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Finitely presented modules and stable theory Gentle, Ronald Stanley 1976

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FINITELY PRESENTED MODULES AND •STABLE THEORY Ronald Stanley Gentle B.Sc, U n i v e r s i t y of 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 accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1976 (c) Ronald Stanley Gentle, 1976 In present ing t h i s thes i s in part 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 that 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 reference and study. I f u r t h e r agree tha t permiss ion for ex tens i ve copying of t h i s t he s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r ep re sen ta t i ve s . It i s understood that copying or p u b l i c a t i o n of t h i s t he s i s f o r f i n a n c i a l gain s h a l l not be al lowed without my w r i t t e n permis s ion. Department of /V/l~TH^^4T^ S The Un i ve r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1WS Date ^ /~? 6 (ii) ABSTRACT This thesis i s a two pronged a f f a i r . Part one i s a study of f i n i t e l y presented modules using the techniques of homological algebra. 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 highly e f f i c i e n t i n dealing with the theory of f i n i t e l y presented modules. An attempt has heen made to u n i f y many of the r e s u l t s found i n the l i t e r a t u r e , (with the i n c l u s i o n of some o r i g i n a l r e s u l t s ) . Part two, which can he read independently of part one, i s a study of the category of short exact sequences modulo s p l i t sequences. Special a t t e n t i o n i s paid to projectives i n t h i s category; an e x p l i c i t construction of a proj e c t i v e r e s o l u t i o n , with i t s ' consequences, f o r an a r b i t r a r y object i s given. Part two i s related to part one i n providing a c a t e g o r i c a l bedding, thereby enriching the theory of f i n i t e l y presented modules. ( i i i ) TABLE OP CONTENTS Introduction 1 Part One: F i n i t e l y Presented Modules 1/ Pr e l i m i n a r i e s Ie 2/ Du 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/ Generators and Relations ' 27 Part Two: Stable Theory 6/ The Category of E x a c t / S p l i t Sequences 32 7/ Pro j e c t i v e Homotopy 39 8/ K i l l i n g P r o j e c t i v e s 44 9/ Syzygy Functor 51 10/ Auslander's f.p. D u a l i t y Functor 57 11/ Odds and Ends 60 Bibliography 63 ( i v ) Acknowledgements Dr. S. Page receives my acknowledgement f o r h i s advice, guidance, encouragement and f o r planting a seed of an idea. I also wish to thank Dr. J.L.MaeDonald f o r h i s reading of t h i s t h e s i s . R e a l i z a t i o n of the thesis would not of been possihle without the f i n a n c i a l support of the National Research Council. My appreciation to the authors, M. Auslander, P. Preyd and P. H i l t o n f o r f i l l i n g my head with ideas. F i n a l l y my gratitude to Barbara f o r i n s p i r a t i o n and typing. (1), Introduction For a r i n g R, one has the abelian categories of r i g h t and l e f t R-modules. How does the structure of one of these categories a f f e c t the other? For instance, how does the structure of the subcategory of 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 basic device, i n making the passage from l e f t to r i g h t modules, i s the d u a l i t y functor * (Hom(-,R)). Unfortunately however the d u a l i t y functor gr e a t l y a l t e r s s t r u c t u r a l properties; f o r example a f i n i t e l y generated l e f t module i s not necessary car r i e d to a f i n i t e l y generated r i g h t module. A f i r s t attempt, to overcome t h i s i n s t a b i l i t y of the d u a l i t y functor, i s to consider f o r each f i n i t e l y generated l e f t module N, a p r o j e c t i v e presentation P -^N and l e t U° be coker : N* •» P*. N° i s then a f i n i t e l y generated r i g h t module. However the corresponding presentation P* —s*N°, has kernel N* which i s a dual module; so that the set of modules [N°3 i s only, a s p e c i a l subset of f i n i t e l y generated r i g h t modules. This assignment i s thus d e f i c i e n t , and f a c t s to e f f e c t i v e l y r e l a t e the subcategories of f i n i t e l y generated r i g h t and l e f t modules. A l l i s not l o s t , however i f one passes to the more r e s t r i c t i v e subcategory of f i n i t e l y presented modules (the importance of f i n i t e l y presented (f.p.) modules ( l a ) cannot be under estimated, as every module i s the d i r e c t l i m i t of f.p. modules). This t h e s i s demonstrates how the structure of the category of f.p. l e f t modules determines the structure of 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 ) . I w i l l b r i e f l y o utline the contents of t h i s t h e s i s . In section two, Thm. 2.1 establishes the connection between f.p, l e f t and f.p. r i g h t modules. This theorem sets up the dominoes; a l l r e s u l t i n g propositions of t h i s section (and indeed f o r a l l of part one) f a l l e a s i l y . This section incorporates some r e s u l t s of Jans (13) and Bass (3) (who work with noetherian rings) concerning d u a l i t y , and McRae (14) (15) on p r o j e c t i v e dimensions of f.p, modules, but the use of Thm. 2.1 means the proofs are s u b s t a n t i a l l y d i f f e r e n t . A few o r i g i n a l r e s u l t s are also present. • In section three, pure theory i s introduced following Fielhouse (7). Use of Thm. 2.1 immediately shows t h i s pure theory coincides with that of Conn's (6) (using tensor product). Absolute p u r i t y as defined by Maddox (16) i s shown to be equivalent to both the homological concept of f.p. i n a c t i v i t y and copure i n a c t i v i t y of Fieldhouse (7). The basic properties of absolutely p u r i t y are established. Section four covers some r e s u l t s of Stenstrom (18) and J a i n (12) on f.p. i n j e c t i v i t y and coherence, and 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 >for 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 . Both of the above authors, r e l y h e a v i l y on the character functor (-) 0 = HonigC-^/Z), with d u a l i t y formulas (a) ExtJi(p,L°) = Tor n(F,L)°, (b) Ext n(F,M)° = Tor n(P,M°), R r i g h t coherent, P f.p. and with Tor^(M,P) =" Hom(Ext1(P,R),M), P f.p., M i n f e c t i v e , R l e f t coherent. (Cartan and Eilenberg ( 4 ) ) . I f e e l the introd u c t i o n of character modules i n t o t h i s theory i s somewhat a r t i f i c i a l and obscures the r e s u l t s ; once again Thm. 2.1 enables alternate proofs. Flatness can be interpreted i n terms of r e l a t i o n s ( l i n e a r equations!) (Chase ( 5 ) ) . Section f i v e i s an analysis of t h i s r e s u l t , (using 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 of the the s i s i s of a more c a t e g o r i c a l nature, and i s a synthesis of r e s u l t s of Auslander ( 1 ) (2), H i l t o n (9) ( 10 ) and Freyd (8). Part two originated out of an e f f o r t to make Thm. 2.1 (a) (b) f u n c t o r i a l (which I subsequentially found to be h i s t o r y , Stable Module Theory (2), hence t i t l e of thesis) Section s i x contains two major r e s u l t s . Thm. 6.7 (Preyd (8)) gives one a home f i e l d (an abelian category i n which to study the functors of sections eight and ten. Prop. 6.4 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 sect i o n three concerning p u r i t y . I t i s also of utmost importance ( ie) because i t al lows one to determine when morphisms ( i n are zero maps. S e c t i o n seven i s a study of H i l t o n ' s p r o j e c t i v e homotopy (9) (11). A l l r e s u l t s are known W i t h p o s s i b l e exception of Prop. 7.5. S e c t i o n eight puts some r e s u l t s of H i l t o n and Ree (10) and Auslander and B r i d g e r (2) i n t o the c a t e g o r i c a l framework e s t a b l i s h e d by Preyd (8). S e c t i o n nine contains Thm. 9.3, a r e s u l t of my own, concerning p r o j e c t i v e r e s o l u t i o n s i n the a b e l i a n category £/A> ; i t s ' many c o r o l l a r i e s i n d i c a t e i t i s of some value but I wish f o r a more elegant p r o o f . In s e c t i o n ten , we r e t u r n to the subject matter of part one, p u t t i n g Thm. 2.1 (a) (b) i n t o c a t e g o r i c a l language. F i n a l l y s e c t i o n eleven i s of a miscellaneous nature and poses some problems. Through somewhat for tunate circumstances (more enjoyment on my p a r t ) , although most of the r e s u l t s of t h i s t h e s i s are known a l l the proofs given are o r i g i n a l (with exception of parts of Thm. 6.7). E i t h e r I f e l t my own proofs were s i m p l i c a t i o n s , or because the given proofs were bound up i n too much theory (heavy c a t e g o r i c a l machinery) and would take the reader too f a r a f i e l d or simply because no proofs were g i v e n . (Id) I f no reference i s given i n t h i s t h e s i s , the r e s u l t i s o r i g i n a l ; unless I overlooked i t s author, then being only a second creator. A f i n a l note :>—* denotes monic, — » epic, (which i t i s not. Good reading). (le) 1/ P r e l i m i n a r i e s . Let R be an ass o c i a t i v e r i n g with i d e n t i t y , RM the category of 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 ac t i o n ( f r ) (x) = ( f ( x ) ) r fcM*, xeM, reR. The assignment Mi—>M* i s a contravariant functor. The basic facts concerning t h i s functor are stated i n the following theorem. Theorem 1.0. (a) There i s a natural isomorphism 9:Hom(N,M*) = Hom(M,N*) where (6(f)(m))(n) = (f(n))(m). (b) Regarding * as a covariant functor RM—>MR (the opposite category), then * i s i t s ' own l e f t a djoint. As a r e s u l t * preserves c o l i m i t s , but being contravariant t h i s means * transforms c o l i m i t s to l i m i t s (sums to products, cokernels to kernels, pushouts to pullbacks). (c) The un i t (and counit) of t h i s adjunction i s 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 i s i n j e c t i v e (an isomorphism). (d) 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 adjunction i s Hence we have, op njy[:M—y M' (n M(m))(f). = f(m) feM* M i s a s p l i t monic.