dualize  t o o b t a i n K and i , such t h a t 0  Dualize  again,  0 *  P,P* f . g . and p r o j e c t i v e A*  P*-* P'*-* k  p «**.#. P**, but P**=* P and ?•**= p»  so coker o f t h i s sequence i s l a g a i n A. (a) and shows A =  so a l s o g i v e s  This  establishes  (b) by a l e f t - r i g h t  switch.(ii)  F o r any r i g h t module N and l e f t  module M  consider  N8M + Hom(N?M) where (n8m)f = f(n)m, feN*; t h i s i s an isomorphism i f N i s f . g . and p r o j e c t i v e ( s i n c e i t i s t r u e f o r the r i n g R). 0 -» K  P**-» £  sequence  0 to o b t a i n  P'0M  ?>P0M  Hom(P'?M) 0  U s i n g t h e exact  >A®M-*0  -  (#1)  Hom(K,M) + Hom(P,M) > Hom(A*M) * Ext1(K,M) * 0 Ext1(A,M)  »0  E x t ( K , M ) = Ext (X.,M) amd e v e r y t h i n g 1  i s natural with  2  respect  t o M; so a p p l y  the snake lemma (w the c o n n e c t i n g  homomorphism i s a c t u a l l y an isomorphism) t o o b t a i n 0 •? E x t (A,-) 1  -> A0- ^ H o m ( A t - ) -> E x t ( A , - ) + 0 2  (6)  (iii)  Now  c o n s i d e r the map  y  N*0Q -> Hom(N.Q) g i v e n by  ( f 8 q ) n = f ( n ) q , which i s an isomorphism i f N i s f . g . and p r o j e c t i v e ( a g a i n because i t i s t r u e f o r R). (Remark:  The maps i n ( i i ) and  the f o l l o w i n g commutative  ( i i i ) a r e connected by  diagrams:  N*9Q_  M8Q—>M**8Q  Hom(N,Q)«— Hom(N**,Q)  Hom(M*Q)  induced by N -> N** and M  M**;  f o r f.g. projectives,  when P i s i d e n t i f i e d w i t h P * * t h e are the same. )  maps o f ( i ) and  Using 0 - v L - » P - > A - > O t o 0  0 -> Tor-j (L,Q)  *A*®Q  (iii)  obtain  >Tor-,(A,Q)  *P'8Q  >L®Q—>0  ~  P0Q  0—>Hom(X,Q) —yHom(P'vQ) ->Hom(P*Q) A g a i n the c o n n e c t i n g homomorphism i s an isomorphism, Tor-|(L,Q) = Tor2(A,Q) and w i t h n a t u r a l i t y the snake lemma  >Hom(i,-) ->Tor-|(A,-)  g i v e s 0 -> Tor2(A,-) -> thus ( c ) has been (iv)  Now  0  established.  suppose 0 ^ M  N  Q  0, then  Hom(A\Q)-*Ext1(A\M) T o r i (A,Q)  >A8M  h o r i z o n t a l maps from homology, (#1) and (#2).  v e r t i c a l maps from diagrams  Commutivity must be v e r i f i e d .  To compute  Hom(i,Q) - » E x t ( £ , M ) , use the p r e s e n t a t i o n 0-*K—>P ->& 1  o f A*.  The r e q u i r e d map  ,f  0  i s the c o n n e c t i n g homomorphism o f :  (7)  (Hom(A\Q))  I  (#3)  Hom(P'tM)—^Hom(P'tN)—>Hora(P'tQ) -> 0  I  I  I  Hom(K,M)-^-i Hom(K,N)—»Hom(K,Q)  0—  :  I .(Ext1(Jt,M)) S i m i l i a r l y Tor-|(A,Q)  A8M i s the c o n n e c t i n g  homomorphism  of the f o l l o w i n g diagram, u s i n g the p r e s e n t a t i o n 0 * L > P - > A - » 0  0  (Tor-| (A,Q)) + L8M —>L8N —> L0Q —> 0 J . 1 1 P8M P8N P8Q  (#4)  (A8M) There  i s a n a t u r a l map from  (#3) t o (#4) which  induces  maps i n t o the k e r n e l s o f (#4) out o f the c o k e r n e l s (#3)» c o n n e c t i n g the two c o r r e s p o n d i n g k e r - c o k e r naturally.  Explicitly:  sequences  Hom(P'*-) —> P'8  » L8-  Hom(K,-) ->Hom(P*-)—>P8f i l l ' . i n M, N, Q f o r  So we have w Hom(A,Q)—>Ext'(A,M) \ w ^ Tor-} (A,Q) >A6M A  The  first  v e r t i c a l map induced  into  commuting  the k e r n e l T o r i  of the k e r c o k e r sequence o f (#4) and the second map  (A,Q)  vertical  induced out o f the c o k e r n a l Ext1(A\M) o f the k e r - c o k e r  sequence o f (#3).  Finally  same as those a r i s i n g from  to show these maps a r e the (#1) and ( # 2 ) i  L e t X be e i t h e r  (8)  M, N o r Q, the maps c o n n e c t i n g  the two  ker-coker  sequences are then the unique maps making the f o l l o w i n g diagram commute: Hom(i,X)—>Hom(P'TX)—^Hom(K,X)—*Ext1(i\X)  :  i  Y  Tor-j(A,X)-  (#5)  P'0X  Hom(PtX)  I  I ->P0Xs  ->L0X -  -> A0X  Examine diagram (#1) where Ext1(A,M) > A0M  i s defined  v i a w , and the "snaking"shows i t i s r e q u i r e d map (#5) commute a t the r i g h t end f o r X = M. f o r (#2) w i t h X = Q l o o k i n g a t the l e f t (v)  Also:  V2  The same a p p l i e s end of (#5).  Hom(&,Q)-^Ext1(A,M) Ton(A,Q)  { Y-j  making  >A8M  the map  induced  out o f the c o k e r n e l Tor-|(A,Q) and  the map  induced  i n t o the k e r n e l Ext1(A,M).  i = i _^, and e p i c s can be c a n c e l l e d so  But  A -—hence  v^ = V 2 .  This gives  (d) u s i n g  ker-coker  lemma. (vi)  Tori(A,Q) A"  U  Hom(A,N)—>Hom(A,Q)  » A0M—»A0N  Hxt1(i,M) Im u =;,fcIm(Tori (A,Q)  A0M)  because Hom(A\Q)  T o r i (A,Q)  = ker(A0M * A0N). k e r u = ker(Hom(i,Q) * Ext 1  (A*, M )  = Im(Hom(A\N) * H o m ( A \ Q ) ) .  because Ext1  (A\M)>—¥A0M  (9)  This and  establishes  (e) and ( i v ) , ( v ) and ( v i ) g i v e ( f ) .  (g) comes from (a) t o ( f ) and t h e snake lemma a g a i n .  Corollary  2.2  L e t A he as i n the theorem,  0 •> Ext1(A*,R) -> A  A** + Ext2(A\R)  0 ->» Ext1(A,R) * X + 1** + E x t ( A , R ) 2  Corollary are  2.5  ((13) page 81)  then  0 ((11) page 142.} 0//  A l l f.p. l e f t  modules  t o r s i o n l e s s i f and o n l y i f a l l f . p . r i g h t modules a r e  W-modules ( B i s W-module i f E x t ( B , R ) = 0 ) . / / 1  Corollary  2.4  I n the s i t u a t i o n o f t h e theorem, where  L and K a r e d e f i n e d 0 -> 1* •» P  f  so t h a t :  * P -» A -» 0 and 0 -v A* -v P* L  1  0 * 1 + 1 *  and (b) P< -v K* -y P  0  + 0  Ext1(£,R) -> 0 4 Ext1(A,R) * 0 (b')P* -v L* * P«* commute.  L Proof  t  K  then ( a ) 0 -v L * K* (a )  P** -»  K  A* •> P' -> L -V 0  H  I | 0 ^ A M  h \  P'** -» K* + E x t ( A , R ) -V 0 1  h i s induced out o f t h e c o k e r n e l and t h e snake lemma gives  (a).  The l e f t  h a l f side  d e f i n i t i o n o f L. f o r r i g h t h a l f P  f  K*—>P  o f ( b ) commutes by side  P —»K*—>P 1  P'—>P  L P*—#L can he c a n c e l l e d , (a')  P*  P L  so t h e r i g h t h a l f commutes a l s o .  and ( V ) by l e f t - r i g h t  symmetry.  (10)  Corollary  2.5 £(13) page 71)  I f A i s a torsionless  f . p . r i g h t module t h e r e e x i s t s K a f i n i t e l y left  submodule o f a f r e e module such  0 •) A M 0 -> A  P M A**  K ^ 0 ,  generated  that:  0->K**P-*A^0  Ext1 (K,R) -v 0. ,  0 -> K .-•» K**  Ext1 (A,R) -v 0  C o n v e r s e l y such K g i v e r i s e t o A t o r s i o n l e s s and f . p . P r o o f Apply C o r o l l a r y Prom Cor. 2.2 a l s o * and 0 -> K  0  2.2 and Cor. 2.4 t o g e t L = K* A •» A** -» E x t ( A , R ) = E x t 1 ( K , R ) * 0 2  L* = K»* -» Exf1(A,R) (from Cor. 2.\).//  Remarks (a) A  r  Bi  This  establishes  a correspondence  between  the c l a s s o f f . p . t o r s i o n l e s s r i g h t modules and the c l a s s .of f . g . modules i s o m o r p h i c t o eubmodules  of f r e e l e f t modules. and a l s o  This  coherent B i <= A l ,  duals of f.p. r i g h t s are f . p . l e f t  Prop. 4.1 ). 0 -» A  Now i f R i s l e f t  A**  So f o r AeAr, l e t P - ^ A *  t  (see ahead  Pf.g. free,  P* embeds A i n a f r e e module so A  i s the a p p r o p r i a t e g e n e r a l i z a t i o n  r  then ^ Br.  of : R l e f t  n o e t h e r i a n , a f . g . t o r s i o n l e s s r i g h t module can be embedded i n a f . g . f r e e module  ( ( 3 ) 4.5). F o r R l e f t  and r i g h t coherent A r = B , A l = B i . r  (b)  The r e s t r i c t i o n t h a t A he f . p . i s n o t n e c e s s a r y  here i n the f o l l o w i n g Proposition that  2.6  then ( i ) 0  sense:  L e t A he f . g . , d e f i n e  0->L->P-»A->0  ,  K and L such  , 0->A*-»P*-»K->0  K -» L* -> E x t ( A , R ) -> 0 1  ( P f . g. p r o j . )  (11)  ( i i ) 0 * L ^ K* + Ker(A -» A**) + 0 ( i i i ) K i s f . g . and t o r s i o n l e s s . P r o o f F o r ( i i i ) K i s a submodule o f L* which i s t o r s i o n l e s s and  P*-»K. 0-vL->P-*A->-0 h/ n 0 -»K* P**-*A**  i  h i s induced i n t o the k e r n e l K*, snake lemma then g i v e s ( i i ) . 0 -v A* -V P M  II  0  A*  II  P*  k i s induced out o f c o k e r n e l Corollary  2.7  K + 0 | k , L* Ext1(A,R) -> 0 snake lemma g i v e s  I f A i s f . g . and t o r s i o n l e s s t h e r e  a f . g . t o r s i o n l e s s K, such t h a t : 0  A -y A** * Ext1(K,R) -> 0  Proof  I f A i s torsionless,  then ( i ) g i v e s 0 Then:  K  is  ((13) page 71)  0  K -> K** -> Ext!(A,R)  0  then L = K* ( ( i i ) o f Prop.)  K** •» Ext1(A,R) * 0  0-*L-»P-*A-»0 0 * K»-?-P**->A**  gives 0  (i).//  A  A** •+ Ext1(K,R)  Thus we have a correspondence r i g h t and f . g . t o r s i o n l e s s  Ext1(K,R)  0  0 by snake lemma a g a i n . / / between f . g . t o r s i o n l e s s  lefts.  F o l l o w i n g two r e s u l t s  extend r e s u l t s o f J a n s , ( ( 1 3 ) page 7 3 ) . Proposition  2.8  A f . p . r i g h t , A* = 0, then:  ( i ) A®- =?Ext1(A\-)  ( i i ) Hom(A,-) = T o r i (-,£)  i n p a r t i c u l a r A = Ext1(£,R) and i f A / 0 then p.d.A* = 1 ( p r o j e c t i v e  dimension).  (12)  P r o o f a p p l y ( c ) o f Thm. 2.1 t o g e t ( i ) , ( i i ) a n d E x t ( A % - ) = 0, hut i f A i s p r o j e c t i v e 2  then A = Ext1(£,R) = 0.  Hence p.d.1 = 1.// Proposition  2.9 A f . p . r i g h t , p.d.A = 1 then:  A = Ext1(A,R) and E x f 1 ( A , R ) *  =0.  