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On chain maps inducing isomorphisms in homology Nicollerat, Marc Andre 1973

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ON CHAIN MAPS INDUCING ISOMORPHISMS IN HOMOLOGY by MARC-ANDRE NICOLLERAT Licencie" es Sciences Mathematiques, U n i v e r s i t y of Lausanne, 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the .Department of MATHEMATICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA February, 1973 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Br i t ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of MATHEMATICS The University of Br i t ish Columbia Vancouver 8, Canada Date February 3, 1973 i i i . ABSTRACT Let A be an a b e l i a n category, J the f u l l subcategory of A c o n s i s t i n g of i n j e c t i v e o b j e c t s of A , and K(A) the category whose o b j e c t s are cochain complexes of elements of A , and whose morphisms are homotopy c l a s s e s of cochain maps. In ( 5 ) , lemma 4.6., p.. 42., R. Hartshorne has proved t h a t , under c e r t a i n c o n d i t i o n s , a cochain complex X' e. |K04)| can be embedded i n a complex I* €. |K(J) [ i n such a way that I" has the same cohomology as X*. In Chapter I we show that the c o n s t r u c t i o n given i n the two f i r s t p a r t s of Hartshorne's Lemma i s n a t u r a l i . e . there e x i s t s a f u n c t o r J : K(A) y K ( J ) and a n a t u r a l t r a n s f o r m a t i o n i : I d ^ ^ »• EJ (where E : K ( J ) >• K(4) i s the embedding fu n c t o r ) such t h a t i . i s i n j e c t i v e and induces isomorphism i n cohomology. The question whether the c o n s t r u c t i o n given i n the t h i r d p a r t of the lemma i s f u n c t o r i a l i s s t i l l open. We a l s o prove that J i s l e f t a d j o i n t to E, so that K ( J ) i s a r e f l e c t i v e subcategory of K.C4). In the s p e c i a l case where A i s a category T^Tc of l e f t A-modules, and C(^Tfc) the category of cochain complexes i n jTf\ and cochain maps (not homotopy c l a s s e s ) , we prove the e x i s t e n c e of a f u n c t o r c<.Am) C ( I ) . In Chapter I I we study the n a t u r a l homomorphism v : Hom.(M,L)<a N »• Hom.(M,L® N) A a A B where A, B are r i n g s , and M, L, N modules or chain complexes. In p a r t i c u l a r we g i v e s e v e r a l s u f f i c i e n t c o n d i t i o n s under which v i s an isomorphism, or i v . induces isomorphism i n homology. In the appendix we give a d e t a i l e d proof of Hartshorne's Lemma. We think that t h i s i s u s e f u l , as no complete proof i s , to our knowledge, to be found i n the l i t e r a t u r e . TABLE OF CONTENTS Chapter I : Co-chain complexes c o n s i s t i n g of i n f e c t i v e s Chapter II : The group homomorphism v.: H o m . ^ L ) ® N • Horn. (M,L<2>_,N) Appendix : Proof of Lemma I. 1 Proof of Proposision I. 6 Bibliography v i . ACKNOWLEDGEMENTS - The author wishes to express h i s thanks to h i s supervisor Dr. A. F r e i f o r suggesting the topic of t h i s thesis and f o r h i s h e l p f u l advice and encouragement during i t s preparation. The most generous f i n a n c i a l support of the U n i v e r s i t y of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. 1. CHAPTER I. COCHAIN COMPLEXES CONSISTING OF INJECTIVES. We f i r s t introduce some notations which w i l l be used i n t h i s chapter. If A denotes an abelian category, we denote K(A) the category whose objects are cochain complexes of objects of A, and whose morphlsms are homotopy classes of cochain maps. A cochain complex X* i s said to be bounded below i f there e x i s t s an integer N such X n - 0 i f n > N. We denote by K (A) the f u l l subcategory of KG4) c o n s i s t i n g of the complexes bounded below. According to (1) and (4), a subcategory A1 of an abelian category A i s t h i c k (or i s a Serre subcategory of ^ 4) i f (i ) /!' i s a f u l l subcategory of A. ( i i ) A1 i s non empty. ( i i i ) For any short exact sequence 0 >- B' • B >• B" >- 0 i n A, B t \A' I <^> B' , B" £ \A< | ( 2 ) ( ( i i i ) i s equivalent to ( i i i ) ' : i f X > Y >• Z i s exact i n AS then X, Z 6 \A' I ~Y 6 \A' |). If J4' i s a th i c k subcategory of A we define K^,C4) to be the f u l l subcategory of K(/l) c o n s i s t i n g of those complexes X' whose cohomology objects H^(X') are i n A1. We define K^,(A) by taking complexes bounded below. ^^Note that the d e f i n i t i o n of a thick subcategory i s not absolutely c l e a r i n (5) We denote by |^ 4'| the cla s s of objects of the category A1. 2. If i4' i s a f u l l subcategory of i4, I i s an A - i n j e c t i v e object of A 1 i f I e \A* I and i f I i s i n j e c t i v e , considered as an object of A . A cochain map f : X* > Y* i s c a l l e d a quasi-isomorphism (abbreviated quiso) i f H 1 ( f ) : H 1(X') • H^Y') are a l l isomorphisms. S i m i l a r l y , we c a l l a homotopy cla s s a quasi-isomorphism i f one (hence a l l ) of i t s representatives induces isomorphisms i n cohomology. I f A i s isomorphic to B, we write A = B, and i f A i s homotopic "\> to B, we write A - B. We now state the lemma we are going to discuss ((5), Lemma 4.6, p.42) : Lemma 1. Let A be an abelian category. a) Let IP be a subclass of \ A \ , and assume (i ) every object of A admits a monomorphism into an element of TP . Then every X' & |K 04)| admits a monic quasi-isomorphism i in t o a bounded below complex I" of objects of IP. b) Assume furthermore that IP s a t i s f i e s ( i i ) If 0 • X > Y-—y Z > 0 i s a short exact sequence with X IP , then Y e IP <?=t> Z <= TP . ( i i i ) There e x i s t s a p o s i t i v e integer n such that, i f X° — y 7} — > . .. — y X n * — y X n — > 0 i s an exact sequence, and X° ,XX ,. . . ,X n _ 1 e. TP , then X n & TP . Then every X" £ |K(/1)| admits a quasi-isomorphism into a complex I' of objects of TP . c) Let A 1 be a thick subcategory of A , and assume that A } has enough 4 - i n j e c t i v e s . Then every X' e |K , (A)\ admits a q u a s i -isomorphism i n t o a bounded below complex I" of , 4 - i n j e c t i v e o b j e c t s of A\ We give the c o n s t r u c t i o n of the quiso i : X* *• I " of Lemma 1 a), (For a complete proof of Lemma 1, see the appendix). We may assume X P = 0 f o r p < 0. Embed X° > *• 1°, w i t h 1° i n TP . d°i i 4 Having d e f i n e d I >• I *• . . . we choose l p + ^ to be an element of IP c o n t a i n i n g the pushout of jP-1 ± > I P and i k : X k -> I (k=0,.. ,p) , Then, d P and i P + 1 * * x p + 1 i P p-1 Cok d are defined by the f o l l o w i n g commutative diagram : xp _ J U x ^ 1 C o k d p - i kP+I \ Now we prove t h a t , i f IP i s the f u l l subcategory of A c o n s i s t i n g of a l l i n j e c t i v e o b j e c t s of A, then the c o n s t r u c t i o n given i n Lemma 1 a) i s f u n c t o r i a l . More p r e c i s e l y : l e t A be an a b e l i a n category w i t h enough i n j e c t i v e s . Let I be the f u l l subcategory of A c o n s i s t i n g of a l l i n j e c t i v e o b j e c t s of A, and l e t E + : K +(I) K (A) be the embedding f u n c t o r . Then P r o p o s i t i