// (2) The basic example 0—>• L-—»R—>R/L—> 0 , L a l e f t i d e a l , gives 0—>• ( R / L ) * — r R* exact, and (R/L)* = Ann(L) v i a f n ) f (1 ). A module M i s said to 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—yO with P, P' f i n i t e l y generated p r o j e c t i v e . The f u l l sub-category of f i n i t e l y presented (f.p.) modules i s closed under cokernels but not ne c e s s a r i l y closed under kernels. L e f t Noetherian rings are p r e c i s e l y those rings f o r which the category of f i n i t e l y generated (f.g.) modules i s abelian; a r i n g i s c a l l e d l e f t coherent i f the category of f.p. modules i s abelian. A r i n g i s coherent i f and only i f f i n i t e l y generated submodules of f.p. modules are i n turn f.p. modules. (3) 2/ Duality and PiniteJLv_JPresejited_Mod^l^es For reference the following i s a s l i g h t extension of the standard Snake-lemma. Snake Lemma - Ker-Coker Sequence ((4) Lemma 10.1 page 101) 0 —>B1—^2—^B3-^B4-^B5-^... commutative with exact rows, = ker f^, = coker f±, then we have an exact sequence .. * A5 -* A4 * Ki * K2 * K3 * Ci * 02 * C3 *B4 B5 ...// Theorem 2.1. Given a f i n i t e presentation P' P -*• A of a r i g h t module A, there e x i s t s a f i n i t e l y presented l e f t module A such that (a) 0 * 1* + P* * P -» A '+ 0 and 0 •> A* * P* -> P 1* £ * 0 (b) Any f i n i t e l y presented l e f t module B, i s of the form A* f o r some f i n i t e l y presented r i g h t module A (and vice'" versa). (c) 0 * Ext 1 (A,-) -y A0- + Hom(A*,-) Ext2(A,-) -> .0 0 * Tor2(-,A) + -8A* * Hom(A, - ) -» Tor-} (-,A) -» 0 I f 0 - » M - * * N - ? Q - > 0 i s exact then there e x i s t s u, v such that (d) 0 -» Hom(A,M) Hom(A.N) Hom(A,Q) £ ASM + A0N * A0Q v 0 (e) ... Tor! (A,N) -> Tor-j (A,Q) ^  Ext 1(A,M) Ext 1 (A>N) .. (f) Following commutes i n every possible way Hom(l,Q) * Ext 1(A,M) Tori(A,Q) 9 AOM (g) a l l of t h i s i s expressed i n following commutative diagram of exact sequences 0 i 0 V Tor3(A,Q) -^Tor2(A,M)^Tor2(A,N)-^Tor2 Tor* (A?Q) >A*8M >A*8N > A*8Q > 0 * i t I [ G 0 w 0 >Hom(A,M)^Hom(A,N)^Hom(A,Q)-^Ext 1U,M)^Ext1U,N)-^Ext1(A,Q) w -» Tor! (A,M)->Tor1 (A,N)->Tor-| (A,Q) • ASM • ->A8N-V -*A8Q ->0 y 0 0 0 *Hom(AtM)—»Hom(A*N)^Hom(A*Q)-*Ext 1(AtM) > -^Ext 2(A,M)^Ext 2(A,N)->Ext2(A,Q)->Ext3(A,M) > v 0 o o (5) Remarks; Part (a) i s p a r t i a l l y used by everyone, but mostly the At— ? A correspondence,, that (b) A4 - ) i i s generally neglected. Part (c) can be found i n (1): M. Auslander: Coherent Functors, but the .approach i s d i f f e r e n t , more c a t e g o r i c a l . The proof of the theorem w i l l be divided into sections; the only great d i f f i c u l t y i s part ( f ) . Proof: ( i ) Let P'-^-P—-»A—>0, P,P* f.g. and projective Dualize to obtain K and i , such that 0 A* P*-* P'*-* k Dualize again, 0 * p «**.#. P**, but P**=* P and ?•**= p» so coker of t h i s sequence i s l a g a i n A. This establishes (a) and shows A = so also gives (b) by a l e f t - r i g h t switch.-( i i ) For 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 projective (since i t i s true f o r the r i n g R). Using the exact sequence 0 -» K P**-» £ 0 to obtain P'0M ?>P0M > A ® M - * 0 Hom(P'?M) - (#1) 0 Hom(K,M) + Hom(P,M) > Hom(A*M) * Ext1(K,M) * 0 Ext1(A,M) »0 Ext 1(K,M) = Ext2(X.,M) amd everything i s natural with respect to M; so apply the snake lemma (w the connecting homomorphism i s a c t u a l l y an isomorphism) to obtain 0 •? Ext 1 (A,-) -> A0- ^ Hom(At-) -> Ext 2(A,-) + 0 (6) y ( i i i ) Now consider the map N*0Q -> Hom(N.Q) given by (f8q)n = f ( n ) q , which i s an isomorphism i f N i s f.g. and pro j e c t i v e (again because i t i s true f o r R). (Remark: The maps i n ( i i ) and ( i i i ) are connected by the following commutative 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. pr o j e c t i v e s , when P i s i d e n t i f i e d with P * * t h e maps of ( i ) and ( i i i ) are the same. ) Using 0 - v L - » P - > A - > O t o obtain 0 >Tor-,(A,Q) 0 -> Tor-j (L,Q) *A*®Q *P'8Q > L ® Q — > 0 ~ P0Q 0—>Hom(X,Q) —yHom(P'vQ) ->Hom(P*Q) Again the connecting homomorphism i s an isomorphism, Tor-|(L,Q) = Tor2(A,Q) and with n a t u r a l i t y the snake lemma gives 0 -> Tor2(A,-) -> >Hom(i,-) ->Tor-|(A,-) 0 thus (c) has been established. ( i v ) Now suppose 0 ^ M N Q 0, then Hom(A\Q)-*Ext1(A\M) Tori (A,Q) >A8M hor i z o n t a l maps from homology, v e r t i c a l maps from diagrams (#1) and (#2). Commutivity must be v e r i f i e d . To compute Hom(i,Q) -»Ext 1(£,M), use the presentation 0-*K—>P,f->& 0 of A*. The required map i s the connecting homomorphism of: (7) (Hom(A\Q)) I (#3) Hom(P'tM)—^Hom(P'tN)—>Hora(P'tQ) -> 0 I I I 0 — Hom(K,M)-^- i :Hom(K,N)—»Hom(K,Q) I .(Ext1(Jt,M)) S i m i l i a r l y Tor-|(A,Q) A8M i s the connecting homomorphism of the following diagram, using the presentation 0 * L > P - > A - » 0 (Tor-| (A,Q)) + (#4) L8M —>L8N —> L0Q —> 0 J . 1 1 0 P8M P8N P8Q (A8M) There i s a natural map from (#3) to (#4) which induces maps into the kernels of (#4) out of the cokernels (#3)» connecting the two corresponding ker-coker sequences n a t u r a l l y . E x p l i c i t l y : Hom(P'*-) —> P'8 » L8-Hom(K,-) ->Hom(P*-)—>P8-f i l l ' . i n M, N, Q f o r So we have A w Hom(A,Q)—>Ext'(A,M) commuting \ w ^ Tor-} (A,Q) >A6M The f i r s t v e r t i c a l map induced into the kernel Tori (A,Q) of the kercoker sequence of (#4) and the second v e r t i c a l map induced out of the cokernal Ext1(A\M) of the ker-coker sequence of (#3). F i n a l l y to show these maps are the same as those a r i s i n g from (#1) and (#2) i Let X be ei t h e r (8) M, N or Q, the maps connecting the two ker-coker sequences are then the unique maps making the following diagram commute: Hom(i,X)—>Hom(P'TX)—^Hom(K,X)—*Ext1(i\X) : i Y Tor-j(A,X)-P'0X I ->L0X -(#5) Hom(PtX) I s ->P0X- -> A0X Examine diagram (#1) where Ext1(A,M) > A0M i s defined v i a w , and the "snaking"shows i t i s required map making (#5) commute at the r i g h t end f o r X = M. The same applies f o r (#2) with X = Q looking at the l e f t end of (#5). (v) Also: Hom(&,Q)-^Ext1(A,M) Ton(A,Q) >A8M { Y-j the map induced out of the cokernel Tor-|(A,Q) and V2 the map induced into the kernel Ext1(A,M). But i = i _^, and epics can be cancelled so A - — h e n c e v^ = V2. This gives (d) using ker-coker lemma. ( v i ) 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) Tori (A,Q) = ker(A0M * A0N). ker u = ker(Hom(i,Q) * Ext 1 (A*, M) because Ext1 (A\M)>—¥A0M = Im(Hom(A\N) *Hom ( A\Q)). (9) This establishes (e) and ( i v ) , (v) and ( v i ) give ( f ) . and (g) comes from (a) to ( f ) and the snake lemma again. C o r o l l a r y 2.2 Let A he as i n the theorem, then 0 •> Ext1(A*,R) -> A A** + Ext2(A\R) 0 ((11) page 142.} 0 ->» Ext1(A,R) * X + 1** + Ext 2(A,R) 0// Cor o l l a r y 2.5 ((13) page 81) A l l f.p. l e f t modules are t o r s i o n l e s s i f and only i f a l l f.p. r i g h t modules are W-modules (B i s W-module i f Ext 1 ( B,R) = 0 ) . / / Cor o l l a r y 2.4 In the s i t u a t i o n of the theorem, where L and K are defined so that: 0 -> 1* •» P f * P -» A -» 0 and 0 -v A* -v P* P** -» t + 0 L K then (a) 0 -v L * K* Ext1(£,R) -> 0 (a 1 ) 0 * 1 + 1 * 4 Ext1(A,R) * 0 and (b) P< -v K* -y P (b')P* -v L* * P«* commute. L K Proof 0 A* •> P' -> L -V 0 I | H h \ 0 ^ A M P'** -» K* + Ext 1(A,R) -V 0 h i s induced out of the cokernel and the snake lemma gives (a). The l e f t h a l f side of (b) commutes by d e f i n i t i o n of L. f o r r i g h t h a l f side P f K*—>P P 1 — » K * — > P P'—>P P* P L L P*—#L can he cancelled, so the r i g h t h a l f commutes al s o . (a') and ( V ) by l e f t - r i g h t symmetry. (10) C o r o l l a r y 2.5 £(13) page 71) I f A i s a t o r s i o n l e s s f.p. r i g h t module there e x i s t s K a f i n i t e l y generated l e f t submodule of a free module such that: 0 •) A M P M K ^ 0 , 0 - > K * * P - * A ^ 0 0 -> A A** Ext1 (K,R) -v 0. , 0 -> K .-•» K** Ext1 (A,R) -v 0 Conversely such K give r i s e to A t o r s i o n l e s s and f.p. Proof Apply C o r o l l a r y 2.2 and Cor. 2.4 to get L = K* Prom Cor. 2.2 also* 0 A •» A** -» Ext 2(A,R) = Ext1(K,R)* 0 and 0 -> K L* = K»* -» Exf1(A,R) (from Cor. 2.