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UBC Theses and Dissertations

Derived categories and functors Loo, Donald Doo Fuey 1971

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DERIVED CATEGORIES AND FUNCTORS by Donald Doo Fuey Loo B . S c , Un i v e r s i t y of B r i t i s h Columbia, 1970 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 thes i s as conforming to the requ i red standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1971 In presenting th i s thes is in pa r t i a l fu l f i lment o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f r ee l y ava i l ab le for reference and study. I fu r ther agree that permission for extensive copying of th i s thes is for scho la r l y purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i ca t ion o f th is thes is fo r f inanc ia l gain sha l l not be allowed without my wr i t ten permiss ion. Department The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada i i . ABSTRACT For each abe l i an category A, there i s a category D(A), c a l l e d the der ived category of A, whose objects are complexes of objects of A, and whose morphisms are formal f r a c t i o n s of homotopy c lasses of complex morphisms having as denominators homotopy c lasses induc ing isomorphisms i n cohomology. I f F : A > IS i s an add i t i ve functor between abe l i an ca t ego r i e s , then under su i t ab l e cond i t ions on A, there i s a functor RF : D(A) >• D(B) with the property that i f objects X of A are considered as complexes concentrated at degree 0, then there are isomorphisms H nRF(X)~R nF(X) f o r a l l n, where R U F i s the ord inary n t h r i g h t der i ved functor of F. RF i s c a l l e d the der ived func tor of F, and one may look upon i t as a k ind of extension of F. i i i . CONTENTS Page Chapter I. In t roduct ion 1 Chapter II. Ca lcu las of L e f t F r ac t i ons 7 § 1 . D e f i n i t i o n s of l o c a l i s a t i o n s and extensions 7 §2. Ex istence and d e s c r i p t i o n 10 Chapter II I. Pseudo S p l i t Sequences and Mapping Cones 19 Chapter IV. The Derived Category 28 Comparison with c l a s s i c a l theory 36 Chapter V. Ext 38 Appendix 1. Ca l cu las of l e f t - f r a c t i o n s 41 2. A note on F-acyc l i c subcategor ies 53 3. Quas i -sp l i t sequences 55 4. Quisos and the der ived category 63 5. Ext 70 References 72 Acknowledgements I would l i k e to thank Dr. K. Hoechsmann f o r h i s guidance and ass i s tance dur ing the w r i t i n g o f t h i s t h e s i s . In a d d i t i o n , I am g r a t e f u l to the Nat iona l Research Counc i l f o r t h e i r f i n a n c i a l support . - 1 -CHAPTER I INTRODUCTION Let F: A > 15 be an add i t i ve functor between abe l i an ca tegor ies where A has enough i n j e c t i v e s . The " r i g h t der ived func to r " of F i s an exact 6 functor or an exact connected sequence {R n F ,6 n : n>0} of functors from A to B together with a na tu ra l map n: F > R ° F . I ts d e f i n i t i o n i s as fo l l ows : For X £ Object A, take an i n j e c t i v e r e s o l u t i o n I*: 0 > X — ^ 1° " > I 1 " > I > < • • and set R nF(X) = H n F ( I * ) . n : F > R ° F i s induced by the u n i v e r s a l i t y of Ker d° = R ° F . 0 > F(X) £ > F ( I ° ) ^ F d 1 ) > V ' F ( d O ) = R ° F ( X ) Given a map X — - — > Y the n i ce mapping p roper t i es of i n j e c t i v e s y i e l d a morphism a:I* ^ J 0 > X > 1° > I 1 > 0 > Y > J ° > J 1 — > i* 1«° |- ] - 2 -of r e s o l u t i o n s , which i s unique up to homotopy of complexes. Since the cohomology functors H n do not d i s t i n g u i s h homotopic morphisms, t h i s produces a we l l-def ined map R n F ( f ) : R nF(X) *- R nF(Y) The connect ing morphisms 6 n are def ined v i a the " Snake Lemma " from the f a c t that every shor t exact sequence i n A can be i n j e c t ed i n to a short exact sequence & 4e & o > i > J y K > o of i n j e c t i v e r e so lu t i ons and the l a t t e r stays exact a f t e r an a p p l i c a t i o n of F . I f F i s l e f t exact , n i s an isomorphism and { R n F , 8 n } co inc ides wi th the sequence of " r i g h t s a t e l l i t e s " of F — i . e . an exact connected sequence of functors whose 0 ^ term i s isomorphic to F. The disadvantage of the theory ou t l i ned above i s i t s heavy r e l i an ce on i n j e c t i v e s . But i n case F i s l e f t exact , R nF can be computed from any F-acyc l i c r e so lu t i on ( X £ 0 b A i s F-acyc l i c i f R nF(X) = 0 f o r n>_l ) f o r i n th i s s i t u a t i o n , i f * 0 1 2 X : 0 > A r X > X. > X -> • • • i s an F-acyc l i c r e so lu t i on of A, then R nF(A) = H nF(X*) - 3 -and even i f i n j e c t i v e s are not a v a i l a b l e , the r i g h t s a t e l l i t e s of F can be def ined i n terras of F-acyc l i c r eso lu t ions i f there i s a f u n c t o r i a l way of ass ign ing such reso lu t ions to objects o f A and F i s exact on these r e s o l u t i o n s . Th is i s usua l l y done i n group-cohomology with coinduced modules and i n ord inary sheaf-cohomology with f labby sheaves. In a l l cases the fo l low ing pat te rn emerges: We have a c lass 1^  of objects of A with the fo l l ow ing p r o p e r t i e s : (AC1) Each object of A admits a monomorphism i n to an object of JE. (AC2) .1 i s c losed under f i n i t e d i r e c t sums and cokernels of monomorphisms. (AC3) Short exact sequences of objects