P r o o f I f 0 •> P'- -» P •> A -> 0 i s exact, then 1* = 0. Hence by l a s t Proposition  p r o p o s i t i o n A" = E x t ( A , R ) . / / 1  2.10 F o l l o w i n g  are equivalent:  ( i ) R i s Regular. (ii)  A l l f.p. l e f t  modules a r e p r o j e c t i v e .  ( i i i ) A l l f . p . r i g h t modules a r e p r o j e c t i v e . In f a c t A i s p r o j e c t i v e i f and o n l y Proof  C l e a r l y ( i ) ^ ( i i ) and ( i i i ) .  i f A* i s p r o j e c t i v e . ( f . p . f l a t s are  projective) ( i i ) 4 (i) I f L i s f.g. l e f t  i d e a l then 0  L -> R •> R/L -> 0  s p l i t s by ( i i ) , hence L i s a d i r e c t summand. ( i i ) <V» ( i i i ) by ( c ) o f Thm. 2.1 A 0 - ^ H o m ( A f - ) Ext^A*,-) = 0, hence r e s u l t f o l l o w s For  implies  by (b) o f Thm. 2.1.//  the moment we jump a dimension.  Proposition  2.11  of f . p . l e f t Proof  right] £ 2 ^  duals  are projective.  0 -» X*  C o r o l l a r y 2.12 left  Sup [p.d.A: A f f . p .  -> P'  P -v A •> 0 g i v e s  the r e s u l t . / /  ( ( 3 ) 5^2) I f R i s r i g h t and l e f t  g l o b a l dimension R ^ 2  noetherian,  duals of f . g . r i g h t i s  projective. Proof  f.p.coincides  w i t h f . g . and g l o b a l dimension can  be c a l c u l a t e d  over  f.g.//  More g e n e r a l l y : P r o p o s i t i o n 2.13 ( i ) Sup ( i i ) Dual  ( ( H ) 3.1) F o l l o w i n g a r e e q u i v a l e n t :  Jp.d.A; A r i g h t f.p.] = n+2 o f any f.p. l e f t  has p . d . ^ n . / /  P r o p o s i t i o n 2.14 L e t P* •> P -» A and Q' * Q finite and  p r e s e n t a t i o n s o f A.  C be coker Q * -H-Q>**.  Proof  A be two  L e t B (=£) be coker P*—• p'*,  Then E x t ( C , - ) n  = Extn(B,-).  F o r n = 1,2, f o l l o w s because they a r e k e r and  coker o f A®- -> Hom(A*,-) by ( c ) o f Thm. 2.1. F o r n > 2, 0  A* -> P*  P'* -> B * 0 g i v e s E x t n ( B , - ) = E x t n - 2 ( A * , - ) .  P r o p o s i t i o n 2..15 F o l l o w i n g a r e e q u i v a l e n t : (i) R i s l e f t ( i i ) Sup  semihereditary.  [p.d.B; B f . p . l e f t ] £ 1.  ( i i i ) any f.p. r i g h t A, i s o f the form A = A'$P where P i s f.g. proj., Proof  A  f  f . p . and A'* = 0.  ( i ) & ( i i ) i s standard.  ( i i ) => ( i i i ) f o r A f . p . r i g h t , we have as b e f o r e 0 - y A M P* -»JP'* -> B  0. ' p.d.B $ 1 $ K i s p r o j e c t i v e  =7* A* i s p r o j e c t i v e so 0 4 A M  P M  remains ( s p l i t ) exact when d u a l e d , Then  0 •> I  K •> 0 s p l i t s ,  so  ( A l s o A* i s a l s o f . g .  P -yA 4 0  0 -» KMP**->A**->0 Snake lemma g i v e s A—*»A**, but A** i s p r o j e c t i v e . A sr A**$ k e r ( A +A**) = A**© Ext1(A,R) , and  Ext1(A,R)*  = 0 by Prop. 2.9.  Hence:  (Cor. 2.2)  (  (iii)  H  )  ^ ( i i ) Given B f.p. l e f t ,  A = A'©P where A » * = 0.  L e t Q'  Q*, Q» f.g. p r o j . then Q«  0 -» P* -> Q*$P*  Q  A = B.  Then  A* -> 0 be exact  * Q0P-* A*©P s A  t o compute k,  this presentation  construct  0 use  s i n c e A'* = 0,  Q»* -y % •» 0 t h i s g i v e s  0 * Q* -y Q»*-*£ + 0  (the P* j u s t r i d e s a l o n g , note & = A * ) , so p.d.A* £ 1 but  by prop.2414, p.d.B = p.d.A* <i 1.//  C o r o l l a r y 2.16 (i)  semihereditary.  A** = P i s a f . g .  (ii)  A f.p.  (iv)  projective.  A ® - = P8- 9 E x t * l ( ^ , - ) . For l e f t  functorial, Proof  f.p.B, the assignment B v-*-B can he made  by s e t t i n g B = E x t 1 ( B , R ) .  F o r ( i ) , ( i i ) and ( i i i ) use ( c ) o f Thm. 2.1,  the f a c t t h a t  last  proposition  i f A = A'©P then 1 =  , and p r o o f o f  t h a t A** i s f . g . p r o j e c t i v e .  choose f o r each B. an exact sequence 0 •> Q' Q»,  2.17  Q -» B -» 0,  A f . p . , A i s p r o j . 4* Ext1 (A,L)  a l l f.g. l e f t  Proof  Using 0  so A i s a l s o Proposition  = 0  i d e a l s L. L  Tor-| (A\R/L) = 0.  are  For (iv)  Q f . g . p r o j . , then B = E x t 1 ( B , R ) . / /  Proposition for  right  Hom(A,-) = Hom(P,-) © Tor-|(-,&).  (iii)  and  R left  R  R/L  0 i n ( e ) o f Thm. 2.1  Thus 1 i s f l a t ,  gives  hence p r o j e c t i v e , and  projective.// 2.18  equivalent:  R left  coherent, A f . p . l e f t .  Following  ( i ) A i s projective. (ii) (iii)  Ext1(A,B) = 0 f o r a l l f . p . B. Ext1(A,B) = 0  f o r a l l c y c l i c f . p . B.  (15)  Proof  (i)  ( i i ) L e f t coherence  implies f.g. l e f t  ideals  are f . p . , a p p l y l a s t p r o p o s i t i o n . (ii)  ( i i ) Induct on minimal  number of g e n e r a t o r s .  F o r B f . p . l e t x be an element of a minimal set.  Use  0 -» Rx -y B -VB/Rx  0 and  generating  induction i n long  exact Ext sequence. / / C o r o l l a r y 2.19 left. (i)  ((15) Prop.2)  R left  coherent, A f . p .  F o l l o w i n g are e q u i v a l e n t :  p.d. A ± n  ( i i ) T o r i ( C , A ) = 0. f o r a l l f . p . C n +  ( i i - t ) T o r i ( C , A ) = 0, f o r a l l f . p . c y c l i c  C  n +  ( i i i ) Extn+1(A,B) = 0, f o r a l l f . p . B ( i i i ) Ext*+1(A,B) = 0, f o r a l l f . p . c y c l i c 1  Proof  (i)  (ii)  ( i i ' ) gives f l a t  A i s f . p . and R i s l e f t  coherent,  B.  dimension  so f l a t  A - n,  dimension  but  = proj.  dimension. ( i ) 4$ ( i i i ) dimension  shifting  P r o p o s i t i o n 2.20 Sup  (R i s l e f t  p r o p o s i t i o n and  coherent).  ((13) page 74)  £p.d.B ; B f . p . l e f t ,  A* = o i m p l i e s A = Proof  ( i i i ' ) By the l a s t  R left  // coherent,  p.d. B <°°j = 0  then  f or A f . p . r i g h t ,  0.  (=>) by P r o p o s i t i o n 2.8 (4=) by P r o p o s i t i o n 2.9-  F o r i f B has p.d.  then |» = 0 ^ t = 0 ^ B i s p r o j e c t i v e .  //  - 1  (16)  P r o p o s i t i o n 2.21  ( ( 3 ) 5.3)  Sup £p.d. B : B f . p . l e f t ,  R left  coherent,  p.d. B < » j ^ 1  then  f or A f . p  t o r s i o n l e s s , A* p r o j . i m p l i e s A p r o j . Proof  (=^) Suppose A* i s p r o j . , then p.d. JUCD, hence  p.d. A - 1, hence K (as i n Cor. 2.4) i s p r o j .  Then by  Cor. 2.5, A = A** i s p r o j .  (4=) I f t h e r e i s a f . p . modul  with f i n i t e  p r o j . dimension  g r e a t e r than 2, t h e r e i s a  torsionless  f . p . A w i t h p.d. e q u a l T , ( l e f t  coherence).  By Cor. 2.4 o -» A* ^ P' •» P ^> A ^ 0 so K* i s p r o j . , K* by assumption K i s p r o j e c t i v e . and  0  A*  P*  K  Then hy Cor. 2.5 A=  A**  0, so A* i s p r o j . , hence A i s p r o j  (17)  3/  Absolute P u r i t y  (f.p. I n a c t i v i t y )  I n the c a t e g o r y £ o f s h o r t e x a c t s e q u e n c e s , c o n s i d e r the s e t of f i n i t e  presentations % ; that i s short exact  sequences G : 0 -> G" -> G -vG*  0, G f . g . pro;). G" f . g . .  A sequence A i s pure i f f o r any G — v A over a s p l i t so t h a t i n  s h o r t exact sequence, , f = 0;  , Ge^then f f a c t o r s  (f  0, see ahead Prop. 6.4,  (G,A) = 0 f o r a l l Ger»- L e t  (R be the c l a s s o f pure sequences.  Then H i s copure i f f o r  f  any H-=> A , Ae(P = f 0.  A module A ( r e s p e c t i v e l y C) i s  a b s o l u t e l y pure ( f l a t ) i f whenever  E : 0 4 A - > B ^ C ^ 0  i s e x a c t , t h e n E i s pure.  M  If C o n s i d e r t h e s i t u a t i o n E : 0 - * A - * B - } C - > 0 , where M i s f . p . and E pure, embed t h i s i n t h e diagram G:0->K-»-P-»M->-0 P f . g . p r o j . , f induced by 1  ft E  if" if  it  A ->B 4 C ^ 0  : 0  p r o j e c t i v i t y o f P.  S i n c e G i s copure and E i s pure, f ~ 0 so by-Prop. 6 . 4 fills  B—*C  i n , t h a t i s Hom(M,B) — » Hom(M,C), f o r a l l f . p . M.  P r o p o s i t i o n 3.1  E : 0-»A>B-»C-»0is  i f Hom(M,B) —>Hom(M,C) -> 0 f o r a l l Proof  ,M  pure i f and o n l y  f . p . M.  (^) by above. (<£) by d e f i n i t i o n o f p u r i t y and Prop. 6 . 4 . / /  C o r o l l a r y 3-2  A i s a b s o l u t e l y pure (C i s f l a t ) i s and o n l y  i f whenever 0 - > A - » B - y C > 0 i s  exact.,  Hom(M,B) ->Hom(M,C) > 0 i s a l s o e x a c t , f o r a l l f.p. M. / / Lemma 3-3  A % B $ C % » e x a c t , a e p i c ^ b = 0 # c monic. / /  (18)  F o l l o w i n g p r o p o s i t i o n shows f l a t n e s s and p u r i t y as d e f i n e d on p r e v i o u s page, c o i n c i d e s w i t h the standard concepts. P r o p o s i t i o n 3.4 (i)  E : 0-»A-»B-»C>0  E pure & E0M i s exact f o r any f . p . M (hence, (E®M  (ii)  any M),  = 0 -> ASM > B®M -> C0M -v 0)  C flat<#>E0M exact f o r a l l such E i n v o l v i n g C & M f . p . ty T o r i  (iii)  (C,M) = 0 f o r any f . p . M (hence  any M)  A a b s o l u t e l y pure 4^ E®M exact f o r any such E  i n v o l v i n g A and a l l f . p . M. # E x t 1 ( M , A ) = 0 f o r a l l f . p . M. Proof ( i ) Apply  p a r t (d) o f Thm. 2.1, Lemma 3-3 and Prop. 3.1.  Hom(fi,B) % Hom(M,C) 4 A®M £ B8M (ii)  First  e q u i v a l e n c e from  ( i ) above.  e q u i v a l e n c e , take B p r o j e c t i v e 0  Tor-|(C,M) -VA9M  F o r second  then:  B8M ^ C0M * 0  So i f A ^ B ^ C i s pure (C f l a t by d e f i n i t i o n on p r e v i o u s page)  then ( i ) i m p l i e s Torl(C,M)  Conversely  = 0.  Hom(M,B) ^Hom(M,C) ^ E x t 1 ( f t , A ) Tori(M,C) -> M0A  Tor-|(M,C) = 0 ^ b = 0 ^ a i s (iii)  First  onto f o r a l l M, a p p l y Cor. 3.2.  equivalence*by ( i ) .  