\).// Remarks (a) This establishes a correspondence between A r the cla s s of f.p. t o r s i o n l e s s r i g h t modules and Bi the class .of f.g. modules isomorphic to eubmodules of free l e f t modules. Now i f R i s l e f t coherent B i <= A l , and also duals of f.p. r i g h t s are f.p. l e f t (see ahead Prop. 4.1 ). So f o r AeAr, l e t P-^A* t Pf.g. free, then 0 -» A A** P* embeds A i n a free module so A r ^ Br. This i s the appropriate g e n e r a l i z a t i o n of : R l e f t noetherian,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. free module ((3) 4.5). For R l e f t and r i g h t coherent Ar = B r, A l = B i . (b) The r e s t r i c t i o n that A he f.p. i s not necessary , here i n the following sense: Proposition 2.6 Let A he f.g., define K and L such that 0 - > L - > P - » A - > 0 , 0 - > A * - » P * - » K - > 0 (Pf. g. p r o j . ) then ( i ) 0 K -» L* -> Ext 1(A,R) -> 0 (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 . Proof For ( i i i ) K i s a submodule of 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 i 0 -»K* P**-*A** h i s induced in t o the kernel K*, snake lemma then gives ( i i ) . 0 -v A* -V P M K + 0 II II | k , 0 A* P* L* Ext1(A,R) -> 0 k i s induced out of cokernel snake lemma gives ( i ) . / / C o r o l l a r y 2.7 I f A i s f.g. and t o r s i o n l e s s t h e r e i s a f.g. t o r s i o n l e s s K, such that: ((13) page 71) 0 A -y A** * Ext1(K,R) -> 0 0 K -> K** -> Ext!(A,R) 0 Proof I f A i s t o r s i o n l e s s , then L = K* ( ( i i ) of Prop.) then ( i ) gives 0 K K** •» Ext1(A,R) * 0 Then: 0 - * L - » P - * A - » 0 0 * K»-?-P**->A** Ext1(K,R) 0 gives 0 A A** •+ Ext1(K,R) 0 by snake lemma again.// Thus we have a correspondence between f.g. t o r s i o n l e s s r i g h t and f.g. t o r s i o n l e s s l e f t s . Following two r e s u l t s extend r e s u l t s of Jans,((13) page 73). Proposition 2.8 Af.p. r i g h t , A* = 0, then: ( i ) A®- =?Ext1(A\-) ( i i ) Hom(A,-) = Tori (-,£) 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 ( projective dimension). (12) Proof apply (c) of Thm. 2.1 to get ( i ) , ( i i ) a n d Ext 2(A%-) = 0, hut i f A i s pro j e c t i v e 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 Exf1(A,R)* =0. Proof I f 0 •> P'- -» P •> A -> 0 i s exact, then 1* = 0. Hence by l a s t proposition A" = Ext 1(A,R).// Proposition 2.10 Following are equivalent: ( i ) R i s Regular. ( i i ) A l l f.p. l e f t modules are p r o j e c t i v e . ( i i i ) A l l f.p. r i g h t modules are p r o j e c t i v e . In f a c t A i s pro j e c t i v e i f and only i f A* i s p r o j e c t i v e . Proof C l e a r l y ( i ) ^  ( i i ) and ( i i i ) . ( 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) of Thm. 2.1 A0-^Hom(Af-) implies Ext^A*,-) = 0, hence r e s u l t follows by (b) of Thm. 2.1.// For the moment we jump a dimension. Proposition 2.11 Sup [p.d.A: Aff.p. r i g h t ] £ 2 ^  duals of f.p. l e f t are p r o j e c t i v e . Proof 0 -» X* -> P' P -v A •> 0 gives the res u l t . / / C o r o l l a r y 2.12 ((3) 5^2) I f R i s r i g h t and l e f t noetherian, l e f t global dimension R ^ 2 duals of f.g. r i g h t i s pr o j e c t i v e . Proof f.p. coincides with f.g. and global dimension can be calculated over f . g . / / More generally: Proposition 2.13 ( ( H ) 3.1) Following are equivalent: ( i ) Sup Jp.d.A; A r i g h t f.p.] = n+2 ( i i ) Dual of any f.p. l e f t has p.d.^n.// Proposition 2.14 Let P* •> P -» A and Q' * Q A be two f i n i t e presentations of A. Let B (=£) be coker P*—• p'*, and C be coker Q * -H-Q>**. Then Ext n(C,-) = Extn(B,-). Proof For n = 1,2, follows because they are ker and coker of A®- -> Hom(A*,-) by (c) of Thm. 2.1. For n > 2, 0 A* -> P* P'* -> B * 0 gives Extn(B,-) = Extn-2(A*,-). Proposition 2..15 Following are equivalent: ( i ) R i s l e f t semihereditary. ( i i ) Sup [p.d.B; Bf.p. l e f t ] £ 1. ( i i i ) any f.p. r i g h t A, i s of the form A = A'$P where P i s f.g. proj., A f f.p. and A'* = 0. Proof ( 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 before 0 - y A M P* -»JP'* -> B 0. ' p.d.B $ 1 $ K i s pr o j e c t i v e =7* A* i s pr o j e c t i v e so 0 4 A M P M K •> 0 s p l i t s , so remains ( s p l i t ) exact when dualed, (Also A* i s also f.g. Then 0 •> I P -yA 4 0 0 -» KMP**->A**->0 Snake lemma gives A—*»A**, but A** i s p r o j e c t i v e . Hence: A sr A**$ ker(A +A**) = A**© Ext1(A,R) , (Cor. 2.2) and Ext1(A,R)* = 0 by Prop. 2.9. ( H ) ( i i i ) ^ ( i i ) Given B f.p. l e f t , construct A = B. Then A = A'©P where A»* = 0. Let Q' Q A* -> 0 be exact Q*, Q» f.g. p r o j . then Q« * Q0P-* A*©P s A 0 use t h i s presentation to compute k, since A'* = 0, 0 -» P* -> Q*$P* Q»* -y % •» 0 t h i s gives 0 * Q* -y Q»*-*£ + 0 (the P* just r i d e s along, note & = A * ) , so p.d.A* £ 1 but by prop.2414, p.d.B = p.d.A* <i 1.// Co r o l l a r y 2.16 R l e f t semihereditary. A f.p. r i g h t ( i ) A** = P i s a f.g. p r o j e c t i v e . ( i i ) Hom(A,-) = Hom(P,-) © Tor-|(-,&). ( i i i ) A®- = P8- 9 E x t * l ( ^ , - ) . ( i v ) For l e f t f.p.B, the assignment B v-*-B can he made f u n c t o r i a l , by s e t t i n g B = Ext1(B,R). Proof For ( i ) , ( i i ) and ( i i i ) use (c) of Thm. 2.1, and the f a c t that i f A = A'©P then 1 = , and proof of l a s t proposition that A** i s f.g. p r o j e c t i v e . For ( i v ) choose f o r each B. an exact sequence 0 •> Q' Q -» B -» 0, Q», Q f.g. p r o j . , then B = Ext1(B,R).// Proposition 2.17 A f.p., A i s proj. 4* Ext1 (A,L) = 0 fo r a l l f.g. l e f t i d e a l s L. Proof Using 0 L R R/L 0 i n (e) of Thm. 2.1 gives Tor-| (A\R/L) = 0. Thus 1 i s f l a t , hence p r o j e c t i v e , and so A i s also p r o j e c t i v e . / / Proposition 2.18 R l e f t coherent, A f.p. l e f t . Following are equivalent: ( i ) A i s p r o j e c t i v e . ( i i ) Ext1(A,B) = 0 f o r a l l f.p. B. ( i i i ) 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 i d e a l s are f.p., apply l a s t proposition. ( i i ) ( i i ) Induct on minimal number of generators. For B f.p. l e t x be an element of a minimal generating set. Use 0 -» Rx -y B -VB/Rx 0 and induction i n long exact Ext sequence. // C o r o l l a r y 2.19 ((15) Prop.2) R l e f t coherent, A f.p. l e f t . Following are equivalent: ( i ) p.d. A ± n ( i i ) T o r n + i ( C , A ) = 0. f o r a l l f.p. C (ii- t ) T o r n + i ( C , A ) = 0, f o r a l l f.p. c y c l i c C ( i i i ) Extn+1(A,B) = 0, f o r a l l f.p. B ( i i i 1 ) Ext*+1(A,B) = 0, f o r a l l f.p. c y c l i c B. Proof ( i ) ( i i ) ( i i ' ) gives f l a t dimension A - n, but A i s f.p. and R i s l e f t coherent, so f l a t dimension = p r o j . dimension. ( i ) 4$ ( i i i ) ( i i i ' ) By the l a s t proposition and dimension s h i f t i n g (R i s l e f t coherent). // Proposition 2.20 ((13) page 74) R l e f t coherent, then Sup £p.d.B ; B f.p. l e f t , p.d. B <°°j = 0 f or A f.p. r i g h t , A* = o implies A = 0. Proof (=>) by Proposition 2.8 (4=) by Proposition 2.9- For i f B has p.d. - 1 then |» = 0 ^ t = 0 ^ B i s p r o j e c t i v e . // (16) Proposition 2.21 ((3) 5.3) R l e f t coherent, then Sup £p.d. B : B f.p. l e f t , p.d. B<»j ^  1 f or A f.p t o r s i o n l e s s , A* p r o j . implies 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 there i s a f.p. modul with f i n i t e p r o j . dimension greater than 2, there i s a t o r s i o n l e s s f.p. A with p.d. equal 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 . Then hy Cor. 2.5 A= A** and 0 A* P* K 0, so A* i s p r o j . , hence A i s proj (17) 3/ Absolute P u r i t y ( f . p . I n a c t i v i t y ) In the category £ of short exact sequences,consider the set 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—vA , Ge^then f fa c t o r s over a s p l i t short exact sequence, ( f 0, see ahead Prop. 6.4, so that i n , f = 0; (G,A) = 0 f o r a l l Ger»- Let (R be the cl a s s of 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 absolutely pure ( f l a t ) i f whenever E : 0 4 A - > B ^ C ^ 0 i s exact, then E i s pure. M If Consider the 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 the diagram G : 0 - > K - » - P - » M - > - 0 P f.g. p r o j . , f 1 induced by ft if" if it E : 0 A ->B 4 C ^  0 p r o j e c t i v i t y of P. ,M Since G i s copure and E i s pure, f~ 0 so by-Prop. 6 .4 B—*C f i l l s i n , that i s Hom(M,B) — » Hom(M,C), f o r a l l f.p. M. Proposition 3.1 E : 0 - » A > B - » C - » 0 i s pure i f and only i f Hom(M,B) —>Hom(M,C) -> 0 f o r a l l f.p. M. Proof (^) by above. (<£) by d e f i n i t i o n of p u r i t y and Prop. 6 . 