o n 2. + + + There e x i s t s a f u n c t o r J : K (A) —»• K (J) and a n a t u r a l t r a n s f o r m a t i o n 1 : I DK +C4) — E + J + such t h a t , f o r each X* <& |'K+0Q | : ( i ) i . i s a quiso ( i i ) i . i s monic. A. (K +(J) denotes the f u l l subcategory of K +(4) whose o b j e c t s I* are such t h a t I n e | j | f o r a l l n ) . I f A has enough i n j e c t i v e s , Lemma 1 a) c l e a r l y holds f o r IP =? . We -prove that the c o n s t r u c t i o n i s -functo-ri-a-1. I t s u f f i c e s to prove that i f X', Y" ^  |K+C4)| and i f I" (resp. J') i s a complex cons t r u c t e d from X' (resp. Y'), then, f o r any K +04)-morphism f : X' >• Y* , there e x i s t s a unique <j> : I" »• J " such t h a t the diagram X* > —> V I J commutes. Y* •> ^ »• J * ( i : X'>—>• I" and j : Y"' *• J' are the embeddings of Lemma 1 a ) ) . Indeed, l e t L* be a complex constructed from Z' e. |K+G4)|, and l e t g : Y" *• Z' be a morphism. One checks t h a t ( 1 ) *JL°*IJ = *IL and that Id . : X' • X' y i e l d s .A. (2) <j>II = I d ] [ < Moreover, i f I j and I ^ are two complexes constructed from X", there e x i s t s a unique n = <j> : I j —>• I ' such that the diagram 1 I 1 I 2 1 / ± l X* > I J n i X a £ • —. a * " X2 commutes. But i ^ and i ^ are q u i s o ' s , so that n^ . i s a quiso from a bounded below complex c o n s i s t i n g of i n j e c t i v e s i n t o 1^. Hence n i s an isomorphism ( ( 5 ) , p. 41). Thus, we are allowed to i d e n t i f y I J and 1^ v i a the isomorphism n j-» and we w r i t e J + X ' f o r I * . F i n a l l y , we have to d e f i n e , f o r a given f : X' *• Y" , an induced morphism J f : J X' > J Y" . Or course, we d e f i n e J f = <(> : I > J * . I f we do so, then the formulas (1) and (2) w i l l ensure that J i s a f u n c t o r . The only thing- l e f t to check i s that the d e f i n i t i o n of J + f i s compatible w i t h the i d e n t i f i c a t i o n made under n, i . e . t h a t , i f I j , I ^ (resp. J j , J'^) are constructed from X' (resp. Y * ) , then ( J , ^ = " j w h i c h ± g c l e a r s i n c e both make the f o l l o w i n g diagram commutative : a) Existence_of We c o n s t r u c t <J>P by i n d u c t i o n : as J° i s i n j e c t i v e , there e x i s t s a map <j>° : 1° *• J° such that .o „o 1 T o X > >- I i i ! 6° I <P Y° >-J > J° commutes. Suppose now that <J>°, <f>P are d e f i n e d , and consider where I P = Cok d P 1 , J P = Cok d P 1 K P + 1 = p u s h o u t ( d P ; i P ) , 1 J , X L P + * = pushout(d P; j P ) . Then, cj>P ^, (j)P induce <j>P on cokernels. By u n i v e r s a l property of the pushout, we get a map ^ : K P +* >• L P + * such that the upper cube commutes. As J p " ^ i s i n j e c t i v e , ip induces a map <j>P*^  : I P * ^ *• J^+^ such that the square K P + 1 • L P + 1 I P + 1 v J P + 1 commutes. I t i s then easy to check that the bottom square of the diagram p. 6 commutes, so that <J> i s the r e q u i r e d map. b) Uniqueness. Suppose that there e x i s t ^ and ^ such that <f>^ i - ( J ^ i . Consider the short exact sequence (*) : 0 >• X' I" — ^ Y* > 0. From the long exact sequence of (*) i n cohomology, one sees that Y" i s a c y c l i c . In the diagram we have (<f> - ( f ^ ) ! - 0, so that <j>^  - §^ f a c t o r s through Y* : there e x i s t s a map f : Y' > J ' such that fp - <j>^  - c)^. As Y" i s a c y c l i c , and J " c o n s i s t s of i n j e c t i v e s and i s bounded below, we have f - 0 ( ( 5 ) , p.40). Thus, <j>j - if^ - fp - 0, and t h i s concludes the proof of Prop. 2. The f u n c t o r J + has the other n i c e property to send q u a s i -isomorphisms i n t o isomorphisms. More p r e c i s e l y : P r o p o s i t i o n 3. I f f : X >• Y' i s a quasi-isomorphism i n K + C 4 ) , then + + . + . J f : J X" • J Y* i s an isomorphism. Proof. From Prop.2, we have a commutative diagram i x - E + J + X ' E + J + f Y* — — • E + j V where i , i and f are quiso's. Thus J f i s a quiso. But J X" i s a A. I + bounded below complex c o n s i s t i n g of i n f e c t i v e s , so that J f i s an isomorphism ( ( 5 ) , p.41). We are now able to s t a t e a f i r s t u n i v e r s a l property f o r j " * C o r o l l a r y 4. + + + The f u n c t o r J : K (A) • K (J), s a t i s f i e s the f o l l o w i n g u n i v e r s a l property : For each quiso f : X" > Y" there e x i s t s a unique homotopy c l a s s + + <j> : Y'->E J X' such that the diagram X* f E+ J V commutes, Proof. a) E x i s t e n c e . As J f i s an isomorphism by previous p r o p o s i t i o n , i t i s s u f f i c i e n t to take <f> = ( E + J + f ) _ 1 i v . 9. b) Uniqueness. Suppose <J>' : Y' >• E + J + X * i s such t h a t <j>'f i , and consider the diagram X" > ^ • E + J + X * > l E + J + X " + ' CD Y" •+ E + J + E + J + X ' E +J +(f) E + j V E +J*V + + + + As <j>f = (j>'f, we have ( J <J>) ( J f ) = ( J <f>')(J f ) . But f i s a q u i s o , so that + + + J f i s an isomorphism; thus J <j> = J <f>'. By commutativity of ( 1 ) , we get ^"Vx** = i E + J + X ' * ' ' i E + J + X " b e i n g m o n i c > w e h a v e <f> = P r o p o s i t i o n 5. Notations being as i n Prop.2_, the fu n c t o r J i s l e f t -adjoint to the embedding f u n c t o r E : K (J) *• K (.4). In other words, K (J) i s a r e f l e c t i v e subcategory of K (A) Proof. By Prop.2 , we have a n a t u r a l t r a n s f o r m a t i o n 1 : I DK + G 4 ) - > E V Furthermore, f o r each Y* e. |K (J)|, we have J E Y* = Y' by d e f i n i t i o n of J . Hence, we have a n a t u r a l t r a n s f o r m a t i o n 6 : J V — l d K + ( I ) which i s the i d e n t i t y . Thus, i t i s s u f f i c i e n t to prove t h a t ( i ) 6 + o J + i = Id + ( i i ) E 6 o i E + = Id E+ ( i ) • Apply Cor o l l a r y 4 for Y" = E + J + X ' X V E +J +X* s 10. E +J +X* The unique <j>' such that the diagram commutes i s defined by • « ( E + J ^ . ) " 1 ! ^ . But, also <J> = Id +_+ . makes the diagram commutative. Thus £i J A E J i . = i_+_+ v., and, by construction of J , i + •+ . = Id +T+ . , so that E + J + i v . = Id„+ T+ V. = E + I d T + v . . Thus : J + i . = Id T+ v. . As 6_+v. = I d J + X ' » X L J A J X A J A . J A we get 6 T+ V. o J i . = Id T+ v., and ( i ) i s proved. S i m i l a r l y , one proves ( i i ) J X X J X Remark. Cor o l l a r y 4 and Proposition 5 give two d i f f e r e n t u n i v e r s a l properties f o r i X' quiso -> E + J + X ' X' X E +J +X" + E I ' C o r o l l a r y 4 Proposition 5 (Note that, i n general, Prop.5 Cor.4). Up to now, we have only discussed a p p l i c a t i o n s of the f i r s t part of Lemma 1. Looking at the second part, we get the following r e s u l t : Let A be an abelian category of f i n i t e i n j e c t i v e homological dimension, l e t J be the f u l l subcategory of A c o n s i s t i n g of i n j e c t i v e objects of A, and l e t E : K(J) >- K(A) be the embedding functor. Then : 11. Proposition 6. There e x i s t s a functor J : Y.(A) *• K ( I ) and a nat u r a l transformation i : Id w.