Hom(M,B) ^ Hom(M,C) -*» Exf1(M,A)  Exf1(M,B)  I f A i s a b s o l u t e l y pure a i s e p i c , take B i n f e c t i v e ,  (19)  A > » B.  C o n v e r s e l y Ext1(M,A) = 0, i m p l i e s a e p i c f o r any  M, hence A a b s o l u t e l y pure by Cor. 3.2.// A i s called  copure i n f e c t i v e  i f given G :  G : 0^G"•>G->G 0 copure f\ /f A then f has an e x t e n s i o n f . ,  v  P r o p o s i t i o n 3.5 Proof  A copure i n j . 4$ A i s a b s o l u t e l y  pure.  (=^) G : 0-s> G "-» G G ' — > 0 , G any copure. f+ f " t f| f'| "E : 0 —>A — * B —>C — > 0  s i n c e A i s copure i n j . f " extends t o G, by Prop.6.4,  f~0,  hence E i s pure, and A i s a b s o l u t e l y pure, «=) g i v e n  G:0->G -»G->G -»0 f«• l A H  ,  C o n s i d e r a sequence E : 0 - » A - » B e > C * 0 then G : 0 -> G E  with B i n j .  - > G ^ ^ 0  n  : 0 -» A -> B  f " induces f : G  C  0  E , s i n c e B i s i n f e c t i v e ; E i s pure  s i n c e A i s a b s o l u t e l y pure, hence f^O, and t h e r e q u i r e d extension o f f " e x i s t s . // Remark called  The c o n d i t i o n Ext1(M,A) = 0 f o r a l l f . p . M i s f.p. i n j e c t i v i t y .  Thus a b s o l u t e pure = f p. i n f e c t i v e = copure C o r o l l a r y 3.6  infective.  ((17) Cor.2 ,page 562)  A i s a b s o l u t e l y pure  g i v e n N>—»P  Iff A  w i t h N f . g . and P p r o j e c t i v e , Proof  f has an e x t e n s i o n t o P.  (^) embed P £ F, F f r e e ,  then N - F* = F f o r some  (20)  f.g.  d i r e c t summand o f F.  is a finite  Then 0 ~i> N - » F  p r e s e n t a t i o n hence i s copure.  an e x t e n s i o n t o F' by Prop. 5.5, extended  to P.  (note 0-v N  f thus has  t h i s can f u r t h e r by  t o F s i n c e F' i s a d i r e c t  restricted  F*/N -*> 0  T  summand and then  P -> P/N * 0 i s c o p u r e ) .  G : 0 —*-G"—> G —>G' —> 0  IT E  I  r  n  : 0 — » A —>?B -» C —*> 0  G a finite  p r e s e n t a t i o n , by h y p o t h e s i s f " extends t o G,  hence f ~ 0 , s o E i s pure by d e f i n i t i o n , E i s a r b i t r a r y so A i s a b s o l u t e l y  pure.  C o r o l l a r y 5.7 ((16) Thm .2) Suppose ( M i ) i s a d i r e c t e d Mj>-»limMj  =^ M a b s o l u t e l y  Proof  such t h a t  ( f o r i n s t a n c e suhmodules .of some f i x e d  ordered by i n c l u s i o n ) 1  system,  M,  Then M^ a b s o l u t e l y pure f o r  pure.  G : 0 -vG" ->G ->G' -» 0, G a f i n i t e  presentation  limMi s i n c e G" i s f . g . and Mi>Vlim  Mj.,0—^G'^-j-^G  f f a c t o r s over some Mi ,  limMj<—yMj  then f extends t o G, hence f extends to G.// P r o p o s i t i o n 5.8 equivalent  F o r any f a m i l y  ( A i ) , following are  ( i ) A i a b s o l u t e pure a l l i . ( i i ) TTAi ab. pure, ( i i i ) ® A i ab. pure.  Proof  alli  ( i ) ^ ( i i ) Ext1(M,TfAi) S l E x t l ( M , A i ) .  (21)  (i)  ( i i i ) by Cor. 3.7  ( i i i ) =^ ( i ) d i r e c t  summands o f ab. pure a r e a b s o l u t e l y  p u r e . E x t ^ M j A e A ) = Ext1 (M,A)$Ext1 ( M , A « ) . 1  //  P r o p o s i t i o n 3.9 ((17) Thm;2.) R i s semi-hereditary a b s o l u t e l y pure Proof  ^> The homomorphic  image o f an  module i s a b s o l u t e l y  pure.  L e t A f . p . l e f t and 0 - * L - * M - > N - * 0 exact,  Ext1(A,M) ^ E x t 1 ( A , N ) 4 E x t 2 ( A , L ) - » E x t 2 ( A , M ) .  then  (=>) l e t M  be f . p . i n j . , i f R. i s s e m i - h e r e d i t a r y ,  p.d. A £ 1. So  above sequence i m p l i e s Ext1(A,N) = 0 .  A i s a r b i t r a r y so  N i s f.p. i n f e c t i v e .  (<?) F o r any L l e t M be an i n j .  module c o n t a i n i n g L.  Then N i s f . p . i n j . , hence the  sequence i m p l i e s E x t 2 ( A , L ) = 0 so p.d. A < 1 . / /  (22)  4/  Coherence and f . p . I n j e c t i v i t y I n t h i s s e c t i o n , imposing t h e c o n d i t i o n s  o f coherence  and f . p . i n j e c t i v i t y on t h e r i n g R, a r e i n v e s t i g a t e d . ring i s left  A  coherent i f f . g . submodules o f f . p . l e f t s  are f . p ; P r o p o s i t i o n 4.1  R i s r i g h t coherent 44 d u a l s o f f . p . l e f t  are f . g . ( i n which case they a r e a l s o f . p . ) Proof  (=*•) A l e f t  f.p.  0-v  A*-vP*->P'*->l->0 L  L i s f . g. r i g h t , hence f . p . =^ A* i s f . g . ( and coherence g i v e s A* i s f . p . )  (<=} Suppose K » P > P f . g . p r o j e c t i v e ,  K f . g . form 0 -V  P'—> P-> A-* 0  \ f K by h y p o t h e s i s A> i s f . g . hence K i s f . p . I f now K £ M, M f . p . and K f . g . , form  0  I  0 —>L  0  I  0  I  M  '  > L — > L' —> 0  i  l  l  1  - I  I  i  0—>P"—j-P'eP"—*P'-*0 0 —>K  >M  0  0  P , P" f . g . p r o j . f  I  >M/K—> 0  I  0  by what has j u s t been proved L, L* a r e f . p . hence L " i s f.g.,  thus K i s f . p . / /  P r o p o s i t i o n 4.2 equivalent  I f R i s r i g h t coherent t h e n f o l l o w i n g e r e  ( i ) A i s absolutely (ii)  (iii)  pure.  Ext1(M,A) = 0 f o r a l l c y c l i c f . p . M. Iv-*R  V  can be f i l l e d  i n for I f.g. left  ideal.  (23)  Proof  ( i ) (*£) ( i i ) Induct on number o f g e n e r a t o r s and  Ext1(M/xR,A) -» Ext1(M,A) -> Ext1(xR,A) of  an a r b i t r a r y f . p . M.  use  t a k i n g out a g e n e r a t o r  ( w i t h r i g h t coherence  "shifting"  on the f . p . ' s i s a l l o w e d ) , (ii)  («*) ( i i i ) from Hom(R,A) -yHom(I,A) -» Exf1 ( R / l , A ) -> 0.  Theorem  4.3  M i s f l a t Q e : A*®M-^Hom(A,M)  (a)  f o r a l l f . p . A.  A  If  9  i s e p i c f o r a l l A, i t i s n e c e s s a r i l y an  A  //  (b)  M i s a b s o l u t e l y pure 4A 3?  f.p.  A.  A  If R i s l e f t  c o h e r e n t , then i  h  isomorphism.  : A®M>-> Hom(A*,M) f o r a l l  monic f o r a l l A  implies  •3? i s a isomorphism. A  Proof  (a) A*8M -y Hom(A.M) -»Tori(i,M)  and k e r e  A  2.1  = Tor2(i\M).  (b) 0 ^ E x t 1 ( A \ M ) from Thm.  0 from Thm.  ASM  2,1, g i v e s the r e s u l t .  ^Hom(A*,M)  Ext (£,M) 2  0,  Ext (B,M) = 0 implies 1  Ext (B,M) = 0 f o r a l l B f.p., only i f s h i f t i n g i s p o s s i b l e , 2  so  left  coherence i s needed  P r o p o s i t i o n 4.4 (18)). lim  f o r an isomorphism.  //  ( S t a t e d w i t h o u t p r o o f by Stenstrbm, page  I f A i s f . p . t h e n f o r any d i r e c t system (Mi*)  Hom(A,Mi) ° Hom(A,lim M i ) .  lirn^ Hom(A,Mj) —>»Hom(A,lim  Conversely i f  M-s ) f o r any d i r e c t e d  system then  A i s f.p. Proof the  (Remark  unique map  The map  l i m Hom(A,Mi)—»Hom(A,lim  out o f the d i r e c t l i m i t ,  Mi) i s  induced by the  323  (24)  compatible maps Hom(A,Mi) -> Hom(A,lim from M i — > l i m Mi.) P, P  L e t P' ^ P  M i ) which  >  A  arise  0 be exact w i t h  f g - proj.  f  0 — > l i m Hom(A.Mi)—• l i m H o m ( P , M i ) — » l i m Hom(P ,Mi) ,  O  ^  <  ^  *  >  ^  P  &  i m p l i e s h i s an isomorphism. f o r some d i r e c t e d  ^  W  %  *  >  Conversely, A = l i m A j ,  system (k±) o f f . p . modules.  I f l i m Hom(A,Aj) —fr>Hom(A,lim A i ) = Hom(A.A) A i then t h e i d e n t i t y f a c t o r s o v e r some A^ , A i s a direct  summand  A = A, hence  o f a f . p . module A i and i s f . p . / /  Following i s a result  o f Watts ( 2 1 ) .  Let  A c o g e n e r a t o r , 0 -»* A—»-EHom(A,E) n  E be an i n f e c t i v e  0—>B—>EHom(B,E) E(f)  (e ) eHom(A,E) = g  g  (Sgt)geHom(B,E)  n (a) =  (g(a))  A  E i s a f u n c t o r and n i d e n t i t y t o E. in a directed Theorem  A  g e H o m  (  A f E  )  a n a t u r a l t r a n s f o r m a t i o n from the  Hence any d i r e c t  system o f i n f e c t i v e  system c a n be embedded modules.  4.5 ((18) Thm. 3.2)  Following are equivalent (i) . R right  coherent.  ( i i ) The d i r e c t l i m i t o f a b s o l u t e l y pure modules i s absolutely  pure.  ( i i i ) l i m E x f 1 ( A . M i ) _ = ^ E x t 1 ( A , l i j i M i ) f o r every f . p . A and d i r e c t  system ( M i ) .  (25)  Proof  (i)  ( i i ) For A f.p. l e f t ,  Prop. 4.1.  A* i s f . p . r i g h t t y  ( l i m Mi)®A—>Hom(A*,lim — ^  0 —^lim  | =  (Mi0A)—*lim  Horn(A*,Mi)  h i s an isomorphism by Prop.4.4; s i n c e each Mi i s a b s o l u t e l y  bottom row i s i n f e c t i v e  pure, hence  and by Thm. 4.3, l i m Mi i s a b s o l u t e l y L e t A be f . p . and (Mi) a d i r e c t direct  Mi)  h |  t o p row i s i n f e c t i v e  pure.  system.  (ii)£(iii)  Embed  (Mi) i n a  system o f i n f e c t i v e modules ( E i ) (see note b e f o r e  theorem).  L e t N i denote l i m N i -  0 •> Hom(A,Mi) >Hom(A,Ei) * Hom(A,Ei/Mi) -*Ext1(A,Mi) >  | S  * ir=  >  |S  >  |  *0  e  0 * Hom(A,Mi) ^ Hom(AiV-Ei) •> Hom(A,Ei/Mi) ^ E x t 1 ( A , M i ) ->Ext1(A,Ei -> -> > -> -> By h y p o t h e s i s l i m E i i s a b s o l u t e l y  pure, hence l a s t  term o f  the bottom row i s zero, i m p l y i n g Q i s an isomorphism, (iii)  ^ (i)  L e t K be a f . g . submodule o f a f . g . p r o j e c t i v e  module P, i t s u f f i c e s  to prove K i s f . p . (by p r o o f o f Prop.4.1).  I f O - > K - * P - * A - * 0 , and (Mi) any d i r e c t 0 -» Hom(A,Mi) -y Hom(P,Mi) •> Hom(K,Mi) |^ » L~ yg 0 Hom(A,Mi) * Hom(P,Mi) -» Horn(K,Mi) -> -> -> >  system  Ext1 (A,Mi) -> 0 — * |=Ext1 (A,Mi) + 0 —>  By h y p o t h e s i s the l a s t map i s an isomorphism, hence also,  then by Prop. 4.4, K i s f . p . / /  R i s right a right  self-f.p.  i n j e c t i v e i f i t i s f . p . i n f e c t i v e as  module.  