4 . / / C o r o l l a r y 3-2 A i s absolutely pure (C i s f l a t ) i s and only i f whenever 0 - > A - » B - y C > 0 i s exact., Hom(M,B) ->Hom(M,C) > 0 i s also exact, f o r a l l f.p. M. // Lemma 3-3 A % B $ C % » exact, a epic ^ b = 0 # c monic. // (18) Following proposition shows f l a t n e s s and p u r i t y as defined on previous page, coincides with the standard concepts. Proposition 3.4 E : 0 - » A - » B - » C > 0 ( i ) E pure & E0M i s exact f o r any f.p. M (hence, any M), (E®M = 0 -> ASM > B®M -> C0M -v 0) ( i i ) 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 Tori (C,M) = 0 f o r any f.p. M (hence any M) ( i i i ) A absolutely pure 4^  E®M exact f o r any such E in v o l v i n g A and a l l f.p. M. #Ext1(M,A) = 0 f o r a l l f.p. M. Proof ( i ) Apply part (d) of Thm. 2.1, Lemma 3-3 and Prop. 3.1. Hom(fi,B) % Hom(M,C) 4 A®M £ B8M ( i i ) F i r s t equivalence from ( i ) above. For second equivalence, take B pr o j e c t i v e then: 0 Tor-|(C,M) -VA9M 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 previous page) then ( i ) implies Torl(C,M) = 0. Conversely Hom(M,B) ^ Hom(M,C) ^Ext1(ft,A) Tori(M,C) -> M0A Tor-|(M,C) = 0 ^ b = 0 ^ a i s onto f o r a l l M, apply Cor. 3.2. ( i i i ) F i r s t equivalence*by ( i ) . Hom(M,B) ^  Hom(M,C) -*» Exf1(M,A) Exf1(M,B) I f A i s absolutely pure a i s epic, take B i n f e c t i v e , (19) A>» B. Conversely Ext1(M,A) = 0, implies a epic f o r any M, hence A absolutely pure by Cor. 3.2.// A i s c a l l e d copure i n f e c t i v e i f given G : G : 0^G"•>G->G , 0 copure f \ / f Av then f has an extension f . Proposition 3.5 A copure i n j . 4$ A i s absolutely pure. Proof (=^ ) G : 0-s> G "-» G G ' — > 0 , G any copure. f+ f " t f | f ' | "E : 0 —>A —* B —>C —> 0 since A i s copure i n j . f " extends to G, by Prop.6.4, f~0, hence E i s pure, and A i s absolutely pure, «=) given G : 0 - > G H - » G - > G , - » 0 f «• l A Consider a sequence E : 0 - » A - » B e > C * 0 with B i n j . then G : 0 -> G n - > G ^ ^ 0 E : 0 -» A -> B C 0 f " induces f : G E, since B i s i n f e c t i v e ; E i s pure since A i s absolutely pure, hence f^O, and the required extension of f " e x i s t s . // Remark The condition Ext1(M,A) = 0 f o r a l l f.p. M i s c a l l e d f.p. i n j e c t i v i t y . Thus absolute pure = f p. i n f e c t i v e = copure i n f e c t i v e . C o r o l l a r y 3.6 ((17) Cor.2 ,page 562) A i s absolutely pure given N>—»P Iff A with N f.g. and P pro j e c t i v e , f has an extension to P. Proof (^) embed P £ F, F free, then N - F* = F f o r some (20) f.g. d i r e c t summand of F. Then 0 ~i> N-» F T F*/N -*> 0 i s a f i n i t e presentation hence i s copure. f thus has an extension to F' by Prop. 5.5, t h i s can further by extended to F since F' i s a d i r e c t summand and then r e s t r i c t e d to P. (note 0-v N P -> P/N * 0 i s copure). G : 0 —*-G"—> G —>G' —> 0 IT n r I E : 0 — » A —>?B -» C —*> 0 G a f i n i t e presentation, by hypothesis f " extends to G, hence f ~ 0 , so 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 absolutely pure. C o r o l l a r y 5.7 ((16) Thm .2) Suppose (Mi) i s a directed system, such that Mj>-»limMj ( f o r instance suhmodules .of some fix e d M, ordered by i n c l u s i o n ) Then M^ absolutely pure f o r a l l i 1 =^  M absolutely pure. Proof G : 0 -vG" ->G ->G' -» 0, G a f i n i t e presentation limMi since 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 to G, hence f extends to G.// Proposition 5.8 For any family ( A i ) , following are equivalent ( i ) A i absolute pure a l l i . ( i i ) TTAi ab. pure, ( i i i ) ®Ai ab. pure. Proof ( i ) ^ ( i i ) Ext1(M,TfAi) S l E x t l (M,Ai). (21) ( i ) ( i i i ) by Cor. 3.7 ( i i i ) =^  ( i ) d i r e c t summands of ab. pure are absolutely pure.Ext^MjAeA 1 ) = Ext1 (M,A)$Ext1 (M ,A«). / / Proposition 3.9 ((17) Thm;2.) R i s semi-hereditary >^ The homomorphic image of an absolutely pure module i s absolutely pure. Proof Let A f.p. l e f t and 0-*L-*M->N-*0 exact, then Ext1(A,M) ^Ext1(A,N) 4 Ext2(A,L) -»Ext2(A,M). (=>) l e t M be f.p. i n j . , i f R. i s semi-hereditary, p.d. A £ 1. So above sequence implies 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 . (<?) For any L l e t M be an i n j . module containing L. Then N i s f.p. i n j . , hence the sequence implies Ext2(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 In t h i s section, imposing the conditions of coherence and f.p. i n j e c t i v i t y on the r i n g R, are investigated. A r i n g i s l e f t coherent i f f.g. submodules of f.p. l e f t s are f.p; Proposition 4.1 R i s r i g h t coherent 44 duals of f.p. l e f t are f.g. ( i n which case they are also 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 gives 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 hypothesis 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 0 0 ' I I I 0 —>L M > L —> L' —> 0 i l l 0—>P"—j-P'eP"—*P'-*0 P f, P" f.g. proj . 1 - I I 0 —>K > M >M/K—> 0 I i I 0 0 0 by what has just been proved L, L* are 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 I f R i s r i g h t coherent then following ere equivalent ( i ) A i s absolutely pure. ( i i ) Ext1(M,A) = 0 f o r a l l c y c l i c f.p. M. ( i i i ) Iv-*R can be f i l l e d i n f o r I f.g. l e f t i d e a l . V (23) Proof ( i ) (*£) ( i i ) Induct on number of generators and use Ext1(M/xR,A) -» Ext1(M,A) -> Ext1(xR,A) taking out a generator of an a r b i t r a r y f.p. M. (with r i g h t coherence " s h i f t i n g " on the f.p.'s i s allowed), ( i i ) («*) ( i i i ) from Hom(R,A) -yHom(I,A) -» Exf1 (R/l,A) -> 0. // Theorem 4.3 (a) M i s f l a t Q e A : A*®M-^Hom(A,M) f o r a l l f.p. A. I f 9 A i s epic f o r a l l A, i t i s n e c e s s a r i l y an isomorphism. (b) M i s absolutely pure 4A 3?A : A®M>-> Hom(A*,M) f o r a l l f.p. A. I f R i s l e f t coherent, then ih monic f o r a l l A implies •3?A i s a isomorphism. Proof (a) A*8M -y Hom(A.M) -»Tori(i,M) 0 from Thm. 2.1 and kere A = Tor2(i\M). (b) 0 ^Ext1(A\M) ASM ^Hom(A*,M) Ext 2(£,M) 0, from Thm. 2,1, gives the r e s u l t . Ext 1(B,M) = 0 implies Ext 2(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 possible, so l e f t coherence i s needed f o r an isomorphism. // Proposition 4.4 (Stated without proof by Stenstrbm, page 323 (18)). I f A i s f.p. then f o r any d i r e c t system (Mi*) lim Hom(A,Mi) ° Hom(A,lim Mi). Conversely i f lirn^ Hom(A,Mj) —>»Hom(A,lim M-s ) f o r any directed system then A i s f.p. Proof (Remark The map l i m Hom(A,Mi)—»Hom(A,lim Mi) i s the unique map out of the d i r e c t l i m i t , induced by the (24) compatible maps Hom(A,Mi) -> Hom(A,lim> Mi) which a r i s e from Mi—> l i m Mi.) Let P' ^  P A 0 be exact with P, P f f g - p r o j . 0 —>lim Hom(A.Mi)—• lim Hom(P,Mi)—»lim Hom(P,,Mi) O ^ < ^ * > ^ P & ^ W % * > implies h i s an isomorphism. Conversely, A = l i m A j , f o r some directed system (k±) of f.p. modules. I f l i m Hom(A,Aj) —fr>Hom(A,lim A i ) = Hom(A.A) A i then the i d e n t i t y f a c t o r s over some A^ , A = A, hence A i s a d i r e c t summand of a f.p. module A i and i s f.p. // Following i s a r e s u l t of Watts (21). nA Let E be an i n f e c t i v e cogenerator, 0 -»* A—»-EHom(A,E) 0—>B—>EHom(B,E) E ( f ) (e g) geHom(A,E) = (Sgt)geHom(B,E) n A(a) = ( g ( a ) ) g e H o m ( A f E ) E i s a functor and n A a natural transformation from the i d e n t i t y to E. Hence any d i r e c t system can be embedded i n a directed system of i n f e c t i v e modules. Theorem 4.5 ((18) Thm. 3.2) Following are equivalent ( i ) . R r i g h t coherent. ( i i ) The d i r e c t l i m i t of absolutely pure modules i s ab-so l u t e l y pure. ( i i i ) lim Exf1(A.Mi)_=^Ext1(A,liji Mi) f o r every f.p. A and d i r e c t system (Mi). (25) Proof ( i ) ( i i ) For A f.p. l e f t , A* i s f.p. r i g h t t y Prop. 4.1. (lim Mi)®A—>Hom(A*,lim Mi) — ^ | = h | 0 — ^ l i m (Mi0A)—*lim Horn(A*,Mi) h i s an isomorphism by Prop.4.4; bottom row i s i n f e c t i v e since each Mi i s absolutely pure, hence top row i s i n f e c t i v e and by Thm. 4.3, lim Mi i s absolutely pure. ( i i ) £ ( i i i ) Let A be f.p. and (Mi) a d i r e c t system. Embed (Mi) i n a d i r e c t system of i n f e c t i v e modules (Ei) (see note before theorem). Let Ni denote l i m Ni-0 •> Hom(A,Mi) >Hom(A,Ei) * Hom(A,Ei/Mi) -*Ext1(A,Mi) * 0 > | S * ir= > | S > | e 0 * Hom(A,Mi) ^ Hom(AiV-Ei) •> Hom(A,Ei/Mi) ^Ext1(A,Mi) ->Ext1(A,Ei -> -> > -> -> By hypothesis l i m E i i s absolutely pure, hence l a s t term of the bottom row i s zero, implying Q i s an isomorphism, ( i i i ) ^ ( i ) Let K be a f.g. submodule of a f.g. pro 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 proof of Prop.4.1). I f O - > K - * P - * A - * 0 , and (Mi) any d i r e c t system 0 -» Hom(A,Mi) -y Hom(P,Mi) •> Hom(K,Mi) Ext1 (A,Mi) -> 0 > | ^ » L ~ y g — * |=-0 Hom(A,Mi) * Hom(P,Mi) -» Horn(K,Mi) Ext1 (A,Mi) + 0 -> - > - > —> By hypothesis the l a s t map i s an isomorphism, hence g i s also, then by Prop. 4.4, K i s f.p. // R i s r i g h t s e l f - 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 a r i g h t module. Proposition 4.6 ((12) Thm. 2.3) R i s r i g h t s e l f - f . p . i n j . J$ a l l l e f t f.p. are t o r s i o n l e s s . Proof 0 ^ Ext1(A\R) -> A -> A**. // (26) Proposition 4.7 R i s r i g h t coherent and r i g h t s e l f - f . p . i n j . ^ a l l l e f t f.p. are t o r s i o n l e s s and t h e i r duals are f.g. (and i n which case they are r e f l e x i v e and f.p. r e s p e c t i v e l y ) . Proof Combine Prop. 4.1 and 4 . 6 . Reflexive by S s h i f t i j i g . w // Proposition 4 .8 R i s r i g h t 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 i d e a l s 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 ( i ) and ( i i i ) i s equivalent to R being r i g h t coherent ((5) Thm. 2.2) ( i i ) and ( i v ) i s equivalent to property ( i i i ) of Prop 4-2 ( (19),Prop.18,4), hence applying Prop.4.2 gives the r e s u l t . // Proposition 4.9 R r i g h t coherent and r i g h t s e l f - f . p . i n j . I f A i s f.p. r i g h t , then p.d. A = 0 or w. Proof I f B i s f.p. l e f t , and B* = 0 then B = 0 because B i s t o r s i o n l e s s by Prop. 4 . 