i n I remain exact a f t e r an a p p l i c a t i o n of F. 1 i s c a l l e d a c l ass of F-acyc l i c objects i n A. I t i s not unique i n genera l but there i s always a maximal one. In order to def ine something l i k e a " r i g h t der ived func to r " we seem to need a f u n c t o r i a l way of s t a r t i n g with F-acyc l i c r e s o l u t i o n s . One of the po in ts to be made i n the sequel i s that i t i s not necessary . The ex is tence of any c lass 1^  with p roper t i es AC1 - AC3 w i l l enable us to const ruct a der ived func to r . Ou t l i n e : Let a quiso (quasi-isomorphism), denoted by a double arrow, be a morphism X* sf Y* of co-chain complexes induc ing isomorphisms i n cohomology. Thus, i n cohomology, a quiso has an inve rse . - 4 -S ta r t i ng with the category K(A) of co-chain complexes and homotopy c lasses of co-chain maps of A, we form a category D(A), whose objects are objects of A but whose morphisms, c a l l e d "quasi-morphisms" and denoted by broken arrows X >• Y, are formal f r a c t i ons of morphisms of K(A) with quisos as denominators. Every quasi-morphism from X to Y induces a we l l-de f ined map i n cohomology. An F-acyc l i c r e so lu t i on can be viewed as a quiso of complexes. A : : > 0 > 0 >» A > 0 > 0 > "' \ I 1 i I I I* : ' * ' > 0 > 0 > 1 ° > I 1 • * I 2 > • • • Given a map X —^->Y and reso lu t i ons X > I*, Y ~ v J * , we obta in the diagram X > I* 4 which es tab l i shes a quasi-morphism I > J from I to J . As a consequence of the exactness of F on I_, we know that F turns quasi-morphisms between 1 - complexes i n to quasi-morphisms between t he i r images. Taking the map induced i n cohomology by the quasi-morphism F( I ) > F(J ) , we have: H nF(I*) > H nF(J*) Procur ing the long exact sequence from th i s k ind of " d e r i v e d " func tor i s a more subt le task, but i t too, can be done. We note that th i s procedure of de f i n ing the " R n F ' s " make hard ly any d i s t i n c t i o n between objects X of A and complexes X of such ob jec t s . In genera l we w i l l work on the l e v e l of the complexes. Ev ident l y any complex X of objects of A admits a quxso X rr? I i n to a complex I of objects of I , and any morphism f : X ^ Y induces a quasi-morphism of t he i r respec t i ve " F - a c y c l i c " r e so lu t i ons X* = > I* i The func tor RF : D(A) > D(B), i s def ined by s e t t i n g RF(X*) = F( I* ) . The R nF can then be recovered from RF by taking n^ 1 cohomology. Thus RF w i l l be r e f e r r ed to as the " d e r i v e d " functor of F and we may look upon i t as a k ind of extens ion of F from K(A) to D(A). This p o l i c y of s tay ing on the l e v e l of the complexes has the advantage of c ircumventing the usual s p e c t r a l sequences i n F G the study of composite func to rs . Indeed, given A > IS > C., we obta in a na tu ra l map y : R(G*F) > RG»'RF which i s an - 6 -isomorphism under favourable cond i t i ons . Dua l l y , what has been sa id about r i g h t der ived functors app l i es equa l l y w e l l to l e f t der ived functors a f t e r reve rs ing the appropr ia te arrows, in terchanging the terms " l e f t " and " r i g h t " , " p r o j e c t i v e " and " i n j e c t i v e " , e t c . In conc lus ion we sum up some of the strengths and weaknesses of t h i s approach: (1) We can def ine der ived functors whenever we have a c lass 1^  of F-acyc l i c objects as descr ibed above; appropr ia te l y dua l i zed f o r l e f t -de r i v ed func to r s . An example of a n o n - t r i v i a l a p p l i c a t i o n i s the problem of de f i n ing TOR f o r sheaves where we have p lenty of f l a t s but hard ly any p r o j e c t i v e s . (2) Der ived functors f o r complexes — hypercohomology, are handled j u s t as e a s i l y as those f o r ob jec t s . (3) The s p e c t r a l sequence r e l a t i n g to the der ived functors of a composite i s supplanted by a s imple composit ion of der ived func to r s . However, f o r the more de l i c a t e in format ion ex t rac tab le from a s p e c t r a l sequence, i t seems that t h i s ve r s ion w i l l usua l l y be too crude, though the usual s p e c t r a l sequences can always be set up. - 7 -CHAPTER II CALCULAS OF LEFT FRACTIONS § 1 . D e f i n i t i o n s of Loca l i s a t i ons and Extensions Let S be a c lass of morphisms i n a category JC. We would l i k e to " i n v e r t " the elements of S. In other words, f i n d the " s m a l l e s t " category such that every morphism of S i s i n v e r t i b l e . The f a m i l i a r s i t u a t i o n of a r i ng l o c a l i s e d at a m u l t i p l i c a t i v e system gives us a c lue as how to def ine such a category. D e f i n i t i o n . The l o c a l i s a t i o n of C: wi th respect to S should be a p a i r (Cg,Q) where Q : C > Cg i s a functor with Q(s) being an isomorphism f o r a l l s € S and i s un i v e r s a l wi th respect to t h i s proper ty , i . e . given any other s i m i l a r p a i r (D, G) , there i s a unique func tor H : C_ > JJ such that H-Q = G. G \ • H D As i n the case of extending a homomorphism between two r ings to t he i r l o c a l i s a t i o n s , we w i l l study the condi t ions under which a given func tor F : £_ > D w i l l extend to the l o c a l i -sa t ions (Cg,Q) and (Dg , ,Q ' ) . That i s , when we can complete the diagram - 8 -C — - — > D Q £s £s