Proposition J$ a l l l e f t Proof  gis  4.6  ( ( 1 2 ) Thm. 2.3)  f.p. are t o r s i o n l e s s .  0 ^ Ext1(A\R) -> A -> A**. / /  R i s right  self-f.p. i n j .  (26)  P r o p o s i t i o n 4.7 inj. ^ a l l left f.g.  R i s right  coherent and r i g h t  f.p. are torsionless  self-f.p.  and t h e i r d u a l s a r e  (and i n which case they a r e r e f l e x i v e  and f . p .  respectively). Combine Prop. 4.1 and 4 . 6 .  Proof  Proposition 4.8  R i s right  R e f l e x i v e by S s h i f t i j i g .  w  //  coherent and?.;right s e l f - f . p . i n j .  ( i ) I i n i 2 i s f . g . and ( i i ) 1(1-1012) = l ( l i ) + 1(12) for a l l f . g . r i g h t  ideals H , l 2 -  ( i i i ) .r(a) i s f . g . and ( i v ) l r ( a ) = Ra f o r every aeR. Proof ((5)  ( i ) and ( i i i ) i s e q u i v a l e n t t o R b e i n g r i g h t Thm. 2.2)  of Prop  4-2  coherent  ( i i ) and ( i v ) i s e q u i v a l e n t t o p r o p e r t y ( i i i )  ( ( 1 9 ) , P r o p . 1 8 , 4 ) , hence a p p l y i n g P r o p . 4 . 2  g i v e s the r e s u l t . / / P r o p o s i t i o n 4.9  R right  I f A i s f.p. r i g h t , Proof  coherent and r i g h t  then p.d. A = 0 o r w.  I f B i s f.p. l e f t ,  B i s torsionless  self-f.p. i n j .  and B* = 0 then B = 0 because  by Prop. 4 . 6 , now a p p l y Prop. 2 . 2 0 . / /  P r o p o s i t i o n 4.10  ((18)  Lemma 4 . 1 )  r i g h t f . p . i n j . then f l a t  R right  coherent and  r i g h t modules a r e a b s o l u t e l y  pure. ( f . p . i n j . ) Proof  For A f.p. l e f t  0 —*M«A  —M8A**  Hom(A*,M) f i s monic because flat,  A i s torsionless  g i s an isomorphism  (Prop. 4 . 6 ) and M i s  by Thm. 4 . 3 , because  A* i s  f.p.  (Prop.4.1).  M i s absolutely  Hence h i s monic, and by Thm.  pure. / /  P r o p o s i t i o n 4.11  ((18) Prop. 4.2)  coherent, f o l l o w i n g are (i) R i s right  F o r R r i g h t and  left  equivalent:  self-f.p.i n j .  ( i i ) f . p . i n j . l e f t modules a r e (iii)  4.3,  i n j . left  modules a r e  flat.  flat.  Proof  A f.p. l e f t . Horn (A**, M) -^^Ho^m (A, M)  (i)  =^ ( i i )  A = A**by Prop. 4 . 7 . ,  since R i s l e f t  coherent (Thm.  isomorphism and M i s f l a t (iii)  ^ (i)  ¥ i s an  4.3).  by Thm.  Thus 9 i s an  4.3.  I f M i s i n j . , hence f l a t ,  © i s an isomorphism.  isomorphism  then by Thm.4.3,  Thus h i s e p i c f o r any i n j . M,  t a k i n g M an i n j . c o g e n e r a t o r i m p l i e s A » A * * , Prop. 4.6, R i s r i g h t  and  by  s e l f - f . p . i n j . //  Rings f o r which i n f e c t i v e ^ f l a t  are c a l l e d  IP r i n g s . (Jain(12))  C o r o l l a r y 4.12 Left Proof was  ((12) Thm.3.3)  IP r i n g s a r e r i g h t  self-f.p.  inj.  I n ( i i i ) =7" ( i ) o f the p r o p o s i t i o n ,  not used.  //  coherence  (27)  5/  G e n e r a t o r s and R e l a t i o n s A finite  m-relations  p r e s e n t a t i o n o f B w i t h n - g e n e r a t o r s and  i s an exact  sequence o f the form Rm  I f B i s computed u s i n g t h i s p r e s e n t a t i o n ,  R  n  -» B  0.  the r e s u l t i n g  p r e s e n t a t i o n o f B has m-generators and n - r e l a t i o n s . Example 5.1 A b e l i a n Groups For a f i n i t e l y  generated a b e l i a n group, A* = Z , r  where r i s the rank o f A. p r e s e n t a t i o n 0 -» Z by pr.  Z  Z/p^z  0, whsre f i s m u l t i p l i c a t i o n  The d u a l o f f , f * : Z -» Z, i s simply = Z/prz.  Coker f * = A  A = A f o r any f i n i t e l y exact  I f A = Z/p^z, choose t h e  f again.  I t can thus be arranged  generated a b e l i a n group.  that  F o r any  sequence 0 - > M - > N - * Q - * 0 o f a b e l i a n groups, then  by Thm. 2.1 ( g ) ,  0 r  > 0  A*&Q  Hom(A,Q) —>Ext1(A,M) > km  T o r i (A,Q) 0  * Hom(A*,M) 0  I f A = Z $ T ( A ) , T ( A ) t o r s i o n subgroup, t h i s becomes r  0 -Q  r  ^ 0  Q ©Hom(T(A),Q)—^Ext1(A,M) r  Tor.|(A,Q) 0  >Mr©(T(A)©M) M  r  (28)  That i s Tor-|(A,Q) = Hom(T(A),Q) = Hom(T(A),T(Q)). Ext1(A,M) = T(A)®M Example 5.2 C y c l i c Modules This w i l l be a f i n i t e l y  be a s p e c i a l case o f Example 5-5. generated  right ideal.  Let H  Say x-j , X 2 , . . . x  n )  g e n e r a t e s H. L e t X = ( x i , X 2 , . . . x ) e R Xr R — » R , r i g h t m u l t i p l i c a t i o n ( a c t i o n ) by X, yI—>• (yx-| ,yx£, . . . y x ) n  n  n  n  i n d u c e s 0 -* 1(H)  •> R -> R  R /RX * 0  n  , ..  n  \ /  CA)  R/KH) where 1(H) i s t h e l e f t of X  r  annihilator  i s X i l e t multiplication  inducing 0  (R /RX)* » R n  o f H.  \  t—$h y±  (y-j ,y .. . y ) 2  > R * R/H  n  The d u a l map ±  n  0  ,  v  CB)  f  Thus R/E = R V R X and (R»/RX)* = £ Y = ( y i , . . . y ) n  ( YA: C  (X,Y)=  inner  product  : 2*171  38  notation).  In p a r t i c u l a r R/aR = R/Ra Proposition  5-3  H f . g . r i g h t i d e a l . R/H i s t o r s i o n l e s s  4* r l ( H ) = H.  Proof  U s i n g Cor. 2.2 and 2.4, (A) and (B) above, and the  f a c t that  (R/H)*  0 -> H ^ r l ( H )  = r ( H ) (page ( k e r : R/H  (R/H)**).  Corollary  5-4  generated  right ideal i s a right  Proof Prop.  If R i s left  2 ) , one o b t a i n s  R/H i s t o r s i o n l e s s 4.6. / /  self-f.p.  // i n j . every  finitely  annihilator.  f o r a l l f . g . r i g h t i d e a l s H, by  o]  (29)  Example 5.5 Any  General  Case  f . p . a r i s e s as coker  : R —»R n  m  where X = (x^-j) , i = 1 ,.. .m , j = 1 , . . .n, So t h a t X^ i s the l e f t the d u a l o f XT. i s X of  r  x±jeR.  a c t i o n by the m a t r i x X on R , and n  r i g h t a c t i o n on R  (the transpose  m  the m a t r i x X) 0 and  So R V X R  0 •> k e r X  R R  r  R  n  R  m  R /XR  m  m  -* 0  n  R /R X + 0 .  n  n  m  = R / R X and ( R ^ / R ^ X ) * £T k e r X i v i a  n  n  m  f i — » ( f ( e - j ) , . . . f (en), regarded  kerXx  where [ e ^ a b a s i s o f R , and f i s n  as an element o f ( R ) * which a n n i h i l a t e s R X. n  5.6 F l a t n e s s  m  By Thm. 4.3, and Ex. 5-5,M i s f l a t i f  9 : k e r X i ® W — > H o m ( R * y R X , W ) —> 0 i s e x a c t f o r a l l p o s s i b l e m  c h o i c e s o f the m a t r i x X. Horn (Rn/R X,W) = [w : Xw = o] = k e r X i - ^ m  via in  n  f t—»(f(ei),...f(e )), where X a c t s on the l e f t  on W  n  n  the n a t u r a l way.  9 i s then the map y®WK—>yw, y e R ((yi » . . . y ) ® n  W i s thus f l a t  W H  n  -^(yi f«yn ))w  w  o n l y i f g i v e n any w = (w-j,.. .wn)eW*i,  such t h a t Xw = 0, t h e r e i s some k and Y i e k e r X i 9 Rn ; bieW, i = 1...k (or  Y e k e r X i - M i c ( R ) m a t r i c e s ; beW ) k  nt  But under 9,'^Yi®bii—»>Yb;  recapping  : F o r any weW* such  t h a t Xw = 0 t h e r e e x i s t s k, Y e M ] ( R ) , n>  c  with"^Yi®bi»—> w. 1  beW  k  w i t h XY = 0  (30)  and  w = Yb.  F u r t h e r , f l a t n e s s need o n l y be t e s t e d on  c y c l i c modules, t h a t i s when m = 1 (Example 5-2),  so  one h a s : P r o p o s i t i o n 5 7 (Chase ( 5 ) ) F o l l o w i n g a r e e q u i v a l e n t : (i) W i s flat ( i i ) G i v e n any w-j,.,.w eW and xi,...xneR, such t h a t n  ^  i  x  w  - 0> "there e x i s t s k, b-jeW, j = 1 , . . . k,  i  Y i j e R , i = 1,...k ; j = 1,...n, w i t h I ^ x j Y j i = 0 a n d ^ Y - j i b i = w-j. (iii)  G i v e n any w-j,...w eW, and X-yeR, i = 1,...m; j = 1,...k n  such t h a t ^ X ^ j W j  = 0, t h e r e  exists  b-jGW, j = 1,,.,k and Yj^eR, i = 1,.. .n;  k,  with"^X jYjk  = 0 f o ra l l (i,k) and^Y jbj  i;  i ;  5-8 f . p . I n a c t i v i t y  j = 1,.. .k = w  ±  f o r a l l i . //  By Thm. 4.3 and Ex. 5.5,W i s f . p . i n j .  i f ¥ : 0 — ^ " V X R ^ w —*>Hom(kerX ,M) i s exact f o r a l l c h o i c e s r  of  m a t r i c e s X.  R ®¥—-^p /XR^®W—^0 n  i m p l i e s (Rm/XR4®W = wm/XWn.  m  ( R / X R ) * •*= k e r X r . v i a f i - ^ ( f ( e l ) , . . . f ( e ) ) . m  n  n  ¥ i s wi—>(yi—*yw) ( = w w = (wi,.. .w )eW /XW « n  m  m  SP i s i n f e c t i v e  if w  r  From Ex.5. 5  Hence  r i g h t m u l t i p l i c a t i o n by w).  In co-ordinates  ( y i , . . ,y ) n  H ^ y i W i . ,  = 0 i m p l i e s wcXW . n  r  P r o p o s i t i o n 5.9 W i s f . p . i n j . ^ g i v e n W|,..,w eW m  (weW ) and X i j e R , i = 1...m; j = 1...n ( X e M m  that^YiXij  = 0, y e R  then wi = ^ X i j b j  t  m > n  ( R ) ) such  (yX = 0, yeRm) i m p l i e s £ y i i = 0 (yw = 0) w  , f o r some bjcM (w = Xb, f o r some b e M ) . / / n  (31)  5.10  Corollary  W i s f.p. i n j.  If R is left  coherent.  G i v e n w-j,.. .w eW, and xi,...xmeR such m  = 0 , y-^R, i m p l i e s ^ y i w i  that^yixi  = 0 (yx = 0 $ yw = 0 )  then f o r a l l i , Wj[ = x i b f o r some beM, Proof f.p.  For a l l l e f t  Corollary  Proof  f . p . by Prop. 4 . 2 .  Hence  one can take n = 1 i n the p r o p o s i t i o n .  5.11  //  I f W i s f . p . i n f e c t i v e and H a f . g . r i g h t  i d e a l then I*V/1R(H) =  for  coherent r i n g s , one can check f o r  i n a c t i v i t y on c y c l i c  by Ex. 5 . 2 ,  (then w = xb).  HW.  