6 , now apply Prop. 2 . 2 0 . / / Proposition 4.10 ((18) Lemma 4.1) R r i g h t coherent and ri g h t f.p. i n j . then f l a t r i g h t modules are absolutely pure. ( f . p . i n j . ) Proof For A f.p. l e f t 0 —*M«A — M 8 A * * Hom(A*,M) f i s monic because A i s t o r s i o n l e s s (Prop. 4.6) and M i s f l a t , g i s an isomorphism by Thm. 4 . 3 , because A* i s f.p. (Prop.4.1). Hence h i s monic, and by Thm. 4.3, M i s absolutely pure. // Proposition 4.11 ((18) Prop. 4.2) For R r i g h t and l e f t coherent, following are equivalent: ( i ) R i s r i g h t s e l f - f . p . i n j . ( i i ) f.p. i n j . l e f t modules are f l a t . ( i i i ) i n j . l e f t modules are f l a t . ( i ) =^  ( i i ) A = A**by Prop. 4 . 7 . , ¥ i s an isomorphism since R i s l e f t coherent (Thm. 4 . 3 ) . Thus 9 i s an isomorphism and M i s f l a t by Thm. 4 . 3 . ( i i i ) ^ ( i ) I f M i s i n j . , hence f l a t , then by Thm.4.3, © i s an isomorphism. Thus h i s epic f o r any i n j . M, taking M an i n j . cogenerator implies A » A * * , and by Prop. 4.6, R i s r i g h t 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 . C o r o l l a r y 4.12 ((12) Thm.3.3) L e f t IP rings are r i g h t s e l f - f . p . i n j . Proof In ( i i i ) =7" ( i ) of the proposition, coherence was not used. // Proof A f.p. l e f t . Horn (A**, M) -^ H^o^ m (A, M) (Jain(12)) (27) 5/ Generators and Relations A f i n i t e presentation of B with n-generators and m-relations i s an exact sequence of the form Rm R n -» B 0. I f B i s computed using t h i s presentation, the r e s u l t i n g presentation of B has m-generators and n - r e l a t i o n s . Example 5.1 Abelian Groups For a f i n i t e l y generated abelian group, A* = Z r, where r i s the rank of A. I f A = Z/p^z, choose the presentation 0 -» Z Z Z/p^z 0, whsre f i s m u l t i p l i c a t i o n by pr. The dual of f, f * : Z -» Z, i s simply f again. A = A f o r any f i n i t e l y generated abelian group. For any exact sequence 0->M->N-*Q-*0of abelian groups, then Coker f * = = Z / p r z . I t can thus be arranged that A by Thm. 2.1 (g), 0 Tori (A,Q) > km 0 * Hom(A*,M) r A*&Q > 0 Hom(A,Q) —>Ext1(A,M) 0 I f A = Z r$T(A), T(A) t o r s i o n subgroup, t h i s becomes 0 -Q r ^ 0 Q r©Hom(T(A),Q)—^Ext1(A,M) Tor.|(A,Q) >Mr©(T(A)©M) 0 M r (28) That i s Tor-|(A,Q) = Hom(T(A),Q) = Hom(T(A),T(Q)). Ext1(A,M) = T(A)®M Example 5.2 C y c l i c Modules This w i l l be a s p e c i a l case of Example 5-5. Let H be a f i n i t e l y generated r i g h t i d e a l . Say x-j , X 2 , . . . x n ) generates H. Let X = ( x i , X2,...x n) eR n Xr R — » R n , r i g h t m u l t i p l i c a t i o n (action) by X, yI—>• (yx-| ,yx£, . . .yx n) induces 0 -* 1(H) •> R -> R n Rn/RX * 0 , .. \ / CA) R/KH) where 1(H) i s the l e f t a n n i h i l a t o r of H. The dual map of X r i s X i l e t m u l t i p l i c a t i o n (y-j ,y 2 .. .y n) t—$h±y± inducing 0 (R n/RX)* » R n > R * R/H 0 , v \ f CB) Thus R/E = RVRX and (R»/RX)* = £ Y = ( y i , . . . y n ) : 2*171 3 8 o] ( C Y A : ( X , Y ) = inner product nota t i o n ) . 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 Using Cor. 2.2 and 2.4, (A) and (B) above, and the fa c t that (R/H)* = r(H) (page 2), one obtains 0 -> H ^ r l ( H ) (ker : R/H (R/H)**). // Cor o l l a r y 5-4 I f R i s l e f t s e l f - f . p . i n j . every f i n i t e l y generated r i g h t i d e a l i s a r i g h t a n n i h i l a t o r . Proof R/H i s t o r s i o n l e s s f o r a l l f.g. r i g h t i d e a l s H, by Prop. 4.6. // (29) Example 5.5 General Case Any f.p. a r i s e s as coker : R n — » R m where X = (x^-j) , i = 1 ,.. .m , j = 1 , . . .n, x±jeR. So that X^ i s the l e f t a c t i o n by the matrix X on R n, and the dual of XT. i s X r r i g h t a c t i o n on R m (the transpose of the matrix X) 0 kerXx R n R m R m/XR n -* 0 and 0 •> kerX r R m R n R n/R mX + 0 . So RVXR n = R n/R mX and (R^/R^X)* £T kerXi v i a f i — » ( f (e-j),.. .f (en), where [ e ^ a basis of R n, and f i s regarded as an element of ( R n ) * which a n n i h i l a t e s R mX. 5.6 Flatness By Thm. 4.3, and Ex. 5-5,M i s f l a t i f 9 : kerXi®W—>Hom(R*yR mX,W) —> 0 i s exact f o r a l l possible choices of the matrix X. Horn (Rn/RmX,W) = [w : Xw = o] = kerXi - ^ n v i a f t—»(f(ei),...f(e n)), where X acts on the l e f t on Wn i n the natural way. 9 i s then the map y®WK—>yw, yeR n ( ( y i » . . . y n ) ® W H - ^ ( y i w f « y n w ) ) -W i s thus f l a t only i f given any w = (w-j,.. .wn)eW*i, such that Xw = 0, there i s some k and YiekerXi 9 Rn ; bieW, i = 1...k (or YekerXi - M n tic(R) matrices; beWk) with"^Yi®bi»—> w. But under 9,'^Yi®bii—»>Yb; recapping : For any weW*1 such that Xw = 0 there e x i s t s k, YeM n >] c(R), beWk with XY = 0 (30) and w = Yb. Further,flatness need only be tested on c y c l i c modules, that i s when m = 1 (Example 5-2), so one has: Proposition 5 7 (Chase (5)) Following are equivalent: ( i ) W i s f l a t ( i i ) Given any w-j,.,.wneW and xi,...xneR, such that ^ x i w i - 0> "there e x i s t s k, b-jeW, j = 1 , . . . k, YijeR, i = 1,...k ; j = 1,...n, with I ^ x j Y j i = 0 and^Y-jibi = w-j. ( i i i ) Given any w-j,...wneW, and X-yeR, i = 1,...m; j = 1,...k such t h a t ^ X ^ j W j = 0, there e x i s t s k, b-jGW, j = 1,,.,k and Yj^eR, i = 1,.. .n; j = 1,.. .k with"^X i ;jYjk = 0 f o r a l l ( i , k ) a n d ^ Y i ; j b j = w± f o r a l l i . // 5-8 f.p. I n a c t i v i t y By Thm. 4.3 and Ex. 5.5,W i s f.p. i n j . i f ¥ : 0 —^"VXR^w —*>Hom(kerXr,M) i s exact f o r a l l choices of matrices X. R n ® ¥ — - ^ p m / X R ^ ® W — ^ 0 implies (Rm/XR4®W = wm/XWn. From Ex.5. 5 (R m/XR n)* •*= kerXr.via f i - ^ ( f ( e l ) , . . .f ( e n ) ) . Hence ¥ i s wi—>(yi—*yw) ( = wr r i g h t m u l t i p l i c a t i o n by w). w = (wi,.. .w m)eW n/XW m« In co-ordinates ( y i , . . ,y n) H ^ y i W i . , SP i s i n f e c t i v e i f wr = 0 implies wcXWn. Proposition 5.9 W i s f.p. i n j . ^  given W|,..,wmeW (weWm) and XijeR, i = 1...m; j = 1...n (XeM m > n(R)) such t h a t ^ Y i X i j = 0, y teR (yX = 0, yeRm) implies£yi wi = 0 (yw = 0) then wi = ^ X i j b j , f o r some bjcM (w = Xb, f o r some beM n). // ( 3 1 ) C o r o l l a r y 5 . 1 0 I f R i s l e f t coherent. W i s f.p. i n j . Given w-j,.. .wmeW, and xi,...xmeR such t h a t ^ y i x i = 0 , y-^R, i m p l i e s ^ y i w i = 0 (yx = 0 $ yw = 0 ) then f o r a l l i , Wj[ = xib f o r some beM, (then w = xb). Proof For a l l l e f t coherent rin g s , one can check f o r f.p. i n a c t i v i t y on c y c l i c f.p. by Prop. 4 . 2 . Hence by Ex. 5 . 2 , one can take n = 1 i n the proposition. // Co r o l l a r y 5 . 1 1 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) = HW. Proof Let wertflR(H), and x i , . . . x n generate H. I f y x i = 0 f o r a l l i , then yelR(H), and so yw = 0 , hence by the proposition w = ^ x i b i > bieW. // (32) Part Two 6/ The Category of E x a c t / S p l i t Sequences The assignment At—>i i s not i n general f u n c t o r i a l . One of the obstructions being that f o r P f.g. p r o j . one can choose £ to be 0, P* or any other f.g. p r o j e c t i v e . Prom 0 -> P* e» P A 0, and Schanuel*s Lemma; *k* i s uniquely determined up to pro j e c t i v e summands; t h i s suggests that & i s also 'uniquely 1 determined. This i s indeed the case, and we proceed to set up the machinery to demonstrate t h i s f a c t (and also examine the machinery i t s e l f ! ) S t a r t i n g with an Abelian category Oft one can k i l l the p r o j e c t i v e s <P, to form the quotient category <7/(P. However <=7/ff» i s not a 'nice' category, and i n general w i l l not be Abelian. One can also form category <S. of exact sequences; <£ i s never Abelian ((22)page 375), however k i l l i n g o f f the s p l i t exact sequences £ , r e s u l t s i n an Abelian category $/£ (Thm. 6.7 ahead). I f Of has s u f f i c i e n t p r o j e c t i v e s , one can assign to each object A, a s p e c i f i c p r ojective presentation 0 - ^ K - ^ P ^ A + O . This assignment determines a f u l l embedding of <37/(P into c?