Obvious ly , we have: P ropos i t i on 1. A necessary and s u f f i c i e n t cond i t i on f o r F to extend to Cg i s that Q'F(s) be an isomorphism in Dg' \ / s f e S . F w i l l be c a l l e d (S,S ' ) exact . In case an extension does not ex i s t we consider the weaker not ion of an approximate extens ion . D e f i n i t i o n . A r i g h t approximate extension of F i s a p a i r ( R F , £ ) , where RF : Cg > Dg r and £ : Q'F > RFQ a na tu ra l map s a t i s f y i n g the un i ve r sa l property : f o r any other p a i r (G,<j>) wi th G : Cg > D g» and <J> : Q'F > G«Q there i s a unique map n : RF > G such that HQ* 5 = <j>-F 5 C > D Q'F RFQ Q In conjunct ion with extensions we consider composite - 9 -F G functors A H3 and r i gh t approximate extensions (RF,?) , (RG,?') and ( R ( G F ) , £ " ) . Apply ing the u n i v e r s a l i t y of the p a i r (R(GF), C") to ( RG-RF,RG(5_) . £ J F ) , we obta in a (unique) n a tu r a l map y : R(GF) > RG-RF s a t i s f y i n g the r e l a t i o n Yx' 5'x =RG( CX)' C'(x). A — E - v B — c In case y i s an isomorphism, there i s a na tu ra l map e : RG-Q'F >• R(GF)Q which we w i l l c a l l the edge morphism, and i s def ined by ex - Yx** J^G(?x) where E, : F >- RF-Q i s the na tu r a l map. - 10 -§2. Ex is tence and Desc r ip t ion We assume S contains i d e n t i t i e s and i s c losed under composi t ion. For b r e v i t y elements of S w i l l be denoted by double arrows ^ . ObCg = ObC. Arrows of Cg are equivalence c lasses of reduced words fs'-'-gt-"'" that i s , f i n i t e f s 8 t diagrams of the form X — . . . Y w i th the equivalence r e l a t i o n : X ===£> Y <4=== X rep laced by 1„ A. Y 4 * s Y — Y replaced by l y The composit ion of any two reduced words X-and Y - X ^ : - ^ . . Z i s X - + < = - * ^ . . . Y - » - ^ : - * < = . . . Z . The f f functor Q : C —»> C£ i s de f ined by Q(X —>• Y) « X Y with -1 s Q(s) = X-4= Y f o r s i n S. I f G:C_ —>• D i s any functor wi th the property that G(s) i s an isomorphism i n B f o r every s i n S, then the functor H : _Cg > D s a t i s f y i n g the r e l a t i o n H-Q = G f s g t Gfs ) - ^Gf t')-"'" i s uniquely determined by H(X—><= -><== . . . Y) =? G(X) —> -+ -*»".. .G(Y) GCf) G(g) Remark. This i s too academic and cumbersome to be of much use. We would l i k e to s i m p l i f y the sequence of arrows f s g fc X — > ^ = — Y by success ive "push-outs" of each p a i r and thus reducing the words to the form X — * » ^ = Y. - 11 -f s g t In order to ensure that pa i r s and r e s u l t i n g from the same "pushout" diagram are i d e n t i f i e d we requ i re the a d d i t i o n a l hypothesis on S that i f pa i r s of arrows > are equa l ized by an element s6 S, then they are 6 coequal ized by some t e S . More genera l l y th i s leads us to consider an ax iomizat ion of S. D e f i n i t i o n . The c lass S i s c a l l e d a r i g h t m u l t i p l i c a t i v e system i f : (FRO). S contains a l l i d e n t i t i e s and i s c losed under composi t ion. (FR1). Any diagram >. with s € S can be •If X* 1 > Y X 8 ;> Z completed to a commutative square with t 6 S. X' »• Y a (FR2) Given a p a i r of maps X * Y i f there i s a s £ S such that as = 3s, then there i s a t e S such that ta = tB. Dua l l y we can def ine a l e f t - m u l t i p l i c a t i v e system FRO", F R 1 ° , F R 2 ° by revers ing a l l the arrows. A c lass of morphisms which i s both a l e f t and r i g h t m u l t i p l i c a t i v e system in A i s c a l l e d a m u l t i p l i c a t i v e system in A. - 12 -P ropos i t i on 2. Let £ be a category and S a r i g h t m u l t i p l i c a t i v e system in C_, then the l o c a l i s a t i o n of £ with respect to S ex i s t s and i s unique up to isomorphism. Proof . Appendix. Granted the ex is tence of a r i gh t m u l t i p l i c a t i v e system S i n C we construct the l o c a l i s a t i o n (Cg,Q) by s e t t i n g : ObCg = ObC For any X, Y £ ObCg, a morphism from X to Y, denoted by broken arrows X •> Y, and c a l l e d a quasi-morphism from X f s to Y, i s an equivalence c l ass of morphisms X vY' <" Y f s where the equivalence r e l a t i o n between two pa i r s X >• Y' — ~f - y ~s and X »• Y S Y holds i f there i s a diagram such that the two inner t r i ang l e s are commutative and the outer edges form a commutative square. The composit ion of two f s quasi-morphisms X > Y = X • Y 1 <^  Y and h u Y + Z = Y > Z < Z i s def ined by the diagram X Y' < Y —2—* Z' < Z (1) W - 13 -v/ith (1) obtained by apply ing (FR1) to Y ' <^= Y • h • > Z ' . The functor Q : C > Cg i s def ined by Q(X) = X and Q(X Y) = X -J—> Y < — ^ Y with Q ( s ) - 1 = Y < ~ X f o r s e S. A l t e r n a t e l y one can a lso descr ibe the category C„ by : P ropos i t i on 2' . HomrjgCX.Y) = lira HomrjCXjY') Y ' 6 0b J Y s where Jy i s the f i l t e r e d category of diagrams Y :—> Y' and s whose morphisms are commutative diagrams Y Y 1 Y " We now go on wi th the problem of cons t ruc t ing r i g h t approximate extens ions . Let .1 be a f u l l subcategory sucy that : (RAC1). F|JE i s (S 0 >S' ) exact where S 0 = AR IHS . (RAC2). For every object X of C, there i s a s £ S and an g object I of I such that X :—> I. (RAC2) immediately impl ies that S 0 i s a r i g h t m u l t i p l i c a t i v e system i n I, and moreover, that I j i s a f u l l subcategory of Cg. By P ropos i t i on 1, F |_I has an extension