L e t wertflR(H), and x i , . . . x  n  generate H. I f y x i = 0  a l l i , then y e l R ( H ) , and so yw = 0 , hence by the  proposition  w = ^ x i b i > bieW. / /  (32) P a r t Two 6/  The Category o f E x a c t / S p l i t Sequences The  assignment A t — > i  i s not i n general  One o f the o b s t r u c t i o n s b e i n g one  functorial.  that f o r P f.g. p r o j .  can choose £ t o be 0, P* o r any o t h e r f . g . p r o j e c t i v e .  Prom 0 uniquely  -> P* e» P  'uniquely  1  *k* i s  0, and Schanuel*s Lemma  ;  determined up t o p r o j e c t i v e summands; t h i s  that & i s a l s o case,  A  determined.  suggests  T h i s i s indeed the  and we proceed t o s e t up the machinery t o demonstrate  this fact  (and a l s o examine the machinery  S t a r t i n g w i t h an A b e l i a n c a t e g o r y Of  t  the p r o j e c t i v e s <P, t o form the q u o t i e n t However <=7/ff» i s n o t a ' n i c e ' c a t e g o r y , not be A b e l i a n .  itself!) one can k i l l  c a t e g o r y <7/(P.  and i n g e n e r a l  will  One can a l s o form c a t e g o r y <S. o f exact  sequences; <£ i s never A b e l i a n k i l l i n g o f f t h e s p l i t exact  ((22)page 375), however  sequences £ , r e s u l t s i n an  A b e l i a n c a t e g o r y $/£ (Thm. 6.7 ahead).  I f Of has s u f f i c i e n t  p r o j e c t i v e s , one can a s s i g n t o each o b j e c t A, a s p e c i f i c projective presentation 0 - ^ K - ^ P ^ A + O . assignment determines a f u l l for  This  embedding o f <37/(P i n t o c?/$ ,  which the image o f °7/(P c o n s t i t u t e s 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 f o r £/$ (Thm. 8.5 ahead), ( k i l l p r o j e c t i v e s o n l y t o become p r o j e c t i v e s ) . work i n the A b e l i a n The  Hence i t i s n a t u r a l t o  c a t e g o r y c ^ r a t h e r than Q/<£. 7  next t h r e e lemmas a r e recorded  f o r reference.  (33)  Lemma 6.1 ((11)page 83) Given  then (0->) C^ > ^>A$B a  b  C -£>A bj, is B-=->D  ^ " ^D[—»6] r  s  the square i s a ( p u l l - b a c k ) , Lemma 6.2 ((11)page 84)  i s exact  jpush-outj . //'  0 -> B  E  II  1  I  0 -> B  E  -*> A'  i  -» A  0 0  commutative and exact rows, then the r i g h t hand  square  i s a p u l l - b a c k and a push-out. / / Lemma 6.3 ((23)page 163) Any A ^ B o f s h o r t sequences, 0 ->A  has a f a c t o r i z a t i o n :  A -v A' I II E ^ A'  0  i i 0 ^ B" -> B -> B*  0  M  *  0 ^ B  -ii  M  in<S, the c a t e g o r y  0  A  B  The o b j e c t s o f £ b e i n g exact sequences thought o f as c h a i n complexes naturally  can be  so the n o t i o n o f homotopy  arises.  P r o p o s i t i o n 6.4  Given  0—>A" f"  - V A - ^ A ' /  ->0  A  ->0  B  g/ i B'«  >B  -> B  following are equivalent: (i) (ii) (iii) (iv)  There e x i s t s g such t h a t ga = f " . There e x i s t s h such t h a t b'h = f . There e x i s t s g and h such t h a t bg + h a ' = f . f f a c t o r s through a s p l i t  exact  sequence.  (34)  (v)  f i s c h a i n homotopic t o zero  Note  (i)  (ii) # (iii)  (i) # (iv) Proof  We prove  dually. and  That  (f~o).  Pieldhouse (7),  Freyd ( 8 ) .  (i)  ( i v ) , then ( i i ) & ( i v ) i s proved  ( i ) and ( i i ) combined  (iii)  i s clear,  ( i ) ( i l ) . and ( i i i ) c o n s t i t u t e ( v ) .  ( i v ) =^ ( i )  0 —>A" ,—>A — * A - » 0  The r e q u i r e d g i s  ,  achieved v i a the p r o j e c t i o n C&D-^C. (i) ^ (iv)  Consider  A"—>A  ,  where the square i s a push-out..  k e x i s t s to g i v e a commutative diagram.  Hence B " — » E i s  s p l i t monic, and r e s u l t now f o l l o w s by Lemmas 6.2 and 6.3. /./ Suppose C and  and ID  are a c y c l i c  f : <C—«>(D, a c h a i n map.  s h o r t exact sequences, P r o p o s i t i o n 6.5 homotopic),  ( e x a c t ) c h a i n complexes,  B r e a k i n g the complexes i n t o  f i n d u c e s maps f n between these p i e c e s .  I f <C and "P a r e a c y c l i c , and f ~ o ( c h a i n  then f n ~ o .  Proof  C  n + 1  -S-^C  \K /  n ij  J  e  n-1  c  f  a  c  t  o  r  s  e x i s t s 6 : Kn  n  through coker ( C •Dn,  n + 1  — > C ) = K , so t h e r e n  n  (35)  n+1"  •K n  n •n  n+1  J  d'0c'  Then  •n n  h  = d'(e_«|c) = d'(f -de ) n  n  c' i s e p i c , so d ' e = f  and  induced map f : ( K n  C  n + 1  n  .  n  n  = d'fn  fn'c'  =  Hence by Prop. 6.4 t h e  -v K )  ( L - | -*> D  n  n+  + L ) is  n  n  homotopic t o zero. / / There i s a p a r t i a l converse, whose p r o o f i s the same as showing, two c h a i n maps fP-^(B, i n d u c i n g maps H ((P)  H ((8) a r e then homotopic. ((11)page 127).  0  0  Proposition and  6.6  L e t P be a p r o j e c t i v e r e s o l u t i o n o f A,  B an a c y c l i c complex, suppose f : A  over B.  2  induced c h a i n map P A  B  0  factors  0  ?2  B ->B  P  t h e same  0  1  then the  © i s homotopic t o z e r o , where  A  i s IP w i t h A a d j o i n e d .  Proof  By i n d u c t i o n ,  to construct •n — » Pn Pn-1  Pn+1  fn/B  %+1 B n+T  the homotopy  rt n i l  t n-^ n-1 B  B  n+1  J  M*n  - e d ) = b f  Pirst  e q u a l i t y by i n d u c t i o n ;  ker  b  n  n  n  = Ln , + 1  n  - (f _i  n  but P  n  n  - en-^n.^dn so f  n  - 9 d n  = b f n  -  n  factors  n  f  n  through  i s p r o j e c t i v e and B + i - + > L i , so n  n +  ^ A  n  (36)  to B  t h i s f a c t o r i z a t i o n can be l i f t e d  - |, giving 9 + 1 .  N +  N  F o r each horn s e t ( A , B ) i n £ , those subgroup, and induce with  an e q u i v a l e n c e  the a d d i t i v e s t r u c t u r e o f cf.  Consider  whose o b j e c t s a r e those  hom s e t s a r e  (A,B)/AJ.  Theorem 6.7 is Note:  (Freyd  (8) Thm.  f-vo form a  relation  c a t e g o r y ,  //  compatible the q u o t i e n t  o f <£ but whose  3.3)  Abelian. Following proof  i s adapted  from Freyd's , however  f o r our purposes we need the e x p l i c i t  c a l c u l a t i o n o f the  k e r n e l and c o k e r n e l o f a morphism and i t s c a n o n i c a l f a c t o r i z a t i o n ,for f u r t h e r p r o p o s i t i o n s . Proof will and  <£/-o i s a d d i t i v e because <£ i s a d d i t i v e .  Hence i t  s u f f i c e t o prove t h a t every morphism f has a k e r n e l c o k e r n e l ; and a f a c t o r i z a t i o n f = gh where h i s a  c o k e r n e l and g a k e r n e l ((20)page 8 7 ) . G i v e n f : A 4 B , we w i l l 0 4  show  A" -4 B"©A-4 E  II  0 -> A"-— 4 I 0 4 B" II  4  0  A —y A  1  4  0  A  1  4  0  II  >E  \f  -4  V  B — » B' 4 0 II 0 4 E --4 B©A'-4 B + 0 0 4  B" — >  f  represents  0  4  ker  By Lemma6-3. f = gh.  f 4 A 4 i m f 4 B 4 The exact  coker  f  4  0.  sequences a t top and  bottom r e s u l t from Lemma6-1, u s i n g Lemma6-2 and i t s d u a l .  (37)  We prove  (a) k = k e r f  (a') 1 = c o k e r f  (b) £ = k e r 1 then  ( V ) h = coker k.  (a) ( i ) k i s monic  :  X " — 4 X —  I 0/1 I x A" - ^ > B " © A — > E II*' 4, I *  A" if  dually  >A  >A'  kx = 0 t h e n 6 e x i s t s by Prop. 6.4; the same 9 t h e n  shows x = 0. ( i i ) hk = 0 :  A"—>B ©A,  take 8 t o be the p r o j e c t i o n .  M  —  /  II  A" B"  (iii)  Suppose hx = 0 :  X"—>X—4X',  so t h a t  A" ^ >A—>A*  with the properties  B"—> B — » B '  o f Prop.  | Q/l  |  j  W  then  X"-  *X—>X'  7  I  I  "  W  > A  6.4.  gives a f a c t o r i z a t i o n of x thru  I l l A"—*B ©A —»E II t i  A"  9 exists  >A»  (b) ( i ) £ i s monic, p r o o f same as f o r k. ( i i ) l g = 0 : B"—>E ,  take 9 t o be i d e n t i t y .  /e  II Y' Mr  ( i i i ) Suppose l x = 0 : X"  —  ^X  i  >X  yi  then 9 e x i s t s as  f  I  B"-A>B »B U I II E *B©A' —>B' f  l e t x° be :  X"—>X—*X' B " — » E —»A'  £ II  1  I •  B"—>B—>B  f  i n Prop.  6.4.  , x - x°r*>0 because left  l e g o f x - x° i s  "  the zero map.  ~  k.  (38)  Hence i n  x = x ° , and x can be f a c t o r e d  through g .  (39)  7/  Projective Now  full  Homotopy  assume <7 has enough p r o j e c t i v e s ;  subcategory of  Proposition  7.1  l e t (P he  the  projectives.  Por f : A — * B , the f o l l o w i n g  are  equivalent: ( i ) f can be f a c t o r e d  through some p r o j e c t i v e .  ( i i ) f can be f a c t o r e d  through any p r o j e c t i v e  that  Q-»B.  (iii)  f can be f a c t o r e d  Proof  (9)page  through any  Q,  C—»B.  131. ( p r o o f i s s t r a i g h t f o r w a r d ) .  //  f i s p r o j e c t i v e l y homotopic t o g i f f - g through a p r o j e c t i v e . (A,B)  through a p r o j e c t i v e . 7.2  Proof  L e t 1T(A,B) =  can be  factored  (A,B)/P(A,B).  I f e : Q —«>B, Q p r o j e c t i v e  ?l(A,B) = c o k e r e*  then  : (A,Q) —->(A,B).  lm e* i s the s e t o f f which  can be  factored  through Q - » B ,  hence e q u a l s P(A,B) by Prop. 7.1.  Corollary  ((11)page  7.3  factors  L e t P(A,B) be the subgroup o f  c o n s i s t i n g o f those maps which  Proposition  such  //  135).  The f u n c t o r ?((A,-) i s a d d i t i v e . Proof  To e v a l u a t e 7 7 ( A , B © B ) , take Q © Q + > B © B . f  Let  Q  n  Q  n - 1  -v . . .  