/$ , f o r which the image of °7/(P constitutes a r e s o l v i n g class of projectives f o r £/$ (Thm. 8.5 ahead), ( k i l l p r o j e c t i v e s only to become p r o j e c t i v e s ) . Hence i t i s natural to work i n the Abelian category c 7 ^ rather than Q/<£. The next three lemmas are recorded f o r reference. (33) Lemma 6.1 ((11)page 83) Given C -£>A bj, is B-=->D then (0->) C^a>b^>A$B ^ r" s^D[—»6] i s exact the square i s a (pull-back), jpush-outj . //' Lemma 6.2 ((11)page 84) 0 -> B E 1 -*> A' 0 II I i 0 -> B E -» A 0 commutative and exact rows, then the r i g h t hand square i s a pull-back and a push-out. // Lemma 6.3 ((23)page 163) Any A ^ B in<S, the category of short sequences, has a f a c t o r i z a t i o n : A 0 ->AM A -v A' 0 * I II 0 ^ B M E ^ A' 0 -ii i i 0 ^ B" -> B -> B* 0 B The objects of £ being exact sequences can be thought of as chain complexes so the notion of homotopy n a t u r a l l y a r i s e s . ->0 A Proposition 6.4 Given 0—>A" - V A - ^ A ' f " B'« g/ i >B / -> B ->0 B following are equivalent: ( i ) There ex i s t s g such that ga = f " . ( i i ) There e x i s t s h such that b'h = f . ( i i i ) There e x i s t s g and h such that bg + ha' = f. (i v ) f fa c t o r s through a s p l i t exact sequence. (34) (v) f i s chain homotopic to zero ( f ~ o ) . Note ( i ) ( i i ) # ( i i i ) Pieldhouse (7), ( i ) # ( i v ) Freyd ( 8 ) . Proof We prove ( i ) ( i v ) , then ( i i ) & ( i v ) i s proved dually. That ( i ) and ( i i ) combined ( i i i ) i s cle a r , and ( i ) ( i l ) . and ( i i i ) c o n s t itute ( v ) . ( i v ) =^  ( i ) 0 —>A" ,—>A — * A , - » 0 The required g i s achieved v i a the proj e c t i o n C&D-^C. ( i ) ^ ( i v ) Consider A"—>A , where the square i s a push-out.. k ex i s t s to give a commutative diagram. Hence B " — » E i s s p l i t monic, and r e s u l t now follows by Lemmas 6.2 and 6.3. /./ Suppose C and ID are a c y c l i c (exact) chain complexes, and f : <C—«>(D, a chain map. Breaking the complexes int o short exact sequences, f induces maps f n between these pieces. Proposition 6.5 I f <C and "P are a c y c l i c , and f~o (chain homotopic), then fn~o. Proof C n + 1 -S-^C \ / K n ij Jn e n - 1 c f a c t o r s through coker ( C n + 1 — > C n ) = K n, so there e x i s t s 6 : K n •Dn, (35) n+1" Jn+1 •n n •K n •n n h Then d'0c' = d'(en_«|c) = d'(fn-den) = d'fn = fn'c' and c' i s epic, so d ' e = f n . Hence by Prop. 6.4 the induced map f n : ( K n + 1 C n -v K n) (L n +-| -*> D n + L n ) i s homotopic to zero. // There i s a p a r t i a l converse, whose proof i s the same as showing, two chain maps fP-^(B, inducing the same maps H0((P) H0((8) are then homotopic. ((11)page 127). Proposition 6.6 Let P be a pro j e c t i v e r e s o l u t i o n of A, and B an a c y c l i c complex, suppose f : A B 0 f a c t o r s over B. ?2 B 2->B 1 0 0 then the induced chain map P A © i s homotopic to zero, where P A i s IP with A adjoined. Proof By induction, to construct the homotopy Pn+1 — » Pn %+1 •n B n+T Pn-1 fn/B rt n i l t B n - ^ B n - 1 Jn+1 M*n - e nd n) = b n f n - ( f n _ i - en-^ n.^ dn = b n f n - f n ^ A n P i r s t equality by induction; so f n - 9 n d n factors through ker b n = L n + 1 , but P n i s proj e c t i v e and B n+i-+>L n +i, so (36) 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 to B N + - | , g i v i n g 9 N + 1 . / / For each horn set ( A , B ) i n £ , those f-vo form a subgroup, and induce an equivalence r e l a t i o n compatible with the add i t i v e structure of cf. Consider the quotient c a t e g o r y , whose objects are those of <£ but whose hom sets are ( A , B ) / A J . Theorem 6.7 (Freyd (8) Thm. 3.3) i s Abelian. Note: 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 of the kernel and cokernel of a morphism and i t s canonical f a c t o r i z a t i o n ,for f u r t h e r propositions. Proof <£/-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 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 g a kernel ((20)page 87). Given f : A 4 B , we w i l l show 0 4 A" -4 B"©A-4 E 4 0 II 0 -> A"-— 4 A —y A 1 4 0 I II 0 4 B" > E -4 A 1 4 0 II \f V 0 4 B" — > B — » B' 4 0 II 0 4 E --4 B©A'-4 B f + 0 represents 0 4 ker f 4 A 4 i m f 4 B 4 coker f 4 0. By Lemma6-3. f = gh. The exact sequences at top and bottom r e s u l t from Lemma6-1, using Lemma6-2 and i t s dual. (37) We prove (a) k = ker f (b) £ = ker 1 then dua l l y (a') 1 = coker f ( 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" > A >A' i f kx = 0 then 6 e x i s t s by Prop. 6.4; the same 9 then shows x = 0. ( i i ) hk = 0 : A" — > B M © A , take 8 to be the projection. — II / A" B" ( i i i ) Suppose hx = 0 : X"—>X—4X', so that 9 e x i s t s | Q/l | A" 7^ j>A—>A* with the properties W I I B"—> B —»B' of Prop. 6.4. then X"- * X — > X ' gives a f a c t o r i z a t i o n of x thru k. I l l " A " — * B W © A — » E II t i A" > A >A» (b) ( i ) £ i s monic, proof same as f o r k. ( i i ) l g = 0 : B"—>E , take 9 to be i d e n t i t y . II /e Y' Mr ( i i i ) Suppose l x = 0 : X" ^X >Xf then 9 e x i s t s as — i yi I B"-A>B » B f i n Prop. 6.4. U I II E *B©A' —>B' l e t x° be : X " — > X — * X ' , x - x°r*>0 because B " — » E —»A' l e f t l e g of x - x° i s £ II 1 I • " ~ B"—>B—>B f the zero map. (38) Hence i n x = x ° , and x can be factored through g . (39) 7/ Projective Homotopy Now assume <7 has enough p r o j e c t i v e s ; l e t (P he the f u l l subcategory of p r o j e c t i v e s . Proposition 7.1 Por f : A—*B, the following are equivalent: ( i ) f can be factored through some p r o j e c t i v e . ( i i ) f can be factored through any pr o j e c t i v e Q, such that Q - » B . ( i i i ) f can be factored through any C — » B . Proof (9)page 131. (proof i s straightforward). // f i s p r o j e c t i v e l y homotopic to g i f f - g fact o r s through a p r o j e c t i v e . Let P(A,B) be the subgroup of (A,B) c o n s i s t i n g of those maps which can be factored through a p r o j e c t i v e . Let 1T(A,B) = (A,B)/P(A,B). Proposition 7.2 I f e : Q —«>B, Q projective then ?l(A,B) = coker e* : (A,Q) —->(A,B). Proof lm e* i s the set of f which can be factored through Q - » B , hence equals P(A,B) by Prop. 7.1. // Co r o l l a r y 7.3 ((11)page 135). The functor ?((A,-) i s a d d i t i v e . Proof To evaluate 77(A,B©B f), take Q © Q + > B©B 1. // Let Q n Q n - 1 -v . . . Q , be a pr o j e c t i v e r e s o l u t i o n of B, with n t o syzygy S n. Define 7fn(A,B) =7T(A,S n); since 7T(A,-) k i l l s p r o j e c t i v e s , Schanuels Lemma gives 7Tn(A,-) i s independent of choices (40) of presentations. Proposition 7.4 ((11)page 142) Given 0-^B"-vB-*B' 0 there i s an exact sequence . . . ^?r n(A,B) ^ T r n(A,B«) -v?r n-i(A,B") +?rn^Ut*) ... . . . -> TT(A»B) If (A,B«) -> Ext1(A,B") -» Exf1(A,B) •* ... In p a r t i c u l a r ^(A,-) i s half-exact. Proof Construct 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 l e t = ©'©(Q", then 0 -» (A,Q G») ^ (A,Q 0) (A,Q 0M * 0 I Y 0 -* (A,B") (A,B) ^ (A,B") Ext 1(A,B") -> ... apply snake lemma and Prop. 7.2 to get 0 -» (AfS.,") * (A,S.,) (A.S^) ^?7(A,B H) -v ?f(A,B) -> ^(AjB' ) ->Ext1(A,B") ->Ext1(A,B) * ... S 1 = ker : Q 0 B ( f i r s t syzygy). Going hack another step, using Prop. 7.2 0 -> ( A , C y ) * (A,Q 1) -> (A,Q^ •) -» 0 J/ ^ ^ 0 (A.S.,") (A,S.,) -> ( A ^ ' ) *17(A,B") ^(A,B) + ... tfuts/) ^ T f d s ^ V T T C A . S ^ ) apply snake lemma, noting T/"(A,S^ ) = T f^AjB) then induction completes the sequence. // Proposition 7.5 Let P Q -> P n 1 -* ... -» P Q Z n ( A ) he a pr o j e c t i v e r e s o l u t i o n of A, with n t t syzygy Z n (A). I f Q -^B, Q pr o j e c t i v e then f o r n>0 (41) Ext n(A,Q)—> Extn(A,B)-*tf(Z n(A),B) 0. Proof n = 0 i s Prop. 7.2 (Z Q(A) = A) For n> 1 replace A by Zn_-|(A) to reduce to the case n = 1. Let P = P 0 and Z = Z<|(A), i f 0 S * Q •> B •> 0, apply snake lemma to 0—>(P,S) > (P,Q) > (P,B) > 0 I 1 I 0->(Z,S) >(Z,Q) »(Z,B) >Tf(Z,B)-4 0 I I i Ext1(A,S)— j r Ext1 (A,Q)->(Ext1 (A,B) // Proposition 7.6 ((11)page 142) L n(A,-) =7Tn+l( A»-)» n * 1 -(Ln the n t o l e f t derived functor) Proof Prom ->Q2—4 AQQ. \ / \ / s 2 s 1 one obtains (A,Q 2) (A,Q 1) (A,Q Q) \ / \ V (A,S 2) (A,S^) ker a* = (A,S 2), regarding (A,S 2) - (A,Q^) then im b* are those fe(A,S 2) which f a c t o r through Q 2 •» , so im b* = P(A,S 2) by Prop. 7.1. Then the f i r s t l e f t derived functor of (A,-) : L.,(A,B) = H1((A,<0)) = ker a*/im b* = 7(A,S 2) =7T 2(A,B). Por n*1, L n(A,B) = L ^ A . S ^ ) = -M 2 ( A' Sn-1 ) = " n+lU,B). //. Any f : A 4 B induces a natural transformation Ext1(B,-) Ext1(A,-), which can be computed i n terms of extensions by using pull-backs: (42) 0 4 c — E ' —•» A -V o £. II : I t 0 4 C > 2 > B 4 0 £ Proposition 7 . 7 ((10)Cor. to Thm. 1.3) Every natural transformation 0 : Ext 1(B,-) 4- E x t 1 (A,-), i s induced by a map f : A 4 B. Proof Let 0 * K * > ? * ' A * > 0 and O ^ I » Q ^ B » O b e proj e c t i v e presentations. 0 4 (B,-) 4 (Q,-) -v (I,-) -> Ext 1(B,-) 4 0 P1I P 2 l • F 3 | I 6 (*> 0 4 (A^-) * (PV) (K,-) 4 Ext 1(A,-) -* 0 P^  e x i s t because representables (M,-) are p r o j e c t i v e , (by Yonedas Lemma) i n the functor category Ah0*. Also by Yonedas L.each P i i s induced from some f ^ such that: O ^ K ^ P ^ A - ^ O F 3 0->>L-^Q->B->0 Then f^ induces a natural transformation Ext 1(B,-) ->Ext 1(A,-) which also makes (*) commutative, by the uniqueness of maps induced out of the cokernel, t h i s map must be 0. // Proposition 7 . 