E:(F to lSo. We next note that the i n c l u s i o n functor i : I_ > C c —1> o —1> i s not only f u l l y f a i t h f u l , but a l so y i e l d s an equivalence of ca tegor i es . To see t h i s , we ass ign to each object X of Cg a morphism Vx : x " )> r (X ) , r ( X ) £ o b J E . This def ines a - 14 -functor r : —» I_g and an isomorphism v : Id^ = i ' r —S which i s n a tu r a l s ince any morphism X > Y sets up the diagram X = > r(X) I Y = r > r(Y) represent ing a we l l-de f ined quasi-morphism r(X) » r ( Y ) . Upon c a l l i n g I_ a r i g h t F-acyc l i c subcategory of C., we have P ropos i t i on 3. Let F : £ > D be a covar iant functor and S,S' r i g h t m u l t i p l i c a t i v e system i n (2,1) r e spec t i v e l y . Let I be a r i g h t F-acyc l i c subcategory of C, then the functor RF together wi th the na tu ra l map F ( v y ) £ x : F(X) £-> F(rX) = RF(X) i s a r i g h t der ived functor of F. Moreover, ( R F , £ ) i s unique up to isomorphism of func to rs . Proof . We must prove u n i v e r s a l i t y . Let (G,n) be g iven. G : Co > Dgi n : Q'F *• GQ Want: a unique t, : RF > G such that C^'S = n-- 15 Q'F —£-> RFQ G.Q F i r s t , to see the uniqueness of £, f o r any object X of C_, p i ck X ==£>• r (X ) . F(X) F ( v x ) -> F(rX) = RF(X) RF(X) / RF(v x ) >• RF(rX) r (x) N > C r ( x ) 4, ' r (x) G(X) G(v x ) G(rX) Note that G(v x ) i s an isomorphism s ince v x i s one i n Cg and because £ r ( x ) i s a n isomorphism there i s only one poss ib l e choice f o r C r ( x ) as shown (namely, C r ( x ) = n r ( x ) *? r (x ) ^' But the lower t rapezo id has i sos on top and base, thus there x i s only one poss i b l e choice f o r £ C x = G ( v x ) _ 1 - C r ( x ) - R F ( v x ) or s ince F(rX) = RF(X), we a l so have C x = G(v x ) 1-nr(xy - 16 -N a t u r a l ! t y : n / s G(v ) - 1 C x : F(rX) r W > G(rX) » G(X) i s c l e a r l y na tu r a l with respect to morphisms X • Y i n C_. Next by extens ion , we observe that formal ly there i s a b i j e c t i o n between the sets Horn (RFQ,G'Q) « Hom(RF,G). Hence £ i s n a t u r a l wi th respect to quasi-morphisms X > Y' < Y i n Cg. F i n a l l y to show that the p a i r (RF,5) i s unique, assume another p a i r (RF ' , ? ' ) having the same p r o p e r t i e s , then the u n i v e r s a l i t y of (RF,?) guarantees the ex is tence of a unique n : RF • RF' s a t i s f y i n g HQ•? = £ ' . S i m i l a r l y , there i s a lso a unique n' : RF' > RF with n' = £. But t>..''n' 'K = £; n'n 'V - V- Therefore n -n* = Id ; n' -1 = Id. Q Q Q Q Q Q Q Q Remark. A l e f t approximate extension of F i s a functor LF : Cg • Dg together with a na tu ra l map a : LF-Q > Q'F with the property that f o r any s i m i l a r pa i r (G,x) there i s a unique to : G > L.F s a t i s f y i n g a 'uq = T . By appropr i a te l y reve rs ing the arrows p ropos i t i on 3 can a l so be proved f o r l e f t F-acyc l i c subcategor ies and l e f t approximate extens ions. And although the not ion of an a c y c l i c subcategory do leave us much freedom of cho ice , we w i l l show i n the appendix that there i s , i n f a c t , always a maximal such subcategory. - 17 -Next suppose we have two functors F : C • D G : D »• E and r i g h t m u l t i p l i c a t i v e systems S , S ' , S ' ' i n these respec t i ve ca tegor i es . Suppose fu r the r there i s a r i g h t F-acyc l i c sub -category I of C and a r i g h t G-acyc l i c subcategory V of J). We then have r i gh t approximate extensions RF, RG, R(GF) and P ropos i t i on 4. I f 'F(1)QI,, then the canon ica l morphism Y : R(GF) — — > RG.RF i s an isomorphism. Proof . For X i n ObCg, to obta in RG-RF we have to apply G to I' i n the s i t u a t i o n 5x F(X) —>• RF(X) = F(rX) I V This g ives GF(X) v GF(rX) = R(GF) (X) G( I ' ) = RG-RF(X). Because F ( r X ) £ I_', y i s now an isomorphism i n Eg i i - 18 -F i n a l l y , under the condi t ions of P ropos i t ion 4, we can def ine an edge morphism e : RG-F R.(GF) by the commutative diagram RG-F 5£i£=) ^ RG-RF R(GF) - 19 -CHAPTER III PSEUDO SPLIT SEQUENCES AND MAPPING CONES Let A be an abe l i an category. We denote by C(A) the abe l ian category of a l l co-chain complexes over A and by K(A) the homotopy category thereof . For each object X of C(A) ( K(A) ), def ine the complex T ( X * ) by T ( X * ) N = x n + 1 , d T ( X * ) * " D X * T H E N T : C ( - ^ *" C ( - ) ( R E S P * K ( - ^ * K ^ ) i s an automorphism and i s c a l l e d the t r a n s l a t i o n automorphism. We w i l l o f ten wr i te X * [ l ] ins tead of T ( X * ) and X*[n] ins tead of T N ( X * ) . To every short exact sequence Z : 0 > XQ — i - y X * P > X*. • 0 i n C(A), there corresponds a long exact cohomology sequence n n 6 n . . . Hn(xJ)H-^ 2 H n ( X * ) H - ^ H n ( X | ) — H * + 1 (XJ) — v with the connect ing morphism 6^ furn ished by the Snake Lemma. We consider the fo l low ing condi t ions on short exact sequences under which more r e a d i l y access ib l e desc r ip t ions of the connect ing morphisms are a v a i l a b l e : D e f i n i t i o n . We c a l l a short exact sequence Z : 0 *-X*Q — i-v X * P > X * • 0 - 20 -i n C(A) quas i or pseudo s p l i t i f i t s p l i t s i n each dimension, i . e . 