>. > EJ such that for each X " £ | K ( J4)| ( i ) i i s a quiso X ( i i ) i i s monic. A. The proof of t h i s p r o p o s i t i o n depends h e a v i l y of the construction of the chain complex I* of Lemma l b ) , so that i t w i l l be done i n the appendix. The question whether J i s l e f t adjoint to the embedding E i s s t i l l open. Now we r e s t r i c t our at t e n t i o n to the case where A i s the category T^Tl of l e f t A-modules (we could get s i m i l a r r e s u l t s f o r the category Td^ of ri g h t A-modules). In t h i s category, we prove the existence of a nat u r a l embedding of any module into an i n j e c t i v e module. As a consequence of t h i s f a c t , we s h a l l see that, i n j^d , the functor J + of Prop.2 may be considered •f" + -t as a functor C (^ITc) >• C ( J ) , where C 04) denotes the category whose objects are cochain complexes of objects of A and whose morphisms are cochain maps. Thus here we are working with cochain maps instead of homotopy classes of cochain maps. Lemma 7. For any A <= \^fft |, l e t A be the r i g h t A-module A* = Hom^ (A ; 0}/Z ) Then the map : A >• A defined, f o r a e A, f & A , by <Ka)(f) = f(a) i s a natural embedding. 12. Proof. 1. I t i s clear that I|J e HomA(A;A ). (Note that A e | A"ITl| implies A e. I ^ y j which implies A <£. | ^ T7\ | ) . 2. <£_is a monomorghism. Let a ^ 0 be an element of A, and l e t (a) be the submodule of A generated by a. We define a : (a) > dj/Z as follows : — I f a i s of i n f i n i t e order, a (a) i s any non-zero element of (Q/'Z . — I f the order of a i s n, a(a) i s an element of order d i v i d i n g n. C l e a r l y , a i s a w e l l defined morphism. Considering now the embedding (a) > > A we get, since (Q/2Z i s i n j e c t i v e , a non-zero map f : A *• flj/ZZ such that (a) ? >• A S • s s commutes. Hence, for t h i s p a r t i c u l a r morphism f, we have ijj(a)(f) = f(a) = = a (a) 4- 0, so that a 4 0 implies iKa) 4 0. 3. ^ _ i s n a t u r a l . ~ kk kk kk kk k For any f : A • B, V7e define f : A • B f o r a e. A and <J> £ B by ** f (a) (<!>) = a(<j>f). It i s then easy to see that *A ** A > A f kk f B > B commutes. 1 3 . Theorem 8. Every A-module can be embedded i n a n a t u r a l way i n t o a c o f r e e (hence i n j e c t i v e ) module. In other words, there e x i s t s a f u n c t o r L : ^Tft —»- T^T\ and a n a t u r a l t r a n s f o r m a t i o n u : Id ^  —>- L such t h a t , f o r each A-module M, ( i ) LM i s a cofree module, ( i i ) y w i s a monomorphism. M Proof. We know that every A-module A i s quotient of the f r e e module generated by the u n d e r l y i n g set of A. Hence, there e x i s t s a n a t u r a l epimorphism A, . w- A. Thus the composition ^ x, 0 0 X£ M M ^ U M**= Horn,, (M*;Q/2Z)> , K o m ( © A ) -x-e-M . n . Horn (A, . ;(Q/Z3 ) = LM xs a n a t u r a l embedding and, by A ZZ (x) x e. A d e f i n i t i o n , LM i s a cofree module. We can now apply theorem 8 to Hartshorne's c o n s t r u c t i o n to get the f o l l o w i n g r e s u l t : Let C+(yJR) be the category whose o b j e c t s are bounded below cochain complexes of elements i n , and whose morphisms are cochain maps. I f c J denotes the f u l l subcategory of T^ft c o n s i s t i n g of i n j e c t i v e modules, and i f E : C (I) —* C ( TTi ) i s the embedding f u n c t o r , then 14. Proposition 9. There e x i s t s a functor J + : C +(^ul) >- C + ( J ) together with a natural transformation 1 : I d c + ( A ™ ) — * E + J + such that, for each X" & | c + ( A T i l ) | , ( i ) i i s a quiso A ( i i ) i i s monic. A. Remark. Note that, i f we work i n the category C (^Tft) (and not i n K (^ft.)), the + + functor J i s no more l e f t adjoint to the embedding functor: i f J were l e f t + adjoint to E , then the counit 6 : J + E + — + I d c + ( J ) of the adjunction would be an isomorphism since E + i s f u l l and f a i t h f u l ( ( 8 ) , p.88). As E +6 o i„+ = Id„+ , i„+ would be an isomorphism; whence, to ij b h + + prove that J i s not l e f t adjoint to E , i t i s s u f f i c i e n t to show that i E + i s not an isomorphism. F i r s t , note that i f Hom(B;q/ZZ ) : Hom^ (AjOj/Z ) >• Hom^, (B; fl}/2Z ) i s epic, then 3 : B »- A i s monic. Indeed, suppose that x i s a non-zero element of B such that B(x) = 0. By an argument s i m i l a r to the one used i n the proof of Lemma 7, one shows that there e x i s t s f : B > $I7L such that f(x) 4 0. As Hom(B;Bj/Z) i s epic, there e x i s t s g : A • d)/Z such that g6 = f. Then, f (x) •= g(B(x)) = 0 since 3(x) = 0, c o n t r a d i c t i n g the f a c t that f(x) * 0. Also, i f A i s a non-zero A-module, the canonical epimorphism 15. • • : © A ( X ) — A X £ A i s not monic, so that, for a non-zero A-module M, Hom^ <<f>;q/ZZ > : Hom^ (M*;0J/ZZ) > * Hom^ ( £B ftA(x);Q/Z ) x & M i s not epic, and the map : M — L M of theorem 8 i s not an isomorphism. + 4. . j . . f . < Thus, the map '• E X* >• E J E X i s not an isomorphism. L X In Chapter I I , we s h a l l use the dual of Lemma 1 ; so l e t ' s make some remarks about the dual s i t u a t i o n . As the proof of Lemma 1 invokes only properties of abelian categories and as the dual of any abelian category i s abelian, the dual of lemma 1 holds. We rewrite i t i n d e t a i l s : Lemma f^. Let A be an abelian category. a) Let IP be a subclass of \ A\ , and assume ( i ) For each object X of A, there e x i s t s an element P of IP and an epimorphism P —& A. Then, f o r every X. & | K + 0 4 ) ° P | (where C ° P represents the dual of C), there e x i s t s a bounded below chain complex P. of elements of TP together with a quasi-isomorphism TT : P. > A. , with each TT epic, n b) Assume furthermore that TP s a t i s f i e s ( i i ) If 0 >- Z > Y > X *• 0 i s a short exact sequence with X IP , then Y &. TP Z TP , and ( i i i ) There e x i s t s a p o s i t i v e integer n such that, i f X >• X , >-. . . >- X i s an exact sequence, and n n - 1 o 16. X ,. . . ,X . e, IP , then X <=. IP . o' 'n-1 n Then, for every X. £ |K(4)°^| there e x i s t s a complex P. of elements of TP and a quiso P. >• X. . c) Let A* be a t h i c k subcategory of A, and assume that A'has enough ^ - p r o j e c t i v e s . Then, f o r every X. e. | K ^ , 0 4 ) ° P | there e x i s t s a bounded below complex P. of ^ - p r o j e c t i v e objects of A' and a quiso P. *• X. . Remarks. 1. Consider the p a r t i c u l a r case A - ^[. I f IP i s the class of p r o j e c t i v e modules of jTCL , then the condition ( i ) holds, since T^Tt has enough p r o j e c t i v e s Furthermore, i f the sequence 0 • Z > Y >X >-0 i s exact and i f X i s p r o j e c t i v e , then the sequence s p l i t s , so that Y = Z © X. Thus, Y i s p r o j e c t i v e i f and only i f Z i s p r o j e c t i v e , and ( i i ) holds. Also, by d e f i n i t i o n ( i i i ) holds i f the l e f t homological dimension of A i s f i n i t e . F i n a l l y , the category A' of f i n i t e l y generated A-modules i s a thick subcategory of jpl since i f 0 • A' »- A —-> A" *• 0 i s exact, then A'' and A" are f i n i t e l y generated i f and only i f A i s f i n i t e l y generated. Also, A' has enough" .4-projectives since every f i n i t e l y generated module i s a quotient of a f i n i t e l y generated p r o j e c t i v e module. Hence, we may wri t e Lemma l d c) as follows f o r the case of modules : For every bounded below chain complex X. such that a l l H n(X.) are f i n i t e l y generated, there e x i s t s a quasi-isomorphism P. *• X. , where P. i s a bounded below complex, with P^ p r o j e c t i v e and f i n i t e l y generated. 