8 f : A -> B induces the zero map f° : Ext 1(B,-) 4 Ext 1 (A,-) f fa c t o r s through a p r o j e c t i v e . Proof 0 4 C - » E ' ->A-*0 II J I* 0 * C 4 E B 4 0 (43) f° = 0 ^ top row s p l i t s f o r a l l extensions of B ^ there e x i s t s g such that C — E ' & there e x i s t s h such that A , by Prop. 6.4. >K if E -*B f f a c t o r s over a pro j e c t i v e , by Prop. 7.1. // Proposition 7.9 ((2) Thm. 1.40) There i s an exact sequence 0 *>P(A,B)-* (A,B) 4 JExt1(B,-), Ext1(A,-)] -5> 0. Hence 77(A,B) = [Ext1(B,-), Ext1(A,-)]. Proof The l a s t map i s onto by Prop. 7.7, and exactness at the middle by Prop. 7-8. // (44) 8/ K i l l i n g P r ojectives 8.1 Consider the quotient category <37/(P, whose objects are those ofcy, but with horn sets T f i A ^ B ) . For each A of 07 chose a pr o j e c t i v e presentation 0 + K - » P - * A * 0 , then A 0 - > K - » P - » A - » 0 1 i I [ 1 f * * v B 0 -> L * Q * B V O F(B) 4 P(A) constitutes a functor F : Ol •» c°/^. In f a c t , i f f induces two maps f ' , f " : F(A) F(B), then f ' - f " " 0 , hence f ' = f " ing/&, so P(f) i s well-defined. I f F' were defined using d i f f e r e n t presentations, then 0 -J> K -> P ->> A -> 0 F(A) ft = *A 0 K' •» P' A 0 F 1 (A) 1 - $A®k"Q since r i g h t l e g of 1 - i^QA i s "the zero map A A, hence inCL/^, ¥ A and 6^ are inverses of each other, and determine a natural equivalence between the functors F and F*, To determine ker F, suppose f : A B, F( f ) = 0 P(f )"0 f fact o r s over Q -j- B 4) f = 0 i n °//<P Hence F fact o r s 07-^C?/^ '® which embeds °7/(P as a f u l l subcategory of the abelian category . Following extends a r e s u l t of H i l t o n and Ree ((1-0) Thm.2.1). Theorem 8.2 f : A B, following are equivalent ( i ) Ext1(B,-)>->Ext1(A,-). ( i i ) there i s B* such that Ext1 (B,-)<BExt1 (B',-) ~ Ext1(A,-). (45) ( i i i ) Given Q—^B, Q p r o j e c t i v e , then B i s a d i r e c t summand of A©Q. (i v ) B i s a d i r e c t f a c t o r of A i n <V(P, that i s f i s s p l i t epic i n oi/(P. (v) F(B) i s a d i r e c t f a c t o r of F(A) i n £/l , that i s F ( f ) i s s p l i t epic i n . (v i ) F ( f ) i s epic. Proof ( i i i ) ( i i ) take B 1 to he complement of B i n A@Q. ( i i ) =*> ( i ) i s c l e a r . ( i ) ( i i i ) I f g :. Q B form K E e ' + Q®A >B E-i e Ext 1(A,K) f ° K E 9 e Ext1(B,K) by taking the pull-back, v : A >$> Q©A the i n c l u s i o n . By Prop. 6.4, © e x i s t s (because v does), hence the top ? row s p l i t s , and ( i ) implies the bottom must also s p l i t since f£ ± a i n - j e c t i v e # I^AQQ^-^B where <f,-g>r i s i d e n t i t y map of B. ( i i i ) ^ ( i v ) B In °7/(Pf v and p are isomorphisms, hence B A ft*"§^v> B, gives B as a d i r e c t f a c t o r of A i n ^/(P. ( i v ) (v) ( v i ) c l e a r . ( v i ) =^  ( i i i ) Prom proof of Thm. 6.7, coker P(f) i s 0 -» L —> Q —£-> B ->0 so i f P(f) . i s epic I I II 0 -p- K A©Q gz"^ B 0 then bottom s p l i t s . // (46) This theorem does not 'dualize' s a t i s f a c t o r i l y , f o r example the dual of ( v i ) l P ( f ) being monic: from Thm. 6. ker 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 top row s p l i t s , so Ext 2(B,-) = E x t 1 ( L , - ) = Ext 1(L©P,-) = Ext 1(K,-)©Ext 1(E,-) = Ext 2(A,-)©Ext 1(E,-) and Ext 2(A,-) i s a d i r e c t summand of E x t 2 ( B , - ) , t h i s i s not the corresponding dual of ( i i ) of the theorem.As f o r the dual of ( i ) , the following: Proposition 8.3 ((IQ)Thm. 2.2) Ext 1(B,-) — E x t 1 ( A , - ) there e x i s t s E, and a proj e c t i v e P, and an exact sequence 0 P A©E B •¥ 0. Further P can be chosen to be any proj e c t i v e such that P—**A. Proof (4>) From long exact Ext sequence. (£) Given 0 - * K - > P - » A - » 0 i n Ext 1(A,K), by sur j e c t i v i t y , there i s 0 - > K ^ E ^ B ^ 0 i n Ext 1(B,K) such that 0-^K + P + A ^ O II 0 - » K - » E + B - » 0 then by Lemma 6.2, 0 P -*> A©E B 0 i s exact. // Theorem 8.4 ((12)Thm. 1.44) Following are equivalent ( i ) Ext 1(B,-) = Ext 1(A,-) ( i i ) There e x i s t s p r o j e c t i v e s P and Q such that A©Q ^  B©P ( i i i ) A = B i n q/<P ( i v ) F(A) ^ F(B) i n £/S . (47) Proof ( i i ) =^  ( i ) i s c l e a r . v U-k t^^^ If ( i ) * ( i i ) i f g : Q — » B ; 0 P — ^ Q © A ^ ^ B 0 by Thm. 8.2 t h i s sequence s p l i t s , f u r t h e r P i s pr o j e c t i v e since Ext 1(B©P,-) = Ext(Q©A,-) = Ext(A,-) implying Ext 1(P,-) = 0. ( i i i ) ( i v ) because °f/(P i s a f u l l embedding* ( i ) $ ( i i i ) f : A ^ B fa c t o r s as A A A © Q - % B © P - ^ ? B as i n the proof of ( i ) ^  ( i i ) . But v and p are 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 induces the zero map Ext 1 (A,-) ->Ext 1(A,-) by Prop. 7.8. Hence Ext 1(A,-) = Ext 1(A,-) vh° /f* Ext 1(B,-) . // Theorem 8.5 ((8)Cor. 2.9) Oi-?-* , i s a f u l l embedding of i n , and lm F = C7/(P i s a f u l l subcategory of r e s o l v i n g projectives Proof F i r s t statement i s contained i n section 8 . 1 . (a) a//(j? resolves f o r 0 - ^ K - ? - P ^ A , ^ 0 = F(A') II 0 - ? A " * A ^ A f y O = A can be f i l l e d i n , and the r e s u l t i n g map F(A')-> A i s epic in£/^ (by proof of Thm. 6.7) (b) F(A) i s proj e c t i v e f o r any A. For 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 epic by Thm. 8.2. // Remark 8.6 Given f : A B, i f g : Q - » B, Q pr o j e c t i v e then A>^A©Q , and i n <?/(P, v i s an isomorphism, hence B rep l a c i n g f by ( f , - g ) t one can assume f i s epic i n 07. Proposition 8.7 (Freyd(8)) tf/(P has weak kernels (no uniqueness property required). Proof Let f : A 4 B, by above remark assume f i s epic i n Oj. I f 0 e> K 4 A -> B 4 0 i s exact i n Oj, then K -»• A i s a weak kernel of f i n £7/<P. f g = 0 i n <T/(P, means fg factors over a proj e c t i v e P i n : X-&»A-£->B, p a i s induced since P i s p r o j e c t i v e . Then f ( g - ab) = 0, so g - ab fa c t o r s through K = ker f inCT, but g - ab = g i n <7/P, so g facto r s through K i n Q/(£, Proof 2 (Freyd) Let Q—S4B, take pull-back E—>Q , then E -> A i s a weak kernel of f . A -£>B (use Prop. 7.1). // Example 8.8 Cf/W w i l l not i n general be Abelian. For example i f 0/ = Ab, proje c t i v e s = free s , then i n Ab/^the canonical map i s both epic and monic but i s not an isomorphism. Proof : Hom(<p,Z) = 0 implies Hom((D,F) = 0 f o r F f r e e . Suppose Q -» X in Ab, t h i s map remains epic i n Ab/y, f o r i f Q -» X A i s a f a c t o r i z a t i o n F (49) over a free P, then necessarily <R 4 X 4 A = 0, hence X 4 A i s zero. Thus in particular <Q 4 Q/Z i s epic. By Prop. 8.7 Z » <Q i s a weak kernel of this map, hut this is the zero map Ah/?', hence i t i s actually 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 resolving projectives of an abelian category G, then the inclusion (B 4 C preserves kernels. Proof Suppose K •> C i s the kernel of C -> D in (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 in CB. K 4 C 4 D g exists since L = ker C 4 D i n G, B—->=>L and K 4 C monic implies g i s monic. h exists since K = ker C 4 D i n <B. K C K—»C C B L B B —>~L and the monic L •» C can be cancelled hence K i s commutative, implying /I B-»L that g i s also epic, g i s then an isomorphism and K 4 C is also the kernel of C 4 D i n 6. // Corollary 8.10 F : °7/<P y> preserves kernels. // (50) Corollary 8,11 (a remark of Freyd (8)page 99) If oT/(p has kernels then p.d. - 2 . Proof For N i n , choose F(B) -> F(C) -» N •* 0, B, C InO? , by the proposition 0 4 F(A) 4 F(B) F(C) * N 0 for some A. Since F(A) i s projective by Thm. 8.5, 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 (for convenience i f A i s projective chose A —»A, and i f A i s f.g. chose Pf.g.). Let Z(A) = leer (P -> A). 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 A , l e t Z(f) be the representative of i i f L •=? Q B the induced map K L in1j'(K,L). If Z(f) i s well-defined, then Z i s clearly an additive functor Q •> CT/P. To do this, i t w i l l suffice to consider the case f = 0 and prove Z(f) = 0, that is Z(f) factors over a projective. However even more is true, i f f factors over a projective, so does the induced map. In fact, i f f factors over Q in the above diagram, then by Prop. 6.4, K -> L factors over P. Thus Z factors : 0}K-^h> Q7/(P. \ S The functors P and Z can be related by F(A) =(0 -> Z(A) * P * A * 0) Suppose Z and Z* are different syzygy functors, arising from different presentations chosen. Consider K* = K' ty I K -> N •» P» II ty ty K •> P -> A Let n 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 t t syzygy functor can be defined by Z n(A) = Z(Z Q_ 1(A)), (where Z i s now regarded as a functor °7/(P ~$> C7/CP) 9.2 ((8)Prop. 1.2) For any Abelian category C, i f © i s a f u l l category of resolving projectives, then any functor G : (B CD, fl) abelian,has a unique extension to a right exact functor C "4 CD. E x p l i c i t l y for each CeC, chose B 1 -> B -» C -» 0, B* , B i n (B, and define G(C) = coker : G(B') ^G(B). Theorem 9.3 Let A = 0•*A ^-»•A-»A , e> 0, the following is an exact sequence in , with P, P', P" projective, w 0 K" = K " —*> K ->> K' -*> A" -> 0 0 -> K P" -»P-^P' -» A 0 1 -w I i I (#) 0 K' A -*> A A' = A' -* 0 Proof Let P» — » A ' , and P"—se»A", set P = P'©P", then 0 0 0 0 0 0 1 V * v< * V I -» A' 0 0 0 The maps r , s, u, v result from the s p l i t t i n g of P" -> P P', via canonical projections and injections, and give the properties of Prop. 6.4. (53) (I) Making use of Thm. 6 . 7 , to start a projective resolution of A : K t (b't W ) > A " © P ' < " f ' ^ A w the induced map. b» ^ a* ^ > P* >A' Lower right square I ' £ l £ , ll A " > A > A ' commutes by ( + ). (II) Building on the kernel of (I) and using Thm. 6 . 