0 * Xjj • > X" V 0 s p l i t s f o r each n. t P ropos i t i on 5. I f Z q u a s i - s p l i t s , then one can ex t rac t from the d i f f e r e n t i a l operator [ " ^\ = [ !?x* ^ I of [6 Y j \0 d x * j X 1 , a morphism 3: X 2 c a l l e d the twist of E. 3 has the most important property that i n cohomology the connecting morphism 6£: H n (X 2 ) >- H n + 1 (X Q) one obtains from the Snake Lemma i s induced by the tw i s t . In other words, the long exact cohomology sequence of Z i s obtained from the sequence of complexes ... —«•[„] •x^tn] ^ ^ [ n + l ] m . X * [ n + 1 1 — * ••• by taking 0-cohomology. Furthermore 3 measures the degree of s p l i t n e s s of Z i n the sense that i t i s homotopic to zero i f and only i f the sequence s p l i t s . Proof . Appendix Remark: As to t h e i r a v a i l a b i l i t y , we would l i k e to " t r a c k " down a l l the q u a s i - s p l i t sequences and w i l l show that up to homotopy isomorphism—an isomorphism of short exact sequences in K(A) , every q u a s i - s p l i t sequence i s a " t runca ted mapping cone" sequence. We next construct a q u a s i - s p l i t sequence * whose twis t i s the homotopy c l a ss of a g iven map f : X > Y . F i r s t , the mapping cone of f i s the co-chain complex def ined by c n = Y n e xn+1 or Then the sequence * : 0 Y* 1 > C* P > X*[l] > 0 q u a s i - s p l i t s with twist T ( f ) , which i s equiva lent to f i n the homotopy category. By a mapping cone sequence we s h a l l mean a sequence of the form - 22 -* f * * * T(f) * * T(r>) * ** : X Y -—> C >X [1] ±^±*Y [1] yc [l]^h [2] From the long exact cohomology sequence der ived from * , or apply ing 0-cohomology to * * we immediately have * * P ropos i t i on 6. f : X • > Y induces isomorphisms i n A cohomology i f and only i f the cohomology of i s t r i v i a l . P ropos i t i on 7. Given any homotopy commutative diagram * f * X l » X 2 I | * y* Y > Y 1 g 2 there i s a map a : C.. • C such that each of the f g squares i n X* -V C* • X*[l] T ( a i ) \a2 \a I •k * * Y 2 ^ C g — * Y ± [ 1 ] commutes. P roof . In matr ix no t a t i on , a =( a2 ^ \ where 0 a , hn : x"+^ • Y^  i s the homotopy of the o r i g i n a l square. - 23 -Commutativity i s t r i v i a l . In order to e s t a b l i s h a r e l a t i o n between q u a s i - s p l i t and mapping cone sequences, we f i r s t compare any short * •; * P * exact sequence E : 0 > X Q > X-L X 2 ^ 0 with a t runcated mapping cone sequence * : * i * p * Z : 0 > X Q —±-> X1 > X 2 • 0 * : 0 > X Q i > X1 * C_. >•( X Q[1] By P ropos i t i on 7 there i s a morphism w: C > Xn render ing — i the diagram X 0 — i - ^ X x * -> C . - l -»XQ[1] I p I I ""<p'0) I x 2 >• x 2 -*-0 commutative. Apply ing cohomology and the 5 Lemma, H n ( i ) H n (p) f n H n (xj) — * H n al) — * H n ( X*) 5>H n + 1 (X*) H n + 1 (X* ) -1 Hn(tr) , n , * r H (XQ) *H"(X ) (C_.) » H " ( X o [ 1]) > H" (X 1 [ 1 ])• H n (- i ) - 24 -we conclude that -n induces an isomorphism i n cohomology. Hence up to homotopy and cohomology, every short exact se-& i it D A quence Z : 0 > X Q * X± ——> X 2 )• 0 i s a truncated mapping cone sequence. That i s , the diagram * i * p * X 0 *" X l X 2 i \ t • * -*- * * x o — > x i • C - i represents an isomorphism of complexes i n some su i t ab l e homotopy category where morphisms induc ing isomorphisms i n cohomology are i n v e r t i b l e . Moreover: P r o p o s i t i o n . I f Z q u a s i - s p l i t s wi th twist 6 and sec t ion * * ' s, then ir : C . y X^ has a homotopy inverse given by C ) . Thus we have the very important r e s u l t that i n the homotopy category K(A) every q u a s i - s p l i t sequence i s i s o -morphic to a truncated mapping cone sequence. The morphism TT assoc ia ted with Z measures the dev ia t ion of the sequence from being q u a s i - s p l i t and i s always an isomorphism i n co -homology. - 25 -Remark. There i s a dual s to r y . Suppose once again _ A i * p A we are given E : 0 >• X Q • > X^ *- 0. A A Then there ex i s t s a morphism t : X [1] >» C ( again 0 p evoking p ropos i t i on 7) such that x* > . o »• x* [ i ] • 1 >' X*[ l ] 0 0 0 A P A A A x x • c y x [ l ] 1 2 p l commutes. By the 5 lemma i a lso induces an isomorphism in cohomology. P ropos i t i on . ir and i are r e l a t ed by the homotopy commutative diagram A c . -1 x* x 0*[i] A C P A ». A Proof . The maps C . + C i n matr ix no ta t ion - 1 P appears as i s the requ i red homotopy. Now i t i s c l ea r that: P r o p o s i t i o n . The functors H o m ^ ^ ( - , Z ) and Horn^.^ when app l i ed to any mapping cone sequence X — i + Y C f • X [1] Y [1] f C [1] y i e l d long exact sequences * * £JI * * J# * * H o m K ( A ) ( Z ,X ) — H o m ^ (Z ,Y ) — H o r n ^ (Z , C f ) • H o m K ( A ) ( X * ' Z * ) a ^ H o n ^ ^ C Y * ^ * ) * ^ H o m K ( A ) (C*. Z*) « Although A i s assumed to be an abe l ian category, the category K(A) i s i n general not abe l i an . However, i n view of the developments i n th i s chapter, we can axiomize a l l the s t ruc tu res that carry enough informat ion f o r our pur -poses by the not ion of a t r i angu la ted category, where a t r i angu la t ed category i s a t r i p l e (C, A, T) such that : (1) T i s an automorphism of (2. (2) A i s a c o l l e c t i o n of sextuples (X,Y,Z, u,v,w) c a l l e d t r i a n g l e s of C^ , where i n each t r i ang l e X — ± + Y — Z — T ( X ) . A morphism of t r i ang l e s i s a commutative diagram - 27 -x —±L+ Y - ^ U W —2-*- T(X) f | g j h { T ( f ) | X' Y' — W -^-*- T(X ' ) . The t r i p l e (_C, A,T) s a t i s f i e s the