2. The dual of t h i s statement i s f a l s e since i n general a f i n i t e l y generated module cannot be embedded i n a f i n i t e l y generated i n j e c t i v e module (take for instance A = 7L ). 3. Even i n , the quiso of lemma l d a) i s not a homotopy equivalence i n general, as we see i n the following example : take X. : ... y 0 y 0 >- 0 y 7L/27L y 0 Then, Hartshorne's construction y i e l d s a chain complex P. and a quiso IT as follows : P. : ... •—y 0 — y 7L —y 71 —y 0 P X. : ... y 0 y 0 >• 7L /2ZZ »-0 Then, Hom(P.;2Z ) = P. and Hom(X.\7L ) = 0, since ZZ/2TL i s a t o r s i o n group. Hence, -Hn-(Horn (P. ;-Z ) i s not-isomorphic to Iin-(Hom (X. \7L ) , which -proves that-P. and X. are not homotopy equivalent. As a f i n a l remark, note that the dual of Propositions 2, 3 and Co r o l l a r y 4 hold. There i s also a dual construction f o r theorem 8, since every A-module M i s a quotient of A, Hence, the dual of x e M W Proposition 9 also-holds. 18-CHAPTER I I . THE GROUP HOMOMORPHISM V : Hora^(M;L) ® > HomA(M; L® gN) In t h i s chapter, we denote TU the category of l e f t A-modules, and A T H B the category of (A,B)-bimodules. (M e Ij^Pigl means that M i s a l e f t A-module, a r i g h t B-module, and that (ax)3 = a(x3) f o r a l l a«=.A, xeM, 3 6 B). We f i r s t prove the existence of a group homomorphism v : Hom.(M;L) ® N >• Horn. (M;L ® N) : A is A D Let A,B be two ri n g s , M e | TT\ | , L e | 7T\ | , N £ | Tft | . Then Proposition 1. The map v : HomA(M;L) ® g N • HomA(M;L <2>B N) defined by v(u @y) (x") -='u(x);® y (u c Horn (M";L) , 76N, x e M) I s — a group homomorphism, natural i n M, L, N. Proof. V i s w e l l defined : for y &N and ufeHom A(M;L), define v' (u,y) T M • L «> N by v'(u,y)(x) = u(x)®y (xeM). Then, by d e f i n i t i o n of the l e f t A-module structure of L ® N, v 1 (u,y) £ Horn. (M;L <£> N) . Furthermore, one checks jj A IS d i r e c t l y that v' : Horn (M;L) x N HomA(M;L ® B N) (u,y) 1 »• v'(u,y) i s b i l i n e a r and B-balanced. By the un i v e r s a l property of <S>B, v' induces the group homomorphism v. 19. v i s n a t u r a l i n M. Indeed, f o r each f : M' > M, the commutativity of the diagram VM H o m A ( M ; L ) ® c N • HomA (M;L 05^ N) (1) VM' HomA(M' ;L) ® B N - > H o m A ( M ' ; L ® B N) r e s u l t s immediately of the d e f i n i t i o n s . S i m i l a r l y , one v e r i f i e s the n a t u r a l i t y i n L and N. Convention. To the end of t h i s chapter, M,L,N,A,B,V w i l l have the same meaning as i n Proposition 1. In general, v i s neither an epimorphism, nor a monomorphism. We give a se r i e s of s u f f i c i e n t conditions under which v i s an isomorphism. These conditions w i l l be summarized at the end of the chapter. Pr o p o s i t i o n 2^ -If N i s p r o j e c t i v e (resp. f i n i t e l y generated p r o j e c t i v e ) , then v i s a monomorphism (resp. v i s an isomorphism). Proof. We f i r s t prove the proposition when N i s free (resp. f i n i t e l y generated f r e e ) . In t h i s case, N can be i d e n t i f i e d to a d i r e c t sum of copies of the r i n g B : N = £ D B ( i f N i s f i n i t e l y generated, the set I of indices can be chosen I f i n i t e ) . Horn, (f ;L) <8>_ N A B Horn, (f ;L ® „ N) A a Consider the following diagram 20. (2) Hom.(M;L) (J) B A a j 0 Hom.(M;L) >-I A -» Hom^M;!®,, 63 B) A D T T T Hom.(M;L) I A where a i s the canonical isomorphism, defined by a(u®(y^)^) = (uv^)-£ » j the embedding, and B i s defined, f o r f £ Horn (M;L ® €B B) such that A I f(x) = u(x)(g>(y.) i , by B(f) = ( f ± ) I , with f.(x) = u(x)y.. . M s a monomorphism, as composition of the following n a t u r a l maps : Horn (M;L© © B) Hom.(M; 6 3 (L ® B) -^—r Horn (M; © L) > y I A T a A i ) e m b e d d i n S , H o m ( M ; ] - [ L ) _±_+ f ] Horn (M; L ) . I I One checks that (2) commutes, so that 3v = jot i s a monomorphism. Hence, v i s a monomorphism. If N i s f i n i t e l y generated, the set I i s f i n i t e , thus since X , j and 6 are isomorphisms, and v i s also an isomorphism, I I and the pro p o s i t i o n i s proved when N i s f r e e . Suppose now N p r o j e c t i v e . Then N i s a d i r e c t summand i n a free module, i . e . there e x i s t a module R and a free module F such that F = N © R. But v„ i s an isomorphism i f and only i f v„ and v„ are isomorphisms. Indeed, F 1 N R consider the following diagram 21. Horn. (M;L) <8> (NOR) A Jo [HOID. (M;L) ® B NJ ®[Hom A(M;L) ® B RJ (3) V N e v R H o m A ( M ; L ® B (N©R)) »• [HomA(M;L ® £ N)] e [Hom^(M;L 8>B R)] where <f> and <f>' are the canonical isomorphisms. One checks that (3) commutes, hence v i s an isomorphism i f and r only i f v„,© v_ i s an isomorphism. But v „ © v „ i s an isomorphism i f and only N R N R i f v and v are both isomorphisms. This concludes the proof of Proposition 2. N R We have an analogous r e s u l t when we put the assumptions on the module M Proposition 3. If M i s f i n i t e l y generated and p r o j e c t i v e , then v i s an isomorphism. Proof. As f o r Proposition-2, we f i r s t suppose that M i s f i n i t e l y generated f r e e , n so that we can write M = A. Then, i f a, 8 and y are the canonical i = 1 isomorphisms, the following diagram commutes : Horn ( © A ; L) ®- K A . , JD i = l -> Hom.( © A ; L <S> N) 1=1 (4) n © L i=l • ^ N y 1 — » £B (L ® B N) i= l 22. Hence, v i s an isomorphism i f M i s free . We pass from the case M free to the case M p r o j e c t i v e by the same argument as i n the proof of Proposition 2. Now, we "spread" the two conditions onto M and N. In order to prove our next r e s u l t , we need a Lemma If M i s f i n i t e l y generated, then there e x i s t s a natural isomorphism <f> : Horn (M ; £B L) • © Horn (M;L) I I defined, f o r each f & Horn (M ; L) such that f(x) = ( y . ) T by <j>(f) = (fjO-j- J where f ^ i s defined by f^(x) = y^. "Proof. 4> i s w e l l defined : i f M i s generated by x^,...,x n, and i f f : M • £B L i s such that f(x) = ( y . ) T , then f i s e n t i r e l y determined I by f(x.) = ( y . . ) . T . But, by d e f i n i t i o n of © L, y.. i s nonzero f o r ' j=l»..»n only f i n i t e l y many ie=I. This implies that f ^ i s nonzero for only f i n i t e l y many indices i &I, thus ( f . ) _ e ( B Hom.(M;L). X X .^ A One checks d i r e c t l y that § i s a monomorphism and an epimorphism. Proposition 4. If M i s f i n i t e l y generated and i f N i s p r o j e c t i v e , then v i s an isomorphism. 