7 again K > K ' © P > A " © P ' K >P > A k-| |(-r,p) || K ' T b V ^ A " W <-f,B> > A however this time we must check bottom squares commute. For the lower l e f t , i t i s required that b'k' = pb, which is clear from (+), and wk' = -rb. Por the second equality apply the monic f : f(wk' + rb) = 8b'kr + (a - sp)b = sb*k' - spb = 0 a sim i l i a r calculation for lower right square. ( I l l ) Continuing to project on the kernel, the obvious choice i s K w (°>b"l> P (P»r)>P«eA» I I II K * K'©P >P'©A However, (p,r) : P = P'©P" —^'©A", (p,r) = 1©a" so the top row i s isomorphic, in , to 0 K" -*» P" "•A" 0, and we use this for the projective: ( 5 4 ) K „ ( kr b") > K©P" <l(kSb),-(Q>i)>> K, ©P b" K M > P" (0,i) <-w,-r> (0,-1) •©A" K (k«,b)^ K ,® P <-(w,b«),{-r,p)>P' Lower l e f t commutes by (+). Lower right <- (w V ) , (p,-r)> (0,i) = ( p , - r ) i = (0,-ri) = (0,-a») using (+). (IV) The kernel of (III) i s isomorphic to K" K -» K' via 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 $ this i s clear, and for 0, upper l e f t square commutes directly from ( + ). Finally 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. // Corollary 9.4 (i) (remark of Freyd, (8)page 109) The extension of the syzygy functor Z to i s given by Z(.A« * A -* A') = Z(A") •* Z(A) Z(A* ). Hence 0 Z(A) •> F(A") -» F(A) F(A') ^  A -v 0 i s exact ±n£/&. (55) Proof By the theorem F(A") -» F(A) P(A') -» A •» 0 i s exact. Z(A) = coker (FZ(A) •vPZ(A')) = Z(A") -* Z(A) -*>Z(Af) hy the theorem applied to K = K" K -» K'. (K = Z(A)). // Remarks (a) With this 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 third syzygy functor. Corollary 9.5 (i) P i s a half-exact functor o? <?/g . ( i i ) P i s l e f t exact<?/o = 0 ^ a l l short exact sequences i n Cy s p l i t . ( i i i ) P right exact implies p.d. c?/^ - 2. // Corollary 9.6 (remark of Preyd (8)page 109) Z i s an exact functor £/& -» . Proof Z : ^ -A » i s the unique right exact extension of Z : °7/(P 4 , hence i t suffices to prove Z preserves monies. 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 factorization of f given in Thm. 6.7, one can assume A" = B M and f" i s the identity, but then 0 4 Z(A) -v P(A") -»• ?(A) , hence Z(A) * Z(B) i s monic. // 1~ II 1 ~ 0 -7 Z(B) -> P(A") P(B) Corollary 9.7 Following i s a projective resolution of A ...P(ZnA) ^F(Z nA') ->P(Z n_ 1A«) 4... . P(ZA') F(A") 4 F(A) -> F(A r) -» A 4 0 . // (56) Z(A')—T-P 1 —*A* , l e t f 1 = f 1 , f 0 = f, and f , = -w, A" —^ —^ A -^A f this gives rise to an i n f i n i t e sequence . ...Zn(A) -> Z n(A») -j-^ Z n - 1 ( A W ) - > ••• 3n . . .Z(A) —*r->>Z(k') — A " - 3 - > A — A ' i 4 i j i 2 i 1 Corollary 9*8 (i) If f factors over a projective, then p.d. A - m - 1. ( i i ) In particular i f f^ factors over a projective then A = F(A'), so A is projective. Proof Extend (#) i n Thm. 9.3 to the projective resolution given in Cor. 9.7. Then the sequence of maps f i s the bottom row. If f factors over a projective then the corresponding map between exact sequences i s zero. // Remark 9.9 If f : A •> B in07 , and g : Q —=>*B, Q projective, then <(f ,-g> : A$Q —»->B, (see 8.6); one can define the projective dimension of f, as the projective dimension of 0 •* K -> A©Q B -» 0 in <2/8 . Corollary 9.10 If p.d. A" * n p.d. A ^ 3n+1, n 0 p.d. A ^ n ^ p.d. A & 3n, n 2 0 p.d. Af n ^p.d. A * 3n-1, n £ 1 p.d. A* = 0 ^ A = 0. If p.d.o?^ n ^ i 3n-1, n * 1-p.d.07= 0 » = 0. Proof P k i l l s projectives, apply Cor.9.7. // (57) 10/ Auslander's f.p. Duality Functor 10.1 Let denote the subcategory of f.p. right modules. For each A of M^ , choose P* •> P •* A -> 0, P',P i n P^ (f.g.proj.) (for convenience select 0 4 P = P 4 0 for A = P proj.) Define D(A) = coker P* 4 P f* (k of Part one). If f : A 4 B then P« P 4 A i i 1 1 Q* 4 Q 4 B (I) dualize Q* -> Q«* 4 D(B) , l e t D(f) be the representative I i i g P* 4 P'* •> D(A) of g in1T(D(B),D(A)). As for the syzygy functor, i f D i s well-defined, i t w i l l be an additive functor "^P^'/R^'* 8 0 required that f = 0 implies D(f) = 0. Even more is true, i f f factors over a projective, then D(f) = 0 so D factors through M£/P£. If f factors over Q i n (I) via 0 o, then by Prop. 4.2 the induced chain map f : TP^  >^ i s homotopic to zero, B*—> Q* -» N >—* Q'* — V D(B) so 9^  exists P* M Q* 4 N ^Q'* Q* -> N^.Q»* = Q*. P* 4 M •* P»* —> p»* —> L(A) Q* A* -» P* -*> M 4 P»* (last term i s zero.) = Q1 N M 4 P«* (58) since Q* >—> N and M —^P'*, cancel to obtain N-*M=»N-yQ**->P*-*M, a factorization over Q**, hence D(B) -*> D(A) factors over P'* by Prop. 6.4., and is thus the zero map. f : A -y B induces a unique map $ between kernels of 0 -*> Ext"! (D(A),-) A©- -*>Hom(A*,-) I* v * (II) 0 Ext 1 (D(B),-) B©- >Hom(B*,-) By Prop. 7.9 there i s a unique map of 7T(D(B),D(A)) corresponding to S£, that this i s the map D(f) can be seen from (#1) of Thm. 2.1 (the l e f t leg). Suppose different presentations were used, giving a duality functor D°. Replace B by A and D(B) by D°(A) in (II), then the induced map Ext 1(D(A),-) >Exf1(D°(A),-) i s an isomorphism; hence by Thm. 8.4prises from a natural isomorphism in MR/P^, nA:D°(A)-»D(A). There i s then projectives (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 D1 (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 RM'/RP". ((2)Prop.2.6,page 52.) (b) TT(A,D(B)) Sff(B,D(A)) f ff(D(A),B) =/T(D(B),A), SO ( 5 9 ) D i s i t s * own adjoint on l e f t and right (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 factor through a f.g. projective. Proof If f =i;f j L®b i e A*®B, then A 6^ f)>B T f i l pn 1 Conversely i f f factors over a f.g. proj., i t factors over a f.g. free. Let f± = A Rn R and \>± = h(e i), [e^ a basis of R , where A ^ ^ B » t n e n 0 s2lf j®^** f. // R n Remark Im 9-g - P(A,B) and i f A or B i s f.g. there i s equality. Proposition 11.2 (i) ITU,-) = Tor 1(D(A),-) for A f.p. ( i i ) ' ^ n - 1 ( A ' ~ ) = Tor n(D(A),-) for A f.p. Proof ( i) Both are coker 9, by Thm. 2.1 and Prop. 11.1. (ii)TT n_ 1U,B) =?T(A,Z n - 1(B))= Tor^DCA), Z n - 1 (B)) = To.rn(D(A) ,B)/, For a contravariant functor G, the f i r s t right s a t e l l i t e of G i s S1G(A) = Coker G(Q) •>* G(K) where O-^TC-^Q-^A^O, Q proj. The n& right s a t e l l i t e SnG = S 1(sn-1(G)). We specialize G = P(-,B). Proposition 11.3 (i) 0 4 Slp(A,B) •> Ext1(A,B) >7T(Z(A),B) -> 0 ( i i ) 0 S»P(A,B) ^  Extn(A,B) -^?/(Zn(A),B) * 0 Proof ( i ) 0 * P(Q,B) •* Hom(P,B) * 0 1 1 1 0 •* P(K,B) -» Hom(K,B) -* (Z(A),B) ^ 0 S1P(A,B) Ext 1(A,B) apply Snake Lemma. (61) ( i i ) snp(A,B) = S 1P(Z n_ 1(A),B) Ext n(A,B) = E x t ^ Z ^ U M ) apply ( i ) . // Remarks (a) Since ^ /(P -» i s a f u l l embedding we have 0 ->S1P(A,B) 4 Ext 1 (A,B) * (Z(A),B)-> 0 . So Freyd's Prop. 2.10 ( 8 ) , (Ext n(A,B) ^  < % ( Z n ( A ) , B ) ) i s incorrect, and Prop. 11.3 i s the corrected version (for counter example 07= Ab, then;77(Z(A) ,B) = 0 since Z(A) i s free.^ (b) Prop. 11.3 should be taken i n conjunction with Prop.7.5. Since right 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 0 - > K * P 4 A - > 0 , P projective. S n + 1H = S.,(SnH). Proposition 11.4 [Ext n(A,-), Hj = SNH(A), n>0.((1o)Thm.1.2) Proof If 0 ^ K ^ P > A ^ 0 , then (P,-) 4 (K,-) 4 Ext 1(A,-) 4 0 aPPly [-»H] and use Yoneda's lemma, £(M,-),H} = H(M); 0 4 JExt1(A,-),H] 4 H(K) * H(P), so result holds for n =1. Since Ext n(K,-) = Ext n + 1(A,-) and SnH(K) = S n + 1H(A) result follows by induction. // Corollary 11.5 ( (10)Cor. to Thm. 1.3) S^xt^B,-) (A) = [Ext 1(A,-), Ext 1(B,-)J =//(B,A). Proof By Prop. 11.4 and Prop. 7.9. / / 11.6 Problems (a) In order to make the assignment A Is functorial, one was forced to pass to the quotient category 9/(P, (62) which unfortunately was not in general abelian (Example 8.8). When i s <3>/(P abelain? (Probably only when (P = 07 ) , (b) By Cor. 9-10 p.d.07= n ^p.d.c?/^ * 3n-1 , i s this actually an equality, what i s the exact relationship between projective dimensions of 67 and c^,^? (c) 07/<P , embeds <V<P as a resolving class of projectives. Are there any other projectives i n ^ ? Are direct 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: Stable Module Theory, Memoirs of the A.M.S., #94 H. Bass: F i n i t i s t i c Dimension and a Homological Generalization of Semi-Primary Rings, Trans.Amer. Math. Soc, 95 (1960), 46b-488. H. Cartan & S. Eilenberg: Homological Algebra, Princeton University Press, 1956. S.U. Chase: Direct Products of Modules, Trans. Amer. Math. Soc. 97 (1960) , 457 -73 . P.M. Cohn: On the Free Product of Association Rings I, Math. Zeitschrift, vol. 71 (1959) , pp. 380-398. D. Fieldhouse: Pure Theories, Math. Ann., vol. 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. Hilton: Homotopy Theory & Duality. New York: Gordon & BreacH~T9"63T P. Hilton & D. Rees: Natural Maps of Extension  Functors and a Theorem of R.G. Swan. Proc. Camb. PTT1. s o c Vol. 57 (1961) Pp. 469-502. P.J. Hilton & U. Stammbach: A Course in Homological  Algebra, Springer-Verlag, New York, 1971. S. Jain: Flat 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 Finitely- Presented 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} Vol. 2 (1970), pp. 323-329. B. Stenstrom: Rings and Modules of Quotients, Lecture Notes in Mathematics 237, Springer-Verlag, New York, 1971. B. Stenstrom: Rings of Quotients, Springer-Verlag, New York, 1975. C. E. Watts: Intrinsic Characterizations of Some Additive Functors, Proc. Am. Math. Soc. 11 (1960) ^-g-S. MacLane: Homology, Springer-Verlag, Berlin, 1963. B. Mitchell: Theory of Categories,n-Academic Press, New York,. 1965. 

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