axioms: (TR1). A i s c losed under isomorphisms. Every morphism u : X *- Y can be embedded i n a t r i a n g l e X — Y V > Z W ) T (X) . For any object X i n <C, the sextuple ( X , X , 0 , i d x > 0 , 0 ) i s a t r i a n g l e . (TR2). (X,Y,Z,u,v,w) i s a t r i ang l e i f f (Y,Z,T(X) ,v ,w,-T(u) ) i s a t r i a n g l e . (TR3). Given a diagram x — y —2-* Z —2-* T(X) X* —2-+. Y' — Z ' —2-» T (X ' ) where the f i r s t square i s commutative and the rows are t r i a n g l e s , there ex i s t s h : Z *• Z' such that ( f ,g ,h ) i s a mapping of t r i a n g l e s . In the case of K(A), T : K(A) y K(A) i s the t r ans l a t i on automorphism and a t r i ang l e i n K(A) i s any sextuple isomorphic to a mapping cone sequence of the form * f * i * * X y Y C > X [1] . - 28 -CHAPTER IV THE DERIVED CATEGORY We w i l l now fo l low the programme ou t l i ned i n Chapter II f o r the cons t ruc t ion of the der ived functor RF of any functor F : A B. A Let A be abe l i an . We denote C (A) f o r the category C(A) or any one of i t s f u l l subcategor ies C + (A ) , C (A), and C^(A) whose objects are complexes of A bounded above, bounded below, and bounded on both s ides r e spec t i v e l y , and s i m i l a r l y denote K (A), K (A), K~(A) and K (A) f o r t h e i r corresponding homotopy ca tegor i es . A P ropos i t i on 9. In K (A) the c l ass of morphisms which induces isomorphisms i n cohomology, c a l l e d the c lass of quasi-isomorphisms or qu i sos , form a m u l t i p l i c a t i v e system. Proof . Appendix. For b r e v i t y , quisos w i l l be denoted by double arrows = ^ . D e f i n i t i o n . The der ived category D(A) of A i s the l o c a l i s a t i o n of K(A) with respect to the c lass of a l l qu i sos . S i m i l a r l y , there are l o c a l i s a t i o n s D + (A ) , D (A), and D^(A) ° f •+* — B K (A), K (A) and K (A) r e s p e c t i v e l y . As a comparison of t he i r r e l a t i v e s i zes we have: P r opos i t i on : Each of the functors - 29 -D +(A) A • D (A) D(A) D (A) i s a f u l l embedding. Proof . Appendix. Cont inuing with the programme of Chapter I I , we next look f o r " s u i t a b l e " a c y c l i c subcategor ies . I f I i s any subcategory of A such that every object of A admits a monic in to an object of I then fo r X i n C (A) t r i v i a l i n negat ive dimensions, we can const ruct a quiso 0 —• X —• X —• X 1 V J - 1 J -n-1 . . x n _ 1 -JM- x n n-1 X n+1 0 -+ I I2-. Is -n-1 push-1 out i jn-2 _n cokdj. > P by induc t ion and the f ac t s that i n any pushout diagram B 1 I C P n g induces an epimorphism Kera ^ K e r a ' - 30 -and $' induces a monomorphism cok a > > cok a ' . Thus: P ropos i t i on 11. Let I be a fu l l-subcategory of A such that every object of A admits a mono in to an object of I_, then every object of C + (A) admits a quiso i n to an object of C+(_I) Proof . Appendix. Now l e t J3 be another abe l i an category, and F : A • 13 be an add i t i v e func tor . F induces functors C (A) y C (B) K (A) *> K (B) which w i l l s t i l l be denoted by F. We r e c a l l that i f F preserves short exact sequences of objects of 1 and _I i s c losed under cokernels of monomorphisms, then F takes short exact sequences of C + ( I ) i n to exact sequences of C + (B ) . Moreover, i f 1^  i s c losed under d i r e c t sums, then * - s > * + f o r a quiso 1^  " / of objects of C ( I ) , F(C*) = F(I*[1] © I*) = F( I*[1]) © F(I*) = C * ( s ) and the exactness of F on objects of C + ( I ) impl ies that *• F ( s ) - * and i s i n p a r t i c u l a r , (qu iso ,qu iso ) exact . Summarizing, we have F(I^) \>- F ( I 2 ) i s a qu i so , so F maps quisos i n t o qu i sos , P ropos i t i on 12. Let I be a f u l l subcategory of A which i s c losed under d i r e c t sums and cokernels of monomorphisms, f u r the r , assume F preserves short exact sequences of objects of 1^ , then the r e s t r i c t i o n of F to K (I) maps quisos in to qu isos . - 31 -In view of P ropos i t ions 11 and 12, we say that a f u l l sub-category I of A i s r i g h t F-acyc l i c i f (AC1). Every object X in A admits a mono in to an object of I. (AC2). I i s c losed under ( f i n i t e ) d i r e c t sums and co-kerne ls of monos. (AC3) . F preserves short exact sequences of objects of I_. Then, K + ( I ) w i l l be a r i g h t F-acyc l i c subcategory of K + (A) i n the sense of Chapter II (RAC1, RAC2). Hence Theorem 1; I f A contains a r i g h t F-acyc l i c subcategory I_, then the r i g h t approximate extension ( R F , £ ) of F to D +(A) ex i s t s and i s given by D +(A) D + ( I ) - ^ 2 * D(B) as i n P ropos i t i on 3. ( RF ,£ ) i s c a l l e d the r i g h t der ived functor of F. Granted the presence of a r i g h t F-acyc l i c subcategory A A -f-JE of A, RF(X ) i s found on any object X of K (A) by f i r s t * X v * + taking a quiso X '. ^ I i n to an object of C (I) and s e t t i ng RF(X*) = F(I*) and ? x = F ( V ' - 32 -Dua l l y the presence of a l e f t F-acyc l i c subcategory P_ of A guarantees the ex is tence of a l e f t der ived functor (LF,cr ) with LF : D (A) • D(B) . In p r i n c i p l e i t i s of course poss ib l e to speak of the r i g h t approximate extens ion of F to D(A) or the l e f t approximate extens ion of F to D(A). For i t s ex is tence one would need: (a) . Stronger cond i t ions on I_ as to obta in the appropr ia te gene r a l i z a t i on of P ropos