23. Proof. I f N i s free : N = © B , the proposit i o n follows from the commutativity I of the diagram (5) Hom A(M;L>® © B A B j. © Horn (M;L) «-I A -> HomA(M;LG3B © B) ^ Horn (M: 0 L) If N i s p r o j e c t i v e , we express i t as a d i r e c t summand i n a free module, and we proceed as i n Proposition 2. We would l i k e now to interchange the two conditions on M and N. But i t seems that the condition " f i n i t e l y generated" i s not strong enough to prove that v i s an isomorphism; we sub s t i t u t e to i t the stronger condition " f i n i t e l y presented" (these conditions are equivqlent i f the r i n g i s Noetherian). But, f i r s t , we need a Lemma.: If E e- | y ^ | i s f i n i t e l y presented, and i f 6 l^/J' i £ l > then the natural homomorphism <f> : (FT M.) ® A E y T I (M. ® A E) i s an isomorphism. 24. Proof. n We f i r s t suppose that E i s f i n i t e l y generated free : E = £J} A« Then 1 (TTM. ) ® ® A a e [(T1M.)®. A] a £B (TTM.) = F f c e M . i 1 a i I L I I A J i i 1 i i 1 ) = (M. ® A A)] n (M. (2^ Q3 A) Consid er now the general case, and l e t L, *• L ° 1 c -*• 0 be a f i n i t e presentation of E ( L L • E >• 0 i s an exact sequence with L and o o 1 f i n i t e l y generated f r e e ) . As the functor X — i s r i g h t exact, and as, i n TU , the product of exact sequences i s s t i l l an exact sequence ((2), p.A II 10), we get the following commutative diagram with exact rows : (6) ( T I M.) © Lj_ — * ( T~I M.) ® L q —w ( n M.) ® E — * 0 (M. ® L ) — * (M. ®, E) • 0 l A As <j> and <f>T are isomorphisms, <J> 1 S also an isomorphism. L L, h o 1 Proposition 5. I f M i s p r o j e c t i v e and N f i n i t e l y presented, then v i s an isomorphism. Proof. As i n Proposition 3 , i t i s s u f f i c i e n t to give a proof for the case where M 25. i s free : M = kjj A. But i n t h i s case, the r e s u l t i s immediate i f we I consider the commutative diagram Horn. ( £B A ; L) <8> N A T B v * Horn. ( A ; L '«> N ) Y (7) ( n L ) <&R N > Remark. Over a Noetherian r i n g , " f i n i t e l y presented" i s equivalent to " f i n i t e l y generated" ((3), p.36). Hence, i f B i s Noetherian, by Propositions 4 and 5, v i s an isomorphism as soon as the hypotheses " f i n i t e l y generated" and " p r o j e c t i v e " are d i s t r i b u t e d i n any of the four p o s s i b l e ways on M and N. We state now a l a s t p r o p o s i t i o n without assumptions on the bimodule L : Proposition 6. If M i s f i n i t e l y generated (resp. f i n i t e l y presented), and i f N i s f l a t , then v i s a monomorphism (resp. an isomorphism). Proof. Let F, > F M K) be a free presentation of M, with F , F, free 1 o v o' 1 and F f i n i t e l y generated, and consider the commutative diagram 26. 0 > Horn. (M;L) ® N > > Horn . (F ;L) ®_ N >• Horn. (F 1 ;L) «L N A B A o B A 1 B VM L "F * o (8) 0 y Horn. (M; L <S> N) > > Horn (F ;L ®^ N) > Horn. (F .;L ® N) A B A o B A l B (The top row i s exact since — ® N i s an exact functor, and the bottom row B i s exact since Horn. ( — ; L ®_. N) i s l e f t exact). A B By Proposition 3, i s an isomorphism, hence i s a monomorphism. o If M i s f i n i t e l y presented, F, can be chosen f i n i t e l y generated, so that v 1 F l i s also an isomorphism. In t h i s case, the map induced on the kernels i s an isomorphism. Remarks. 1. As v i s natural i n M, L and N, the propositions 1 to 6 admit an immediate g e n e r a l i z a t i o n to the case where M, L or N are (co)-chain complexes. 2. We obtain i n t e r e s t i n g p a r t i c u l a r cases f o r Propositions 1 to 6 by taking A = B = L = A. Then v becomes \> : HomA(M;A) ® A N • HomA(M;N) and i s defined by ^(f®y)(x) = f(x) y . I f , as usual, we denote M = Horn (M; A) (M = dual of M) , the map \> takes the very simple form ~ : M* ® A N • HomA(M;N) (If A i s a f i e l d , and i f M i s a fini t e - d i m e n s i o n a l vector space over A, then t|> i s an isomorphism which i s well known i n tensor c a l c u l u s ) . 27. We s h a l l now use the previous propositions to prove a few statements about the behaviour of an i n j e c t i v e module with respect to the tensor product : Proposition 7. Let A be a l e f t Noetherian r i n g , L £ I^J^gl i n j e c t i v e as A-module, and N I f l a t . Then, L ®^ N i s an i n j e c t i v e A-module. Proof. We r e c a l l that a l e f t A-module M i s i n j e c t i v e i f and only i f , f o r each l e f t i d e a l I of A, the morphism i * : HomA(A;M) >• HomA(I;M) induced by the embedding i : I> > A, i s an epimorphism. Consider the following commutative diagram : Horn (A;L)'® N • Hom A(I;L) ® g N VA V I Hom A(A;L-® B N) »- Hom A(I;L ® g N) ( v A i s an isomorphism since HomA(A;L ®^ N) = L ® B N = HomA(A;L) ® g N ). As L i s i n j e c t i v e , and as the functor — ® N i s r i g h t exact, ±s an ii epimorphism. But, since A i s Noetherian, I i s f i n i t e l y presented. By Proposition 6, as N i s f l a t , v i s an isomorphism. Thus, i i s an epimorphism, and L N i s i n j e c t i v e . B I f A i s a PID, the proposi t i o n i s true without the assumption N f l a t 28. Proposition 8. Let A be a PID, L £ l^-gl i n j e c t i v e as A-module, and l e t N £ \ . Then L & N i s an i n f e c t i v e A-module. B J Proof. I f A i s a PID, then I i s free and f i n i t e l y generated and, by Proposition 3, i s an isomorphism (even i f N i s not f l a t ) . Remark. If we take the p a r t i c u l a r case A = 7L , we get the following r e s u l t : If L £ | T T c B | i s i n j e c t i v e as abelian group, and i f N <£. | ^ TTL | , then L ® N i s an i n j e c t i v e abelian group. B We also have a sort of sual s i t u a t i o n : Proposition 9. Suppose that B i s a l e f t Noetherian r i n g , M £ | HfTl | i s p r o j e c t i v e , and L £ I ^  i J ^ s f l a t a s r i g h t B-module. Then HomA(M;L) i s a f l a t r i g h t B-module. Proof. By (3), Pro p o s i t i o n 1, p.26, HomA(M;L) i s f l a t i f and only i f , f o r each l e f t i d e a l I of B, the canonical morphism <j> : Horn (M;L) ®„ I * Hom.(M;L) ® B A B A B i s a monomorphism. Again by (3), p.36, i f B i s l e f t Noetherian, then I i s f i n i t e l y presented. Consider the commutative diagram 29. HomA(M;L) ® R I ^ >• HomA.(M;L) ® ' B 3 H o m A ( M ; L ® B I) * »• HomA(M;L ® R B) (3 i s an isomorphism since HomA(M;L) ®^ B = HomA(M;L) = HomA(M;L ® ^ B) ). By Proposition 5, i s an isomorphism. Furthermore, L f l a t e n t a i l s that L <?> I >• L ®„ B i s a monomorphism. As Horn. (M; — ) i s l e f t exact, then B B A i|> i s monic. Hence <|> i s a monomorphism, and Horn (M;L) i s f l a t . As before, i f B i s a PID, then Proposition 9 holds without the assumption that M be p r o j e c t i v e since, by Proposition 2, I p r o j e c t i v e implies that v w i s a monomorphism. Thus : M Proposition 10. Suppose B i s a PID, and L £ I .Tft I i s f l a t as r i g h t B-module. A n Then Horn (M;L) i s a f l a t r i g h t B-module. We conclude t h i s chapter by a b r i e f discussion of the case where M or N are complexes. We give some s u f f i c i e n t conditions f or v to induce isomorphism i n homology. Proposition 11. Let A be a l e f t Noetherian r i n g , M. a bounded below chain complex of objects i n |ATR | such Hn(M.) i s f i n i t e l y generated f o r a l l n. If L i s i n j e c t i v e as A-module, and i f N i s f l a t , then v i s a quasi-isomorphism, i . e . H (v) i s an isomorphism f o r a l l n. 30. Proof. By Remark 1, p.16, we know that f o r every bounded below chain complex M. such that Hn(M.) i s f i n i t e l y generated f o r a l l n, there e x i s t s a quasi-isomorphism TT : P. y M. where P. i s a bounded below chain complex with P n p r o j e c t i v e and f i n i t e l y generated for a l l n. By Proposition 7, L ® N i s an i n j e c t i v e A-module. Hence the B functors Horn.