r e s o l u t i o n o f B, w i t h n  to  D e f i n e 7f (A,B) = 7 T ( A , S ) ; n  Schanuels  n  Q  1  , be a  syzygy  //  projective  S . n  s i n c e 7T(A,-) k i l l s  Lemma g i v e s 7Tn(A,-) i s independent  projectives, of choices  (40)  of p r e s e n t a t i o n s . P r o p o s i t i o n 7.4  ((11)page 142)  Given 0 - ^ B " - v B - * B '  0 t h e r e i s an exact  sequence  ...  ^ ? r ( A , B ) ^ T r ( A , B « ) -v?r -i(A,B") +?r ^Ut*)  ...  -> T T ( A » B )  n  n  n  ...  n  If (A,B«) -> Ext1(A,B") -» Exf1(A,B) •* ...  In p a r t i c u l a r ^ ( A , - ) i s h a l f - e x a c t . Proof let  C o n s t r u c t p r o j e c t i v e r e s o l u t i o n s © * , <D",of B* and = ©'©(Q", then  0 -» ( A , Q » ) ^ (A,Q ) G  (A,Q M  0  I  0  * 0  Y  0 -* (A,B")  (A,B)  ^ (A,B")  E x t ( A , B " ) -> ... 1  a p p l y snake lemma and Prop. 7.2 t o g e t ( A . S ^ ) ^ ? 7 ( A , B ) -v ?f(A,B) ->  0 -» (A S.,") * (A,S.,)  H  f  ^ ( A j B ' ) ->Ext1(A,B") ->Ext1(A,B) * ... S  1  = ker : Q  B (first  0  Going hack a n o t h e r 0 -> ( A , C y )  s t e p , u s i n g Prop. 7.2  * (A,Q ) -> (A,Q^ •) -» 0 1  ^  J/  0  syzygy).  (A.S.,")  tfuts/)  ^  (A,S.,) -> ( A ^ ' ) *17(A,B") ^ T f d s ^  ^(A,B) + ...  VTTCA.S^)  a p p l y snake lemma, n o t i n g T/"(A,S^ ) = T f ^ A j B ) then i n d u c t i o n completes the sequence. / / P r o p o s i t i o n 7.5  Let P  Q  -> P  n  1  -* ... -» P  Q  Z (A) n  he a p r o j e c t i v e r e s o l u t i o n o f A, w i t h n I f Q -^B, Q p r o j e c t i v e then f o r n>0  tt  syzygy  Z (A). n  (41)  Ext (A,Q)—> Extn(A,B)-*tf(Z (A),B)  0.  n  n  Proof  n = 0 i s Prop. 7.2 ( Z ( A ) = A) Q  For  n> 1 r e p l a c e A by Z _-|(A) t o reduce t o the case n = 1.  Let  P = P  n  and Z = Z<|(A), i f 0  0  snake lemma t o 0—>(P,S)  S * Q •> B •> 0, a p p l y  > (P,Q)  I  > (P,B)  I  1  0->(Z,S)  >(Z,Q)  I  >0  »(Z,B)  I  >Tf(Z,B)-4 0  i  E x t 1 ( A , S ) — j r Ext1 (A,Q)->(Ext1 (A,B) P r o p o s i t i o n 7.6 (Ln  the n  Proof  ((11)page 142) L ( A , - ) =7Tn+l( »-)» * 1 A  Prom  ->Q —4 2  functor)  AQQ.  \/\/ s  one o b t a i n s  n  n  l e f t derived  t o  //  2  \  s  2  (A,Q )  /  1  (A,Q )  (A,Q )  1  \  (A,S ) 2  Q  V  (A,S^)  ker  a* = ( A , S ) , r e g a r d i n g ( A , S ) - (A,Q^) t h e n im b*  are  those f e ( A , S ) which f a c t o r through Q  im  2  2  2  b* = P ( A , S ) by Prop. 7.1. 2  •»  2  , so  Then the f i r s t l e f t  derived  f u n c t o r o f (A,-) : L.,(A,B) = H ((A,<0)) = k e r a*/im b* = 7 ( A , S ) =7T (A,B). 1  Por  2  n*1, L ( A , B ) = L ^ A . S ^ ) = M -  n  ( A 2  Any f : A 4 B induces a n a t u r a l Ext1(B,-)  ' n-1 S  )  =  " n + l U , B ) . //.  transformation  E x t 1 ( A , - ) , which can be computed  e x t e n s i o n s by u s i n g p u l l - b a c k s :  2  i n terms o f  (42)  0 4 c  —  E  :  II  >2  0 4 C  A -V o  £.  I  t  —•»  '  >B 4 0  £  P r o p o s i t i o n 7 . 7 ( ( 1 0 ) C o r . t o Thm. 1.3) E v e r y n a t u r a l t r a n s f o r m a t i o n 0 : E x t ( B , - ) 4- E x t ( A , - ) , 1  1  i s induced by a map f : A 4 B. Proof  L e t 0 * K * > ? * ' A * > 0 and  O ^ I » Q ^ B » O b e  projective presentations. 0 4 (B,-) 4 (Q,-) -v ( I , - ) -> E x t ( B , - ) 4 0 1  P  1I  P  2l •  F  3|  0 4 (A^-) * (PV) P^ e x i s t because (by Yonedas  I  *>  6  (  (K,-) 4 E x t ( A , - ) -* 0 1  r e p r e s e n t a b l e s (M,-) a r e p r o j e c t i v e ,  Lemma) i n the f u n c t o r c a t e g o r y Ah *. 0  by Yonedas L.each P  i  i s induced from some f ^ such  Also that:  O^K^P^A-^O F  3  0->>L-^Q->B->0 Then f ^ i n d u c e s a n a t u r a l t r a n s f o r m a t i o n E x t ( B , - ) - > E x t ( A , - ) which a l s o makes (*) commutative, 1  1  by the uniqueness o f maps induced out o f the c o k e r n e l , t h i s map must be 0. / / Proposition 7 . 8  f : A -> B i n d u c e s the zero map  f° : E x t ( B , - ) 4 E x t (A,-) 1  Proof  1  04C-»E'  ->A-*0  0 * C 4 E  B 4 0  II J I*  f f a c t o r s through a p r o j e c t i v e .  (43)  f° = 0  ^  top row s p l i t s f o r a l l e x t e n s i o n s o f B  ^  t h e r e e x i s t s g such t h a t  &  t h e r e e x i s t s h such t h a t  C — E '  A  E f  >K if  , by Prop.  -*B  f a c t o r s over a p r o j e c t i v e , by Prop. 7.1.  P r o p o s i t i o n 7.9  ( ( 2 ) Thm.  6.4.  //  1.40)  There i s an exact sequence 0 *>P(A,B)-* (A,B) 4 Hence Proof at  J E x t 1 ( B , - ) , E x t 1 ( A , - ) ] -5> 0.  77(A,B) = [ E x t 1 ( B , - ) , E x t 1 ( A , - ) ] . The l a s t map  i s onto by Prop. 7.7,  the middle by Prop. 7-8.  //  and e x a c t n e s s  (44)  8/  K i l l i n g Projectives 8.1  Consider  o b j e c t s a r e those each A o f then  c a t e g o r y <37/(P, whose  o f c y , but w i t h horn s e t s  07 chose  4  the q u o t i e n t  a projective presentation P(A)  0->K-»P-»A-»0 i I *[ * v  A  1  0 -> L *  B  1  Q * B VO  F(B) In fact, i f f  f ' = f " i n g / & , so P ( f ) i s w e l l - d e f i n e d . defined using d i f f e r e n t presentations,  then  F (A) 1  1 - $A®k"Q s i n c e r i g h t l e g o f 1 - i^QA  i  s  A,  between the f u n c t o r s F and F*,  To determine k e r F, suppose f : A = 0  "the zero map A  and 6^ a r e i n v e r s e s o f each o t h e r , and  determine a n a t u r a l e q u i v a l e n c e  F(f)  I f F' were  *A  0 A  hence  ft  =  hence i n C L / ^ , ¥  induces  F(A)  0 J-> K -> P >-> A -> 0 A  0+K-»P-*A*0,  F ( B ) , then f ' - f " " 0 ,  two maps f ' , f " : F ( A )  K' •» P'  For  f  c o n s t i t u t e s a f u n c t o r F : Ol •» c°/^.  0  TfiA^B).  P ( f )"0  B,  f f a c t o r s over Q -j- B 4) f = 0 i n °//<P  Hence F f a c t o r s 0 7 - ^ C ? / ^ '®  which embeds °7/(P as a f u l l  s u b c a t e g o r y o f the a b e l i a n c a t e g o r y  .  F o l l o w i n g extends a r e s u l t o f H i l t o n and Ree Theorem 8.2 (i) (ii)  f : A  ((1-0)  Thm.2.1).  B, f o l l o w i n g a r e e q u i v a l e n t  Ext1(B,-)>->Ext1(A,-). there i s B* such t h a t Ext1 (B,-)<BExt1 (B',-) ~  Ext1(A,-).  (45)  (iii)  Given Q — ^ B ,  Q projective,  then B i s a d i r e c t  summand o f A©Q. (iv)  B i s a d i r e c t f a c t o r o f A i n <V(P,  split  epic  that  i n oi/(P.  F(B) i s a d i r e c t f a c t o r of F ( A ) i n £/l  (v)  F(f) i s split (vi)  is f is  epic  in  , that i s  .  F ( f ) i s epic.  Proof (ii)  (iii)  ( i i ) take B  1  t o he complement o f B i n  A@Q.  =*> ( i ) i s c l e a r .  (i)  ( i i i ) I f g :. Q  B form  E  K  E-i e  Ext (A,K) 1  e'  f ° K  + Q®A by t a k i n g  >B  9  e Ext1(B,K)  the p u l l - b a c k , v : A >$> Q©A the  By Prop. 6.4, row  E  splits,  and  since  f£  (iii)  ^ (iv)  ± a  © exists  i  n  - j  (because v d o e s ) , hence the top ?  ( i ) implies e  c  t  i  v  e  the bottom must a l s o  f  split  #  B I^AQQ^-^B  where  <f,-g>r i s  i d e n t i t y map  In °7/(P v and  inclusion.  p a r e isomorphisms,  hence B  o f B.  A ft*"§^ > B, v  g i v e s B as a d i r e c t f a c t o r of A i n ^/(P. (iv)  (v)  (vi)  =^ ( i i i )  0 -» L — >  I 0 -p- K  (vi) clear. Prom p r o o f o f Thm.  Q — £ - > B ->0 I II A©Q gz"^ B 0  6.7,  coker P ( f ) i s  so i f P ( f ) . i s e p i c then bottom s p l i t s .  //  (46)  T h i s theorem does n o t ' d u a l i z e ' f o r example  satisfactorily,  the d u a l o f ( v i ) P ( f ) b e i n g monic: from Thm. 6. l  k e r F ( f ) i s 0 -> K -> L©P -» E -» 0 H I I 0 -> K -> P A 0 I f P ( f ) i s monic then t o p row s p l i t s , so E x t ( B , - ) = E x t ( L , - ) = Ext (L©P,-) = Ext (K,-)©Ext (E,-) 2  1  1  = Ext (A,-)©Ext (E,-) 2  1  1  1  and E x t ( A , - ) i s a d i r e c t summand 2  o f E x t ( B , - ) , t h i s i s n o t the c o r r e s p o n d i n g d u a l o f ( i i ) 2  o f the theorem.As f o r the d u a l o f ( i ) , P r o p o s i t i o n 8.3 ((IQ)Thm. 2.2)  the f o l l o w i n g :  Ext (B,-) — E x t 1 ( A , - ) 1  t h e r e e x i s t s E, and a p r o j e c t i v e P, and an exact sequence 0  P  A©E  B •¥ 0.  F u r t h e r P can be chosen  to be any p r o j e c t i v e such t h a t P—**A. Proof  (4>) From l o n g exact E x t sequence.  (£)  Given 0 - * K - > P - » A - » 0 i n  E x t ( A , K ) , by s u r j e c t i v i t y , 1  t h e r e i s 0 - > K ^ E ^ B ^ 0 i n E x t ( B , K ) such t h a t 1  0-^K + P +  A^O  II  0-»K-»E+B-»0 then by Lemma 6.2, 0 Theorem 8.4  P -*> A©E  B  0 i s exact. //  ((12)Thm. 1.44)  Following are equivalent (i) (ii) (iii) (iv)  Ext (B,-) = Ext (A,-) 1  1  There e x i s t s p r o j e c t i v e s P and Q such t h a t A©Q ^ B©P A = B i n q/<P F(A) ^ F(B) i n £/S .  (47)  Proof  ( i i ) =^ ( i ) i s c l e a r .  v U-k t^^^  (i) * ( i i ) i fg : Q — » B ;  0  If  P—^Q©A^^B  by Thm. 8.2 t h i s sequence s p l i t s ,  0  further P i s projective  s i n c e E x t ( B © P , - ) = Ext(Q©A,-) = E x t ( A , - ) i m p l y i n g 1  E x t ( P , - ) = 0. 1  (iii)  ( i v ) because °f/(P  is a full  embedding*  ( i ) $ ( i i i ) f : A ^ B f a c t o r s as A A A © Q - % B © P - ^ ? B as i n the p r o o f o f ( i ) ^ ( i i ) . But v and p a r e isomorphisms i n 0/CP, hence A = B. f ( i i i ) ^ ( i ) A^z?B, such t h a t ( l - fb)~0 and (1 - hfVo. h Then 1 - f h i n d u c e s the zero map E x t (A,-) - > E x t ( A , - ) 1  by Prop. 7.8.  1  Hence E x t ( A , - ) = E x t ( A , - ) 1  1  v  /f*  h°  Ext (B,-)  . //  1  Theorem 8.5 ( ( 8 ) C o r . 2.9)  Oi-?