i t i on 11, and (b) . Stronger exactness cond i t ions of F on I. For our purposes we w i l l focus our a t t en t i on only on the r i gh t + -der ived functor of F on D , and the l e f t der ived functor on D . For the rest , of t h i s chapter , F: A — * 13 w i l l always denote an add i t i v e func to r , I a r i g h t F-acyc l i c subcategory of A. The higher order cohomology functors can be obtained from RF by s e t t i ng R.nF(X*) = R^RFCX*) and there are na tu ra l maps £ N : H n F > R n F def ined by £ N : H N ( £ ) : H n F(X* ) • H V ( X * ) = R n F (X*) . We next const ruct the long exact sequence of the der ived f unc to r s . - 33 -Let 1 : 0 > X* 1 > X* P > X* » 0 be any short * exact sequence i n C (A) . Up to isomorphisms i n cohomology, E i s e s s e n t i a l l y a truncated mapping cone sequence * C . * i * p A ^ * , A t ^ x 0 - x 1 - 2 » x 2 " *o[1]-+h[1]  A * ? \ c p A More p r e c i s e l y , in, the der ived category D (A), the morphisms if and i are isomorphisms, and the diagrams A i A n A A xQ • x± x2 » X Q [1] 4 4 Jh A i A n A A xn • x. — x _ > c 0 1 2 p and A — i A A A xQ • xx * c_t » x 0[i] A i A A Z A xQ * x± > x2 > X Q [1] A are isomorphisms of sequences i n D (A) . Moreover, these i so -morphisms are na tu ra l i n the sense o f : Lemma 1. For a given morphism of short exact sequences - 34 -* -J A D * xQ 1 > x > x2 > 0 l f o l f i I f 2 * 4 I A p' A E1 : 0 • Y Q 1 > Y 1 >• Y 2 • 0 the diagram * i * p * dE * r , XQ > Xx — X 2 *»X [1] l f 0 | f l l f 2 d 1 A i 1 A p ' A E 1 A Y 0 > Y l Y 2 * Y 0 [ 1 ] i s commutative. A A Proof . By P ropos i t i on 7 there i s a morphism a : C > C , P P render ing square (1) of the diagram \ X2 * C p ^ = X 0 [ 1 ] | f 2 (1) | a (2) { A A , • A ^ ! T 2 ^ C p ' ^ = Y 0 [ 1 ] commutative. Square (2) i s c l e a r l y commutative. Now the quiso r e s o l u t i o n of the mapping cone sequence (*) can be chosen again to be a mapping cone sequence (**), or equ i va l en t l y , under the isomorphism of categor ies D( l ) = D(A), mapping cone sequences - 35 -are preserved. d-* 1 * P * i. • * * * xQ —• x1 —• x2 > X Q [1] —* x 1[i] x 2[i] 1 . I i ^ I 1 A 1 A p A * * * xQ —> x1 x2 > c p • x 1[i] —• x 2[i] 4 U fl 4 U U 4 A a A p A A A A (**) i Q —+ i 1 —*• i 2 )• cg • I.J1] —> i 2 [ i ] Since F preserves mapping cone sequences of K + ( I ) , an a p p l i c a t i o n of i t to ( A A ) y i e l d s the long mapping cone sequence A TT (Q ^  A A A F ( I i ) -JL±H y F ( C g ) » FC^tl]) — • F i n a l l y , t a k i n g 0-cohomology y i e l d s the long exact sequence 0 A + H ° F ( I * ) 0 A H U F (C g ) H ° F ( I * [ 1 ] ) which i s p r e c i s e l y O A O A 1 A 1 A R U F(X 1 ) • R U F(X 2 ) •R J-F(X ( )) * R F(X 1> and we have Theorem 2. {R nF , S n} i s exact on short exact sequences A l * p A I : 0 • X Q -v x± > X 2 -> 0 of C (A) where <$£ = R n F (d z >. - 36 -Comparison with C l a s s i c a l Theory. C l a s s i c a l l y , one i s i n t e res t ed i n R nF(X) f o r X in A. By cons ider ing X as a complex concentrated at degree 0 the quiso r e s o l u t i o n X )> I can always be chosen to be the form • ' • > 0 >• X — — > 0 > 0 • 1 I \ b ••• > 0 + 1 ° > I 1 • I 2 > 0 1 with the sequence 0 > X > I y I > exact . We always have R nF(X) = 0 f o r n<0. The long exact cohomology sequence f o r 0 —> X —»• Y — Z — 0 takes the form 0 —> R ° F ( X ) —»• R ° F ( Y ) —> R ° F ( Z ) —> R^CX) — . . . Hence: (1) The map £° : F >• R^F i s an isomorphism i f and only i f F i s l e f t exact . In that case, the _RnF ' s are the r i g h t s a t e l l i t e s of F. (2) For any X i n ob(I), R nF(X) = 0 f o r n>0. (3) If X i s a c y c l i c (£ i s an isomorphism) then R nF(X) = 0 fo r a l l n>0. And i n case A has enough i n j e c t i v e s , the R n F ' s are isomorphic to the c l a s s i c a l der ived func to r s . F G We conclude with a note on composite functors A — • B^  > C_. I f F sends the r i g h t F-acyc l i c subcategory I of A i n to a r i g h t G-acyc l i c subcategory J_ of then we have an isomorphism Y : R(G-F)| *> (RG.RF)| which gives r i s e to an edge morphism e : RG | B . F R(G-F) | A def ined by the commutative diagram RG| B . F -y R(G.F) | RG(g) (RG-RF) | A . and i s obta ined by apply ing G to F(X ) F(v ) F ( I ) y i e l d i n g GF(X*) GF(I*) = R(G-F)(X*) RG.F(X*) = G(J*) This rep laces the usual s p e c t r a l sequence arguments concerning composite func to rs . - 38 -CHAPTER V EXT We conclude with an example of how the Ext - functors work out i n the language of der ived func to rs . F i r s t we note * A A Lemma a. Any quiso s : I = ^ Y where 1 i s an i n f e c t i v e i n C + (A) has a homotopy i nve r se . A Hence every morphism i n D(A) of a complex X to a complex of i n j e c t i v e s bounded below i s represented by an ac tua l morphism of complexes, and i f A has enough i n j e c t i v e s , there i s a canon i -ca l equiva lence of ca tegor ies : K + ( I ) = D +(A) . Next, we observe that the Hom-functors of A can be extended to a b i f u n c t o r Horn* : C ( A ) ° P P x C(A) > C(Ab) by s e t t i n g Hom n(X*,Y*) = II7/ Hom.