( — ;L) and Horn.( — :L ® N) as well as — '®_ "N are exact. A A B B Moreover, we know that i f a functor T i s exact, and i f f : X. —*• Y. i s a chain map, then T(H ( f ) ) = H (Tf) so that exact functors preserve the n n quasi-isomorphisms. Proposition 11 r e s u l t s now from Proposition 3 and from the commutative diagram VM Horn. (M. ;.L) ®i N — y Hom.(M. ;L®_ N) A o A i i quiso •' Horn. (P. ;L) <8> N — y Horn. (P. ;L ® . N) A ' B s A B By s i m i l a r arguments, using Proposition 9, one proves : Proposition 12. Let B be a l e f t Noetherian r i n g , N. a bounded below chain complex of objects i n | 711 | such that H ^ C N . ) i s f i n i t e l y generated f o r a l l n. If L i s f l a t as B-module, and i f M i s p r o j e c t i v e , then v i s a quasi-isomorphism. quiso Now, we summarize our r e s u l t s i n the following l i s t v : Hom A( AM ; ^  ® BN A B M L N v proj ective mono pr o j e c t i v e + f i n i t e l y generated i s o p r o j e c t i v e + f i n i t e l y generated is o f i n i t e l y .generated p r o j e c t i v e . iso - p r o j e c t i v e f i n i t e l y presented is o f i n i t e l y generated f l a t mono f i n i t e l y presented f l a t i s o l e f t Noetherian M. e K * c m ) i4' A i n j e c t i v e f l a t quiso l e f t Noetherian p r o j e c t i v e f l a t N. K+,(£ft) quiso 1 32. APPENDIX. 1. Proof of Lemma 1.1, For any co-chain complex r : ... _ xP" 1-i^-> x p x p + 1 — ... we use the usual notations B P(X') = Im d P _ 1 B , P(X") = Coim d P Z P(X*) = Ker d P Z , P(X*) = Cok d P _ 1 Thus, as objects, we have B | P(X*) = B P + 1 ( X " ) . We r e c a l l Lemma 1.1 a) : Let A be an abelian category, IP a subclass of |;4| and assume that every object of A admits an i n j e c t i o n into an object of IP . Then every X v £ |K +04)| admits a quasi-isomorphism into a bounded below complex I' of objects of JP . Proof. Given a bounded below co-chain complex X*, we can assume X P = 0 f o r p < 0. Then, there e x i s t I ° £ TP and a monomorphism i° : X°>—> 1 ° . Suppose d° , " d P _ 1 0 — • I O - J U I 1 — , . . . — J — . ! * and i k : X k >• 1^ (k = 0,1,...,p) are defined. Then I P + * " , d P and ± & + ^ a r e defined by the following commutative diagram : 3 3 . r P +l where (1) i s a pushout diagram, s P the canonical p r o j e c t i o n , and j P + * : K P +* •> >- I P + * i s an embedding of K P +* int o an element of IP It i s c l e a r from t h i s diagram that d P d P ^  0 and that i P + * d P = d P i P . Thus, we obtain a complex I* and a chain map i : X' x J. I' i*P_is_a_monomor£hism f o r all_.p_.. o Consider the diagram Ker i * 34. where T i s the map induced on the images, and v' the f a c t o r i z a t i o n of d j through Ker s. i P : X*5 —-> I P induces a map i : Ker i P *• Ker s such that TP Ker i P — £ • X P 3. Ker s commutes-. Furthermore, as v" v 1 «= (j 6") (8' s P ^) are two epi-mono f a c t o r i z a t i o n s of j8s, we can i d e n t i f y Im 8 to Ker s, so that i may be considered as a map Ker i P > Im 8. Hence : j8"i = v " i = i P e = jae. But j i s monic, so that B"i = ae. In the diagram 4. Ker i Im 8 >-the square i s b i c a r t e s i a n (see (10), Theorem 13.4.8), hence there e x i s t s i|> : Ker i P > B P(X') such that KI|I = e. F i n a l l y , consider 35. 5. Ker i P ~+ Ker a i p I P • y dx . vp+l (1) a — * K^ As (1) i s pushout, the map y induced on the kernels i s epic ((10), theorem 13.4.8). But d e = (d <)ip = 0 = yy. Then : y epic implies y = 0, i . e . Ker a = 0, so that a i s monic. As j i s also monic, i P = j a i s monic. The following step i s to prove that Recall, _the d e f i n i t i o n of H P(I") : ^P" 1 ^P rP-1 ± - * I p Tp+1 6. H P ( D > - -*• Cok d P 1 h » Coim d P We have H P(I') = Ker h ' P , and h ' P s p = y p . Then : — 1 Vi ' " Cok d P > Coim d P s Im d P ,P+1 K p+1 :p+l rP+l .p+l„p+l p ,p p p p. t p p . p . commutes : j r 8r s r = d^. = v r y r = v rh r s . As s r i s epic j P + y + i 36. Thus : Ker g P + 1 = Ker f*1 H**1 = Ker v P h ' P = Ker h ' P = H P ( l ' ) . H_ (i)^ _ i s egic. Since i n the diagram 1., the square (1) i s pushout, i P induces an epimorphism n P: Ker d P = Z P(X*) » Ker B P = H P(I') such that zp(x') > —*• x p 8. n p H P(I') > > I P commutes. Chasing the diagram i ? " 1 : * i p one sees that X P _ 1 — * B P(X') >• Z P(X") :—>• H P(I') >->- I P i s the zero map. Hence, B P(X') • Z P(X') —^+ H P(I") i s the zero map, and n factors through Cok(B P(X') y Z P(X')) = H P(X'); thus, there e x i s t s an epimorphism 6 such that 6ir =n. 37. We claim that 9 = H P ( i ) . For t h i s , by d e f i n i t i o n of H P ( i ) , we have to prove that 10. H P(X') H P(I*) -*• Z , P(X") = Cok d. p-1 Z' P(I*) = Cok d p-1 commutes, where y i s defined by the commutative diagram X1 p-1 11. rP-1 *> z,p(x") Z , f ( l ' ) Consider the diagram 12. Z P(X') > • X* (i) P(X*) > v Z' P(X") (4) H p ( D i p -y I1 = Cok d p-1 From previous computations, we know that (2), (3) and the large square of diagram 12. commute. By d e f i n i t i o n of Z, Z' and H, the square (1) also commutes, Then (4) commutes, so that H P ( i ) = 6. The l a s t step to prove the f i r s t part of the lemma i s to show that H (I) xs monic. For t h i s , consider the diagram 38. 13. I p (1) B (2) Bs. (3) d I -jBs -y -K -> I -^-^ Cok d X ! *1 Cok B 1 ^ Cok Bs I I Cok d. where (1) i s pushout by construction, (2) i s pushout since s i s epic, and (3) i s pullback since j i s monic; thus, by (10), theorem 13.4.8 : <j>^  and §^ are isomorphisms, and cf>^  i s monic. Hence, XP XP+1 d T P L I p+1 induces a monomorphism $ : Cok d > > Cok d . X JL F i n a l l y , look at the commutative diagram 0 >• H P(X*) > y Cok d P 1 »• Coim d P »- 0 X X 14. H P ( i ) (1) 0 y H P(I') > • Cok d P 1 » Coim d P »• 0 As (1) commutes and <j> i s monic, so i s H P ( i ) . Hence, H P ( i ) i s an isomorphism and i i s a quiso. This ends the proof of Lemma 1 a). Now, we r e c a l l Lemma I. l b ) 39. Let A be an abelian category, l e t IP be a subclass of |J4|, and assume ( i ) Every object of A admits an i n j e c t i o n into an element of IP . ( i i ) I f 0 >• X > Y >• Z K) i s a short exact sequence with X £ TP , then Y 6 IP 4==^> z «• IP . ( i i i ) There e x i s t s a p o s i t i v e integer n such that i f X ° — > X 1 X n ^ *• X n —>- 0 i s an exact sequence, and X°, X 1,. . . ,Xn 1 e IP , then X n e IP . Then, every X* &|K04)| admits a quasi-isomorphism into a complex I* of objects of IP . Proof. F i r s t step. Let X' e| K 0 l ) | , and i Q£ ZZ . Then, by Lemma 1 a), we can f i n d a quasi-isomorphism i of the truncated complex i o i n + 1 . . . —y 0 — • 0 — • X ° — y X ° — y into a complex I" : .. —>• 0 — » • 0 — y I ° — y I with each i P : X P y I P monic. Define X^ to be the complex j i f ) ~ 2 . , i 0 - l , . „ d u . l-d ° . d . . . . . y x o A — A y x l 0 y I 1 0 — ^ — y I 1 0 — y and consider the diagram x- x i o" 2 x i o _ 1 x i o x i o + 1 ~y , 15. i 1 • r i i , i°d,r d T Then, X 1 £ IP f o r i £ i , and each i P : X P >- X P i s monic. o o o 40. i-i-X^jZTt-X'o i s _ a quasi-isomorghism. I t i s c l e a r from the construction of X' that H n(X*) = H n(X") f o r n ^ i - 1 o o o and f o r n ^ i . o (i ) H ^ C X ' ) s H ^ C X ^ ) : C l e a r l y , B^0""^ (X*) = B^'^X') = Im dj°~ 2. Furthermore, Z ±o~ 1(X*) = Ker d ^ ° _ 1 = Ker i d * 0 " 1 = Z l o _ 1 ( X * ) , since i i s monic. A A O Hence, H i o _ 1 ( X ' ) = H io" 1(X^).' ( i i ) H 1o( X-) £ H ± 0(X^) Consider the diagram 16. which shows r e a d i l y that <|> i s an isomorphism : B °(X*) = B l o(X^) We now look at 17. Z l o(X") = Ker dv > z l o(x*) o 41. (By construction, 1*° = Cok d* 0 1 = i*°, since d*° 1 = 0, and j i s monic. Thus, Ker <$1 = Ker 3). Then : (1) commutes implies i s monic (2) i s pushout implies i s epic (see (10), theorem 13.4.8). Hence, H l o(X") = H l o ( X ^ ) , and i : X' > X^ i s a quasi-isomorphism. Second step. Suppose given X ^ € . | K C 4 ) | with X * £ IP f o r i £ i ^ , and l e t 1^ < i ^ . We want to construct a quiso X'^ *• X ^ such that X ^ £ IP f o r 1 > i^, and X * = X ^ f o r i > i^+n (n=integer of condition i i i ) ) . By the f i r s t step, we can f i n d a quiso X * • X ' " such that X , : LfeIP for i > i 2 , and X * > x' 1 monic. Define 'Y1 = Cok( X * > X ' 1 ) . The long exact cohomology sequence of 0 • Xj • X'' y Y* *• 0 then shows that Y" i s a c y c l i c , i . e . H P(Y') = 0 for a l l p. Furthermore, f o r i 5- i ^ > i-2' ^1 & ^ a n d X' 1^ • Hence, by condition ( i i ) , Y 1 ^ TP f o r i > i ^ , The exact sequence , , i „i+l „i+n~l ^.i+n^'N N Y y Y y . . . y Y »• B (Y ) y 0 together with condition ( i i i ) e n t a i l s that B 1(Y") £IP' f o r i ^ i^+n. We are now ready to define X^ : l e t K. be defined by the pushout square i - l 1 y x x quiso ^y 1 1 ._, a 18. X' 1 1 1 * B I ( X " ) 2 • K 1 42. and l e t X' 1 f o r i < i ^ + n XZ = < K 1 for i = i x + n X* f o r i > i ^ + n Convention : To s i m p l i f y the notations, we set i ^ + n = 0 f o r the proof ths X^ rs quasi-isomorphic to X j . We define the complex X^ as follows : A~2 + Y L * • Y ' 1 A° A1 X2 1 1 •»• K —==-•• x: o 1 -* X, where d * and d ^ are defined by the following diagram X2 X2 19. i . e . d ^ = 8°u° and d° i s the unique map making the diagram commutative x 2 x 2 (universal property of the pushout). It i s immediate from t h i s diagram that d Y ^ d Y? = 0, and that d° d ^ = 0. X2 2 X2 To prove that d d° = 0, we consider the diagram X l X2 43. 20. This diagram commutes, but 0 : K v makes also the diagram commutative, hence d v d v =0; hence Xl i s a co-chain complex. X l X2 l We now construct a chain map X 1 X2 21. x; -y X -2 .-2 -> X ,-1 x: • - 2 — > X'"1 dx' 6 u -+ K -»• X, * x i i T * x ? x l - 2 - 1 where i and i are parts of the quiso i : Xj > >^ X 1'. We claim that i : X! > X' i s a quiso. To prove t h i s , i t s u f f i c e s to show that H P ( i ) i s an isomorphism for p = -1,0,1. F i r s t , we show that a°_and_8°_are_monomorghisms, -1 Consider the diagram 22., where vii i s the epi-mono f a c t o r i z a t i o n of d^, : 44. 22. (X') As d \ i 1 = i°d , the u n i v e r s a l property of pushouts implies the existence X X^ of a map j° : K° > X'° such that j°a° = i° and j°8° = v. But i° and v are monomorphisms, thus a° and 3° are also monomorphisms. H _ 1 ( i ) : H _ 1 ( X j ) y H * (xp i s an isomorphism. As we already know that H *(xp = H ^(X''). we only have to prove that H _ 1(X'') = H - 1 ( X 2 ) . For t h i s , consider 23. X2 X' ... —y r ' 2 r ~ l K ° d.v. -y ... ,r,-2 X 1 X' v l o -> x' >• X' • X 1 y . . Since B 1 (Xp = B - 1 ( X , V ) , we have to prove that Z 1 (xp = Z ^ X 1 * ) . 3 and v are monomorphisms, so that Z - 1 ( x ; ) = Ker 3y = Ker p = Ker vy = Ker d~] = Z _ 1(X'*). 2 X Hence : H 1 (i) : H 1 (Xp >- H 1 (xp i s an isomorphism. 45. H°(i) : H°(XJ) —»• H°(X 2) i s an i s o m o r p h i s m . I n t h e d i a g r a m 24. .-1 X l (1) B°(X') K° Cok d x » Z'°(XJ) I -i a i i Y -*»-' Cok S = Cok d -1 Z M X - ) (1) i s pushout e n t a i l s t h a t a : Z'°(Xj) >- Z' (X^) i s an i s o m o r p h i s m . C o n s i d e r t h e epi-mono f a c t o r i z a t i o n o f d° and d° : 1 A 2 ,o ,o 1—• xj K ->- x: B ^ X J ) B ^ X - ) I n t h e d i a g r a m xT1 — U x ° V (1) -*»• B X(Xp > * , > xj • (2) 25. B°(X')>-^ K° — B ^ X ' J v - ^ xj — 6 u .Cok i > »• Cok a >- Cok <f> (2) commutes i m p l i e s t h a t t h e i n d u c e d map <f> i s monic (1) i s pushout i m p l i e s t h a t 3 i s an i s o m o r p h i s m . 46. Then : d 3 = 0 x2 y 3 = 0 <J> y -B = u 3 i = 0. But 3 i epic = ^ y = 0 y a = <f> y = 0 :—> <j> = 0. Hence : <}> epic = ^ Cok <j> = 0 <f> i s an isomorphism: B'°(X') = B^XJ) ^ B^xp = B'°(xp, and H°(i) : H°(xp -y H"(Xp i s an isomorphism. H*(i) : H 1 (Xp > H 1 (xp i s an isomorphism. Consider the diagram X, X, 26. K x: x, We have Z*(xp = Ker d* = Z X(xp. As we already know that B 1 (xp we conclude that H ( i ) i s an isomorphism. BX(xp, Hence i : Xj y i s a quasi-isomorphism. In order to complete the second step, we have to show that X 2 T P for i 1 i 2 , i . e . that K°e= TP . For t h i s , i t s u f f i c e s to prove that 0 B°(Y*) = Cok a°, since then, as 0 >• X° — • K° — » • B°(Y') —-> 0 i s exact and as X°, B°(Y')elP, we have, by assumption ( i i ) , K ° e I P . In the diagram X 27. (1) -> X, B°(X') K 47. (1) i s pushout implies that Cok a that Cok y ' i £ B°(Y'). Consider o .. = Cok y ' i . Hence, i t s u f f i c e s to prove 28. X X -2 ^ S 1 ,-2 d l _ -1. fy* M B (X'*) >-g N -1 B°(Y')>-i - x'c -»- Y where n and n are induced on the images. "We have Cok y ' i = Cok ny = Cok n. Hence, i t s u f f i c e s to prove that n = Cok n. For t h i s , l e t f : B°(X'*) *• M be a map such that fn = 0. We have to f i n d $ : B°(Y*) > M such that <}> n = f (such a § w i l l necessary be unique. since n i s e p i c ) . We have the following implications : fn = 0 > fny = f y ' i = 0 > fy' factors through Y * = Cok i : 1 there e x i s t s g : Y M with g i = f y ' . Now, g d i = g i d ' = f y ' d' =0. But i i s epic, so that g d =0. As Y" i s a c y c l i c , B°(Y') = Cok d, and g d = 0 implies that g f a c t o r s through B U(Y") : th ere e x i s t s <f> such that <j> y = g. One checks then d i r e c t l y that 48. Third step. F i n a l l y , given a complex X*&|KC4)|, choose a sequence of integers i > i , > i _ >... tending to - °°. Choose X * for i as i n the f i r s t step, and o 1 2 6 o o f> choose X * , X ^ , . . . f o r i ^ , i2»... successively as i n the second step. Then, we have quasi-isomorphisms X * • X ^ »- X ^ >• X ^ • ••• and, f o r each i , i i i i the sequence X > X >• X . *• X „ ... i s e i t h e r constant, or i n TP . o 1 2 Then I* = lim^ X ^ i s the required complex of objects i n TP. This ends the proof of Lemma b). Let's now r e c a l l Lemma I. 1 c) : Let A be an abelian category, l e t A ' be a thick subcategory of A , and assume that A' has enough 4 - i n j e c t i v e s . Then every X ' e. \K*.