-*  , isa full  lm F = C7/(P i s a f u l l  Proof  First  embedding  of  subcategory of r e s o l v i n g  statement i s c o n t a i n e d  (a) a//(j? r e s o l v e s  in  , and  projectives  i n section 8 . 1 .  f o r 0 - ^ K - ? - P ^ A ^ 0 = F(A') ,  II 0 - ? A " * A ^ A can  be f i l l e d  f  y O = A  i n , and the r e s u l t i n g map F ( A ' ) - > A i s  e p i c i n £ / ^ (by p r o o f o f Thm. 6.7) (b) F ( A ) i s p r o j e c t i v e f o r any A. F o r i f N — » F ( A ) j by ( a ) , P(B) — V > N some B ; to s p l i t N — ^ F ( A ) i t s u f f i c e s  (48)  to s p l i t F(B) — » F ( A ) , but t h i s i s a s p l i t  e p i c by  Thm. 8.2. / / Remark 8.6  Given f : A  then A > ^ A © Q  B, i f g : Q - » B, Q p r o j e c t i v e  , and i n <?/(P, v i s an isomorphism, hence  B replacing  f by  Proposition tf/(P has  ( f , - g )  8.7  one can assume f i s e p i c  t  i n 07.  (Freyd(8))  weak k e r n e l s  (no uniqueness p r o p e r t y  required).  Proof  L e t f : A 4 B, by above remark assume f i s e p i c  i n Oj.  I f 0 e> K 4 A -> B 4 0 i s exact i n Oj,  i s a weak k e r n e l fg  o f f i n £7/<P.  then K -»• A  f g = 0 i n <T/(P, means  f a c t o r s over a p r o j e c t i v e P i n  : X-&»A-£->B, p  a i s induced s i n c e P i s p r o j e c t i v e .  Then f ( g - ab) = 0,  so g - ab f a c t o r s through K = k e r f inCT, but g - ab = g i n <7/P, so g f a c t o r s through K i n Q/(£, Proof 2 E—>Q A -£>B  (Freyd)  L e t Q—S4B, take  , then E -> A i s a weak k e r n e l (use  Example 8.8  Cf/W w i l l  Ab/^the canonical  n o t i n g e n e r a l be A b e l i a n .  For  p r o j e c t i v e s = f r e e s , then i n map  i s n o t an isomorphism.  Hom((D,F) = 0 f o r F f r e e . remains e p i c  of f .  Prop. 7.1). / /  example i f 0/ = Ab,  but  pull-back  i s both e p i c and monic P r o o f : Hom(<p,Z) = 0  implies  Suppose Q - » X in Ab, t h i s map  i n Ab/y, f o r i f Q -» X F  A i s a factorization  (49)  over a free P, then n e c e s s a r i l y <R 4 X 4 A = 0, hence X 4 A i s zero. By Prop. 8.7  Thus i n p a r t i c u l a r <Q 4 Q/Z i s epic. Z » <Q i s a weak kernel of t h i s map, hut  t h i s i s the zero map Ah/?', hence i t i s a c t u a l l y the kernel and so <Q 4 &/Z i s monic.  This map could not  be an isomorphism because Hom(<Q/Z,<R) = 0. Proposition 8.9 Let (B be a f u l l subcategory of r e s o l v i n g projectives of an abelian category G, then the i n c l u s i o n (B 4 C preserves kernels. Proof  Suppose K •> C i s the kernel of C -> D i n (B.  K 4 G i s monic i n C.  For i f N ->• K  C i s zero l e t  B -»N, B i n OB, then B - ^ N ^ K ~> C = 0 implies B  N 4 X  implies N •» K = 0. Let L •> C be the kernel of C 4 D i n G and l e t B — » L, B i n CB. K 4 C 4 D B—->=>L  g e x i s t s since L = ker C 4 D i n G, and K 4 C monic implies g i s monic.  h e x i s t s since K = ker C 4 D i n <B. K B  L  C  K—»C  C  B  can be cancelled  B —>~L hence  K /I B-»L  and the monic L •» C  i s commutative,  implying  that g i s also epic, g i s then an isomorphism and K 4 C i s also the kernel of C 4 D i n 6. // C o r o l l a r y 8.10 F : °7/<P y>  preserves kernels. //  (50)  C o r o l l a r y 8,11  (a remark of Freyd (8)page 99)  I f oT/(p has kernels then p.d. Proof  For N i n  , choose F(B) -> F(C) -» N  by the proposition 0 4 F(A) 4 F(B) some A.  -2. •*  0, B, C  F(C) * N  0 for  Since F(A) i s p r o j e c t i v e by Thm.  8.5,  InO?  ,  p.d. N - 2 . //  (51)  9/  Syzygy Functor  9.1  General reference  Auslander and Bridger  ((2)pages 48-51).  For each A, chose P—>->A, P projective ( f o r convenience i f A i s projective chose A —»A, and i f A i s f.g. chose Let Z(A) = leer (P -> A).  Pf.g.).  Z i s not a functor from cv to ty ,  however i f the target becomes <7/(P then Z i s a functor. K  P  i  i  A  , l e t Z ( f ) be the representative  of  f  L •=? Q  B  the induced map K  I f Z ( f ) i s well-defined, functor Q •> CT/P.  L in1j'(K,L).  then Z i s c l e a r l y an additive  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 p r o j e c t i v e .  However even more i s true, i f f f a c t o r s  over a p r o j e c t i v e , so does the induced map.  In f a c t , i f  f factors over Q i n the above diagram, then by Prop. K -> L f a c t o r s over P.  6.4,  Thus Z f a c t o r s : 0} -^h> Q7/(P. K  \  S  The functors P and Z can be r e l a t e d by F(A) =(0 -> Z(A) * P * A * 0) Suppose Z and Z* 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 Consider  K* = K' ty  K -> N II  ty  K •> P Let n  A  chosen.  I  •» P» ty  -> A  : K » K©P' = K'©P  K», n  A  i s an isomorphism i n 3/<P,  and determines a natural equivalence between Z and  Z'.  (52)  The n  tt  syzygy functor can be defined by Z (A) n  (where Z i s now 9.2  regarded as a functor °7/(P  ((8)Prop. 1.2)  ~$>  = Z(Z _ (A)), Q  1  C7/CP)  For any Abelian category C, i f © i s  a f u l l category of r e s o l v i n g p r o j e c t i v e s , then any functor G : (B  CD, fl) abelian,has  a r i g h t exact functor C "4 CD. chose B G(C)  1  Theorem 9.3  Let A = 0 • * A - » • A - » A ^  e> 0, the following  , with P, P',  P" - » P - ^ P ' -» A  1 K'  -w  A  Let P» — » A ' ,  1  I i I  P" p r o j e c t i v e ,  V  0  (#)  A' = A' -* 0  -*> A  and P"—se»A", set P = P'©P", then 0  0  0  ,  K" = K " —*> K ->> K' w-*> A" -> 0  0 -> K 0  define  ^G(B).  is an exact sequence i n 0  to  E x p l i c i t l y f o r each CeC,  -> B -» C -» 0, B* , B i n (B, and  = coker : G(B')  Proof  a unique extension  *  0 *  v<  V I  0  0  0  -» A'  0  0 The maps r , s, u, v r e s u l t from the s p l i t t i n g of P" -> P  P',  v i a canonical projections and  and give the properties of Prop.  6.4.  injections,  (53)  (I)  6 . 7 , to s t a r t a projective  Making use of Thm.  r e s o l u t i o n of A : K  t  ( 't b  W  )>A"©P'  b» I  '  (II)  w the  f  ^ > P*  a*  ^ >A'  l  £,  ll  £  A"  " ' ^ A  <  > A  induced  map.  Lower r i g h t square  >A'  commutes by ( + ).  B u i l d i n g on the kernel of (I) and using Thm. >K'©P  K  K k-| K  >A"©P'  >P |(-r,p)  'TbV^ " A  W  6 . 7 again  > A ||  <-f,B>  >  A  however t h i s time we must check bottom squares commute. For the lower l e f t , i t i s required that b'k' = pb, which i s c l e a r from (+), and wk' = -rb.  Por the second equality  apply the monic f : f(wk' + rb) = 8b'k + (a - sp)b = sb*k' - spb = 0 r  a s i m i l i a r c a l c u l a t i o n f o r lower r i g h t square. (Ill)  Continuing to project on the kernel, the obvious  choice i s  K I K  w  (°> "l> P b  I  (P» )>P«eA»  * K'©P  r  II  >P'©A  However, (p,r) : P = P'©P" — ^ ' © A " , (p,r) = 1©a" so the top row i s isomorphic, i n  , to 0  and we use t h i s f o r the p r o j e c t i v e :  K" -*» P" "•A"  0,  (54)  K  „  ( r " ) © P " <l(kSb),-(Q i)> k  b  >K  >  >  K  , ©P <-w,-r>  b" K  > P"  M  (0,i) K  (k«,b)^ ® K,  P  (0,-1)  <-(w,b«),{-r,p)> '•©A" P  Lower l e f t commutes by (+). Lower r i g h t <- (w V ) , (p,-r)>  ( 0 , i ) = ( p , - r ) i = ( 0 , - r i ) = (0,-a») using (+).  (IV)  The kernel of ( I I I ) i s isomorphic  to K"  K -» K' v i a  Por 9, ¥ to be inverse isomorphisms, one need only check 9 and <f are well-defined, because (l - 93^0 and (l - 59^0 since l e f t legs are the zero maps.  Por $ t h i s i s c l e a r ,  and f o r 0, upper l e f t square commutes d i r e c t l y from ( + ). F i n a l l y t(1,u) = <(k'%b), - (0,p)>(1,u) = (k',b) - (0,p)u = (k',b-pu) = (k' ,vkM = (1,v)k'. (V)  Putting the pieces together gives required sequence. //  C o r o l l a r y 9.4  ( i ) (remark of Freyd, (8)page 109)  The extension of the syzygy functor Z to by Z(.A« * A -* A') = Z(A") •* Z(A) Hence 0  Z(A) •> F(A") -» F(A)  i s given  Z(A* ).  F(A') ^ A -v 0 i s exact  ±n£/&.  (55)  Proof  By the theorem F(A") -» F(A)  exact.  P(A') -» A •» 0 i s  Z(A) = coker (FZ(A) •vPZ(A')) = Z(A") -* Z(A) -*>Z(A ) f  hy the theorem applied to K = K" Remarks  (a)  K -» K'.  (K = Z(A)). //  With t h i s extension of Z (unique up to  natural equivalence) ZF = FZ :  aj*>d/%.  (b)  This extension i s not the syzygy functor associated with <?/4, but the t h i r d syzygy functor. C o r o l l a r y 9.5 (i) (ii)  P i s a half-exact functor o?  <?/g .  P i s l e f t e x a c t < ? / o = 0 ^ a l l short exact  sequences i n Cy s p l i t . (iii)  P r i g h t exact implies p.d. c?/^ - 2. //  C o r o l l a r y 9.6  (remark of Preyd  Z i s an exact functor £/& Proof  Z : ^-A  109)  .  » i s the unique r i g h t exact extension  of Z : °7/(P 4 monies.  -»  (8)page  , hence i t s u f f i c e s to prove Z preserves  Let f : A •» B be monic i n <?/l . A" -» A A' " " f l 1 1 B" -v B 4 B 1  Using the canonical f a c t o r i z a t i o n of f given i n Thm. one can assume A" = B  M  and f " i s the i d e n t i t y , but then  0 4 Z(A) -v P(A") -»• ?(A) 1~ II 1 0 -7 Z(B) -> P(A") P(B) C o r o l l a r y 9.7  , hence Z(A) * Z(B) i s monic. //  ~  Following i s a projective r e s o l u t i o n of A  ...P(Z A) ^ F ( Z A ' ) ->P(Z _ A«) n  6.7,  n  .  n  P(ZA')  1  4...  