(A)\ admits a quasi-isomorphism into a bounded below complex I* -of j4-injective objects of A ' . Proof. We may assume X 1 = 0 for i < 0. Embed H ° ( X " ) into 1°, an ^4-injective object of A ' , which i s po s s i b l e , since H ( X * ) e \ A ' \ . Then we can extend t h i s embedding to a map f° : X ° *• 1° : 0 -* x° • x 1 \ f ? "29. / H°(X') • 1° which i s p o s s i b l e , since 1° i s ,4-injective. d° . d f 1 . . Having defined 1° — I >. . . — v l p a n d f 1 : X 1 >- I 1 f o r i=0,l,, we define I P + 1 , d P and f P + 1 by the diagram 49. B P + 1 ( X ' ) 30. ,P-1 rP-1 z P + l ( r ) v _ x p + 1 s p f P (1) - I P - ^ Cok d P _ 1 = Z' P(I-) K P + 1 CP+1 rP+1 where (1) i s a pushout square. We must check that K P + 1 i s i n A' (to be sure that K P + 1 can be embedded i n an ,4-injective object of A'). R e c a l l that a f u l l subcategory A''"of* an abelian category A i s t h i c k (or i s a Serre subcategory of -4) i f one of the following equivalent conditions hold : (i ) If X y Y * Z i s exact, then X,Z &\A'\ ==#• Y ^ \ A ' | . ( i i ) I f 0 y B' >• B >- B" y 0 i s exact, then B<=|4'| i f and only i f B' €.\A'\ and B"e Thus, i t s u f f i c e s to prove that, i n the exact sequence Z , P ( I * ) -y K p+1 — » Cok g Z,V(I)&\A'\ , and Cok ge\A'\ . But : 0 • B P ( I * ) >—y I P — » Z , P ( I * ) y 0 i s a short exact sequence, and IP.<=|J4'|, so that Z , P ( I * ) £|/4'|. Moreover, as the square (1) of diagram 30. i s a pushout, Cok g = Cok d) = = H P + 1(X") ^-|i4'|. Hence, K P + 1 ^ \A'\, and there i s a monomorphism K P + 1 > y I P + 1 where i P + * i s an /4-injective object of A'. F i n a l l y , as i p + 1 i s i n j e c t i v e , the map j P + \ i extends to f P + 1 and one defines d P = j p + 1 g s P . One checks d i r e c t l y that d P d P 1 50. -> I p+1 = 0 and that I i s a chain map. H__(f) i s § n_i so m2£Ehism. 1. H (f) i s epic : consider the diagram B P + V ) 31. Z P(X") >-H P ( 0 >- —> Z , P ( I * ) A. ( i) g - Z P + 1 ( X - ) p+1 B P + 1 ( I ' ) = B , P ( I " ) One checks d i r e c t l y that Im <j> = B P + 1 ( X ' ) and that Im g = B P + 1 ( l ' ) = B , P ( I * ) . As the square (1) of diagram 31. i s a pushout, the map n induced on the kernels i s epic. With a same reasoning as i n diagram 9. (p.36), one sees that n induces an epimorphism 0 = H P ( f ) : H P(X') — H P ( I * ) . 2. H (f) i a monic. Consider the diagram 51. 32. z , p ( i - ) - Z P + 1 ( X ' ) - ^ — x p + 1 pp+1 p+1 J L _ x P+2 Pp+2 -* I p+2 z p + 1 ( D We have d j (* = f P + 2 d x y = 0, and d^jas = d^d^. = 0, hence, since s i s epic, "dj'j » 0. "By the un i v e r s a l property-of the pushout ('1), -djj = 0. Thus, there e x i s t s a unique monomorphism $ such that - One also checks that the t r i a n g l e (2) of diagram 32. commutes. F i n a l l y , consider 33. B P + V ) B P + 1 ( I * ) B P + 1 ( I ' ) >-(3) -y z p + 1 -y K p+1 + z p 4 V ) -» H P + 1 ( X - ) -*y Cok a' H P + 1 ( I * ) As the square (1) of diagram 32. i s a pushout, the square (3) of diagram 33 i s a pushout, so that u i s an isomorphism. Furthermore, by the short f i v e lemma, as <j> i s monic, y' i s monic. Hence, H P ( f ) = y'y i s a monomorphism. 52. To conclude t h i s appendix, we give a proof of Proposition 1.6. We r e c a l l the statement of t h i s proposition : Let A be an abelian category of f i n i t e i n j e c t i v e homological dimension, l e t I be the f u l l subcategory of A c o n s i s t i n g of i n j e c t i v e objects of A, and l e t E : K(J) > K(A) be the embedding functor. Then : There e x i s t s a functor J : K(4) —>• K(J) and a nat u r a l transformation i : I d ^ ^ >- EJ such that, f o r each X*€ |K(A)| : ( i ) i i s a quasi-isomorphism A ( i i ) i i s a monomorphism. A / Proof. I f we take TP = | j | , we see that the axioms ( i ) , ( i i ) and ( i i i ) of Lemma I. 1 b) are f u l f i l l e d . Hence, i t i s s u f f i c i e n t to prove that the ass o c i a t i o n X* • JX' = I" constructed i n Lemma I. 1 b) i s f u n c t o r i a l . (The fact that i i s monic r e s u l t s from the construction of I * ) . We keep the same x notations as i n the proof of Lemma I. 1 b). 1. The a s s o c i a t i o n X" — y X' i s f u n c t o r i a l . o Given a map f : X' • Y*, l e t I* (resp. J*) be the cochain complex co n s i s t i n g of i n j e c t i v e s quasi-isomorphic to the truncated complex X° y X 1 y... (resp. Y° >• y . . . ) . Then, by the f u n c t o r i a l i t y of the f i r s t part of the lemma (Proposition I. 2), f induces a map f ^ : I" • J * . Hence, f induces a map X' y J* as follows : 53. XXo 2 • x""0 i d. -> . . . 34. f (1) Y 1 © - 2 • Y 1 0 " 1 " Y > J I O * J L O + 1 »• The square (1) of diagram 34. commutes, since both squares commute i n x1 o ~ 1 X—^ x i 0 ^ i v 35. , i o - l Y ^ o , j • -*> Y 2. The_association_Xj 33^_X 2 of_the second_step__is_functori-al. By construction of X'', the a s s o c i a t i o n Xj >- X'* i s f u n c t o r i a l , and X i s defined ( i f we suppose i D+n = 0 as before) by 'X'1 f o r i < 0 X* = I K° for i = 0 I X* f o r i > 0 where K° i s defined by the pushout diagram X -1 X, 36. X ,-1 B ° ( X " ) * K° Then, given a co-chain map f : Xj *• Y* we have to f i n d an induced map f : X- Y'. We already know that f induces f : X'* >- Y ' * . Hence, we have to define f : K° y L° such that 54.--»• X'" 2 >• X' 1 • K° *• x\ • X 2 >- . . .. 37. - I f l I + -* Y'~ 2 + Y'" 1 >• L° — > Y? • Y 2 — > commutes. This map i s given by the following diagram (defining K° and L°), where we make use of the uni v e r s a l property of the pushouts : 38. Hence, we have a sequence of natural maps x* >—»- x" >—-y x; • o 1 39. f, *o -> Y* •» y Y' y o 1 -y lira X = I ^ r If -> l i m Y" = J ' - r By u n i v e r s a l property of l i m , f. induce a unique map > r If : li m X" y l i m Y" . 55. BIBLIOGRAPHY (1) (2) (3) (4) BASS (H.) Algebraic K-Theory, W.A. Benjamin Inc., Amsterdam, New York (1968). BOURBAKI (N.) Algebre I, Chap. 1 a 3, Hermann, P a r i s (1970). Algebre Commutative, Chap. 1,2, Hermann, Paris (1961) GROTHENDIECK (A.) Sur quelques points d'algebre homologique, TokOhu math. J . IX (1957), 119-221. (5) HARTSHORNE (R.) (6) HILTON (P.J.) Residues and Duality, Lecture notes i n Mathematics, yol.20, Springer Verlag, B e r l i n , Heidelberg, New York (1966). Correspondences and exact squares, Proceedings of the Conference on Categorical Algebra, La J o l l a 1965, Springer Verlag, B e r l i n , Heidelberg, New York (1966). (7) HILTON (P.J.), STAMMBACH (U.) A Course i n Homological Algebra, Graduate Texts i n Mathematics., Vol.4., Springer Verlag, B e r l i n , Heidelberg, New York (1970). (8) MACLANE (S.) (9) (10) SCHUBERT (H.) Categories f o r the Working Mathematician, Graduate Texts i n Mathematics, Vol.5, Springer Verlag, B e r l i n , Heidelberg, New York (1971). Homology, Springer Verlag, B e r l i n , Heidelberg, New York (1967). Kategorien I, Heidelberger Taschenbiicher, Vol.65, Springer Verlag, B e r l i n , Heidelberg, New York (1970). 

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