F(A") 4 F(A) -> F ( A ) -» A 4 0 . // r  (56)  Z(A')—T-P —*A*  , let  1  A" —^—^A  -^A  f  =  1  f , 1  f  0  =  f,  and  f,  =  -w,  t h i s gives r i s e to an i n f i n i t e  f  sequence . ...Z (A) -> Z (A») -j-^ 3n n  n  Z n  -1  ( A W )  - >  •••  . . .Z(A) —*r->>Z(k') — A "  i  C o r o l l a r y 9*8  (i) If f  4  ij  i  -3->  i  2  A—A' 1  factors over a p r o j e c t i v e , then  p.d. A - m - 1. (ii)  In p a r t i c u l a r i f f^ f a c t o r s over a projective then  A = F(A'), so A i s p r o j e c t i v e . Proof  Extend  (#) i n Thm.  given i n Cor. 9.7. bottom row.  If f  corresponding map Remark 9.9 then <(f ,-g>  9.3  to the projective  resolution  Then the sequence of maps f  i s the  factors over a projective then the between exact sequences i s zero. //  I f f : A •> B in07 , and g : Q —=>*B, Q p r o j e c t i v e , : A$Q —»->B, (see 8.6); one can define the  projective dimension of f , as the projective of 0 •* K -> A©Q  B -» 0 i n <2/8 .  C o r o l l a r y 9.10  I f p.d. A" * n p.d. A p.d. A  dimension  p.d. A ^ 3n+1,  ^ n ^ p.d. A & 3n, n f  n 2  0 0  n ^ p . d . A * 3n-1, n £ 1  p.d. A* = 0 ^ A = 0. I f p.d.o?^ n ^ Proof  i 3n-1, n * 1-p.d.07= 0 »  P k i l l s p r o j e c t i v e s , apply Cor.9.7. //  = 0.  (57)  10/  Auslander's f.p. Duality Functor  10.1  Let  denote the subcategory of f.p. r i g h t modules.  For each A of M^,  choose P* •> P •* A -> 0, P',P  i n P^ (f.g.proj.)  (for convenience s e l e c t 0 4 P = P 4 0 f o r A = P proj.) Define D(A) = coker P* 4 P *  (k of Part one).  f  I f f : A 4 B then  P 4 A  P« i  i  1  1  (I)  Q* 4 Q 4 B dualize  Q* -> Q«* 4 D(B)  I  i  i  , l e t D(f) be the representative g  P* 4 P'* •> D(A)  of g in1T(D(B),D(A)).  As f o r the syzygy functor, i f D i s well-defined, i t w i l l be an a d d i t i v e functor that f = 0 implies D(f) = 0.  "^P^'/R^'*  8 0  required  Even more i s true, i f  f factors over a p r o j e c t i v e , then D(f) = 0 so D f a c t o r s through M£/P£. I f f f a c t o r s over Q i n (I) v i a 0 , o  the induced chain map so 9^ e x i s t s  f : TP^ ^>  then by Prop.  i s homotopic  4.2  to zero,  B*—> Q* -» N >—* Q'* — V D(B) P*  Q* 4 N ^Q'*  M —> p»* —>  Q* -> N ^ . Q » *  L(A) Q*  A* -» P* -*> M 4 P»* ( l a s t term i s zero.) = Q*. P*  = Q  1  4  M •* P»*  N M 4 P«*  (58)  since Q* >—> N and M — ^ P ' * , cancel to obtain N-*M=»N-yQ**->P*-*M,  a f a c t o r i z a t i o n over Q**,  hence D(B) -*> D(A) f a c t o r s over P'* by Prop. 6.4., and i s thus the zero map. f  : A -y B induces a unique map $ between kernels of  0 -*> Ext"! (D(A),-) 0  I*  E x t (D(B),-) 1  A©- -*>Hom(A*,-) v * B©- >Hom(B*,-)  (II)  By Prop. 7.9 there i s a unique map of 7T(D(B),D(A)) corresponding to S£, that t h i s i s the map D(f) can be seen from (#1) of Thm. 2.1 (the l e f t l e g ) . Suppose d i f f e r e n t presentations were used, g i v i n g a d u a l i t y functor D°. in (II),  Replace B by A and D(B) by D°(A)  then the induced map Ext (D(A),-) 1  >Exf1(D°(A),-)  i s an isomorphism; hence by Thm. 8 . 4 p r i s e s from a natural isomorphism i n M /P^, n :D°(A)-»D(A). There i s R  A  then p r o j e c t i v e s (which can be taken to be f.g.)  P, Q  such that D°(A)©Q = D(A)©P, and the functors D and D° are  equivalent. D has a l e f t inverse D  1  (use starred sequences as  presentations when constructing D'); then again determine D" such that D"TJ' = 1, by the above D = D", hence DD• = D"D' = 1. We now have: Thm. 10.2 (a) The functor D determines a contravariant equivalence between Mg/P^ and M'/ P". ((2)Prop.2.6,page 52.) R  (b) TT(A,D(B)) Sff(B,D(A))  f  R  ff(D(A),B) =/T(D(B),A), SO  (59)  D i s i t s * own  adjoint on l e f t and r i g h t (viewed as  a contravariant functor. //  (60)  11/  Odds and Ends Consider the map 8 : A*®- •> Hom(A,-) (see Thm. 2,1)  Proposition 11.1  Im 9-g are those £Tof Hom(A,B) which  f a c t o r through a f . g . p r o j e c t i v e . I f f =i;f ®b  Proof  jL  e A*®B, then  i  A ^ )>B 6  T  f  i l  f  pn  1  Conversely i f f f a c t o r s over a f.g. p r o j . , i t f a c t o r s over a f.g. f r e e . [e^  Let f  = A  ±  a basis of R , where A ^ ^ » B  R Remark  t  R and \> = h ( e ) , ±  n  e  n  0  i  s 2 l f j ® ^ * * f . //  n  Im 9-g - P(A,B) and i f A or B i s f . g . there i s e q u a l i t y .  Proposition 11.2 ( i i )  Rn  '^n-1  Proof  ( A  '~  )  ( i ) ITU,-) = Tor (D(A),-) f o r A f.p. 1  = Tor (D(A),-) f o r A f.p. n  ( i ) Both are coker 9, by Thm. 2.1 and Prop. 11.1.  ( i i ) T T _ U , B ) =?T(A,Z n  1  n-1  For a contravariant  (B))= Tor^DCA), Z  n - 1  (B)) =  To.r (D(A) ,B)/,  functor G, the f i r s t r i g h t  s a t e l l i t e of G i s S G(A) = Coker G(Q) •>* G(K) where 1  O-^TC-^Q-^A^O, S G = S (sn-1(G)). n  1  Q proj.  The n& r i g h t s a t e l l i t e  We s p e c i a l i z e G = P(-,B).  Proposition 11.3 (i) (ii) Proof  0 4 Slp(A,B) •> Ext1(A,B) >7T(Z(A),B) -> 0 0  S»P(A,B) ^ Extn(A,B) -^?/(Z (A),B) * 0 n  ( i ) 0 * P(Q,B) •* Hom(P,B) * 0 1 1 1 0 •* P(K,B) -» Hom(K,B) -* (Z(A),B) ^ 0 S P(A,B) 1  Ext (A,B) 1  apply Snake Lemma.  n  (61)  (ii)  snp(A,B) =  S P(Z _ (A),B) 1  n  1  Ext (A,B) = E x t ^ Z ^ U M )  apply ( i ) . //  n  Remarks  (a) Since ^/(P -»  i s a f u l l embedding we  have 0 ->S P(A,B) 4 E x t (A,B) * 1  (Z(A),B)-> 0 .  1  Freyd's Prop. 2.10  (8),  So  (Ext (A,B) ^ % ( Z ( A ) , B ) ) i s n  <  n  i n c o r r e c t , and Prop. 11.3 i s the corrected v e r s i o n ( f o r counter example 07= Ab, then;77(Z(A) ,B) = 0 since Z(A) i s free.^ Prop. 11.3 should be taken i n conjunction with Prop.7.5.  (b)  Since r i g h t s a t e l l i t e s have been introduced, we may as well introduce l e f t s a t e l l i t e s of covariant functor H, (from an abelian category to Ab) S.,H(A) = ker H(K) 4 H(P), where P projective.  S  Proposition 11.4 Proof a  n + 1  0->K*P4A->0,  H = S.,(S H). n  [ E x t ( A , - ) , Hj = S H(A), n>0.((1o)Thm.1.2) n  N  I f 0 ^ K ^ P > A ^ 0 , then (P,-) 4 (K,-) 4 E x t ( A , - ) 4 0 1  P P l y [-»H] and use Yoneda's lemma, £(M,-),H}  = H(M);  0 4 JExt (A,-),H] 4 H(K) * H(P), so r e s u l t holds f o r n =1. 1  Since Ext (K,-) = E x t n  n + 1  ( A , - ) and S H(K) = S n  n + 1  H(A) result  follows by induction. // C o r o l l a r y 11.5  S ^ x t ^ B , - ) (A) = [Ext (A,-), 1  Proof  1.3)  ( ( 1 0 ) C o r . to Thm.  Ext (B,-)J 1  =//(B,A).  By Prop. 11.4 and Prop. 7.9. / /  11.6  Problems  (a)  In order to make the assignment A  I  s  functorial,  one was forced to pass to the quotient category 9/(P,  (62)  which unfortunately was not i n general (Example 8.8).  When i s <3>/(P abelain?  abelian (Probably only  when (P = 07 ) , (b)  By Cor. 9-10 p.d.07= n ^p.d.c?/^ * 3n-1 , i s t h i s  a c t u a l l y an equality, what i s the exact  relationship  between projective dimensions of 67 and c^,^? (c)  07/<P  projectives.  , embeds  <V<P as  a r e s o l v i n g c l a s s of  Are there any other projectives i n ^ ?  Are d i r e c t summands of P(A) isomorphic  to P(A')  some A  (63)  Bibliography  M. Auslander: Coherent Functors, Proceedings of the Conference on Categorical Algebra, La J o l l a 1965 (Springer-Verlag) pp. 189-231. M. Auslander & M. Bridger: Memoirs of the A.M.S., #94  Stable Module Theory,  H. Bass: F i n i t i s t i c Dimension and a Homological Generalization of Semi-Primary Rings, Trans.Amer. Math. S o c , 95 (1960), 46b-488. H. Cartan & S. Eilenberg: Homological Algebra, Princeton U n i v e r s i t y Press, 1956. S.U. Chase: Direct Products of Modules, Trans. Amer. Math. Soc. 97 ( 1 9 6 0 ) , 4 5 7 - 7 3 . P.M. Cohn: On the Free Product of A s s o c i a t i o n Rings I, Math. Z e i t s c h r i f t , v o l . 71 ( 1 9 5 9 ) , pp. 380-398. D. Fieldhouse: Pure Theories, Math. Ann., v o l . 184 (1969), pp. 1-187" ; P. Preyd: Representations i n Abelian Categories, Proceedings of the Conference on Categorical Algebra, La J o l l a 1965 (Springer-Verlag) pp.95-120. P.J. H i l t o n : Homotopy Theory & Duality. Gordon & BreacH~T9"63T  New  York:  P. H i l t o n & D. Rees: Natural Maps of Extension Functors and a Theorem of R.G. Swan. Proc. Camb. PTT1.  soc  Vol.  57  (1961) p. P  469-502.  P.J. H i l t o n & U. Stammbach: A Course i n Homological Algebra, Springer-Verlag, New York, 1971. S. J a i n :  F l a t and FP-In.jectivity. Proc. Amer. Math.  Soc. 41(2)-T9T3 pp.437-442.  "  J.P. Jans: Rings and Homology, Holt, Rinehart and Winston, New York, 1964.  (64)  D.G. McRae: Homological Dimensions of FinitelyPresented Modules, Mimeographed Notes. D.G. McRae: Homological Dimensions of Coherent Rings, Mimeographed Notes. B. Maddox:  Absolutely Pure Modules, Proc. Amer.  Math. Soc. 16 (1967), 155-158. C. Megibben:  Absolutely Pure Modules, Proc. Amer.  Math. Soc. 26 (1970), 561-566.  B. Stenstrom: Coherent Rings and FP-Injective Modules, J . London Math. Soc. (2} V o l . 2 (1970), pp. 323-329. B. Stenstrom: Rings and Modules of Quotients, Lecture Notes i n Mathematics 237, Springer-Verlag, New York, 1971. B. Stenstrom: Rings of Quotients, Springer-Verlag, New York, 1975. C. E. Watts: I n t r i n s i c Characterizations of Some Additive Functors, Proc. Am. Math. Soc. 11 (1960) ^-gS. MacLane:  Homology, Springer-Verlag, B e r l i n ,  1963.  B. M i t c h e l l : Theory of Categories,n-Academic Press, New York,. 1965.  

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