( X P , Y P + n ) P € ™ _A d n = n ( d P _ 1 + ( - l ) n + 1 d P + n ) p € 2 x y Under th i s d e f i n i t i o n , the n-cycles of the complex A A A Horn ( X ,Y ) are i n a one-to-one correspondence with morphisms A A of X to Y [n] and the n-boundaries corresponds to those mor-- 39 -phisms which are homotopic to zero . Thus one has a na tu ra l isomorphism (*) H n(Hom*(X*,Y*)) * Hom K ( A ) ( X * , Y * [ n ] ) which, together with Lemma a gives * + Lemma b. For each i n j e c t i v e I i n C (A) the functor A A A A X > Horn (X ,1 ) preserves quisos i n C(A). There fore , assuming that A has enough i n f e c t i v e s , one def ines the Ext-groups by n * * n A A A Ext (X ,Y ) = H (Horn (X ,1 )) A A A f o r X i n C(A) and an i n j e c t i v e r e s o l u t i o n Y > I . For objects X, Y of A and an i n j e c t i v e r e s o l u t i o n Y ^ I of Y, Ex t n (X ,Y ) = H n(Hom*(X,I*)) = H n (Hom(X,I*) ) . Thus the Ex t n (X ,Y ) def ined i s the usual Ext . A A A F i n a l l y f o r any quiso s : Y :—^ I of Y i n t o a complex of i n j e c t i v e s , by Lemma a H o m D ( A ) ( X * , I * [ n ] ) e H o m ^ ^ (X*, I*[n]) and (*) H o m K ( A ) ( X * , ] : * [ n ] ) = Hn(Hom*(xV*)) - 40 -one gets na tu ra l isomorphisms Ext n (X* ,Y* ) e Hom D ( A ) ( X * , Y * [ n ] ) f o r X* i n C(A), Y* i n C + (A ) . A P P E N D I X - 41 -1. CALCULAS OF LEFT - FRACTIONS We use the axioms: (FRO) S contains a l l i d e n t i t i e s and i s c losed under composi t ion. (FR1) Any X wi th sG S can be completed 4 X' • Y to a commutative square X >» Y' wi th t 6 S. *fr i n X' >> Y (FR2) Given X > Y, suppose 3 s £ S with as = 3 s , then there i s t€S such that tot = tg . D e f i n i t i o n . A quasi-arrow from X to Y i s a diagram X —> Z •£= Y. D e f i n i t i o n . A k i t e i n C i s a diagram of the form X W with t r i ang l e s (1) and (2) commuting. 42 -[1] Any diagram (*) Y 1 can be completed to a k i t e . Proof . An a p p l i c a t i o n of FR1 to Y ' ' ft Y Y' y i e l d s a commutative square Y " Y : Z it Y' Add on to (*) Y ' X Y Y' ' D e f i n i t i o n . Two quasi-arrows are sa id to be " ~ " i f they f i t i n to a k i t e whose edges form a commutative " square " . [2] " ~ " i s an equivalence r e l a t i o n on the set of a l l quasi-arrows from X to Y. Proof . Symmetry i s t r i v i a l ; r e f l e x i v i t y fo l lows from X' ^ s . X | Y Y' f S , ^ Y' - 43 -T r a n s i t i v i t y : Assume (X Y' ^ = Y) - (X -» Y " <*= Y) and (X — • Y' ' ^= Y) ~ (X ->• Y " » Y) then there are k i t e s with equal edges: X Y' Z' X By FR1) 3 a commutative square Z' —> W Y Z " But s ince the maps Y =^ Y' Z' z " ^ w a r e e t* u a 1' Z' — ^ Z " = ^ W = ^ Z are a lso equal ( fo r some Z by FR2) I t fo l lows that the outer edges of ^ Y' X Y => Z are equal , D e f i n i t i o n . An equivalence c lass of quasi-arrows f o r X to Y i s c a l l e d a quasi-morphism from X to Y. We w i l l denote quasi-morphisms by broken arrows X -> Y. - 44 -[3] Equa l i t y can be tested on any k i t e . That i s , given a k i t e Y' y % X Y Y " with (X -> Y 1 <= Y) - (X Y) there i s Z = ^ W equa l i s i ng i t s outer edges. Proof . By d e f i n i t i o n of equiva lence, there i s a k i t e Y ' X Y Z' whose edges are equa l , and we have w i th the square on the l e f t commutative. By FR1 we get a commutative diagram Z' 4= Y V • Z W' , there i s a Since the morphism Y Y' equa l izes Y' W1 =^ W " render ing the rows of Y 1 Z l ^ w ' ^ w " equa l . Correspondingly the equa l i s a t i on of Y = ^ Y " on the rows Z 1 Y ' 1 ^ ^ W induces a morphism W =^ W ' " coequa l i s ing them; and an a p p l i c a t i o n of FR1 to W"'<£= W*=$> W' ' y i e l d s - 45 -W " ' £ = W =^ W" . F i n a l l y from W ^ W where (1) (2) and (3) are commutative, i t fo l lows that the k i t e X Y = ^ Z = ^ W has equa l i sed outer edges. Y " Note. [3] works fo r pseudo-kites as w e l l . That i s , k i t e - l i k e Y' diagrams X . Y —V W i n which Y -> W i s a r b i t r a r y . >*. & Y ' ' ——' [4]. Any two quasi-arrows making a f i x ed "ft X' —y Y i n to commutative squares(by FR1 ) are equ iva lent . X — • Y' X —*• Y' 1 Proof . Assuming 4b "rt" and -fr ^ commute, X' —> Y x ! —• Y we use [1] to f i t X X' —• Y i n to a k i t e - 46 -whose outer edges are equa l i sed by X* =^ X ; hence a l so by some Z =^ W. [5]. I f the quasi-arrows X Y are equ iva lent , so are X' X ^ J " Y Y f " >*• Y 1 ' ^ Y ' ^ ^ Proof . Any k i t e X Y whose outer edges are ^ Y ' equa l i sed produces by composit ion a k i t e whose edges are t r i v i a l l y equa l i sed . [6], Composition of quasi-morphisms. Given X Y' Y , Y -* Z' 4 = Z , we def ine t h e i r composite by apply ing FR1 to Y 1 ^ = Y -> Z' Y »- Z' W By [4], [5] th i s does not depend on W. Dependence on Y 1 : Suppose X —> Y 1 are equ iva lent with compositions Y and X - 47 -y > w' x Y — > z' 4 = z Y " * W " Then any k i t e on X ^ Z gives a pseudo-kite on W " ^ Y ^ X - Y . The l a t t e r can be equa l ized by some s € S. Dependence on Z' can be analogously proved. Hence composit ion of quasi-morphisms i s we l l de f ined . A s s o c i a t i v i t y fo l lows from the diagrams X —> Y' 3= Y Z' Z — » W' <£= W 4>d -> B X —> Y '<$=Y —>Z'-4= Z —»• W W \ . ^ ^ £ b # d C a B B B The quasi-morphism X » X = X ^ X ^ = X i s the i d e n t i t y . s s Note that fo r s : X =^Y , X Y •<= X i s a lso the i d e n t i t y by X - 48 -D e f i n i t i o n . We form the category C^: Ob = Ob C — D — Horn (X,Y) = set of a l l quasi-morphisms X > Y. -S f f 1 and def ine Q : C • C g by Q(X Y) = X Y 4= Y . Then Q ( l ) = ln, x and the diagram x Q(x) X Y 4= Y - 8 — • Z ^ = Z shows that Q(g^f) = Q(g) .Q(f ) . [7] . Q(s) i s an isomorphism fo r a l l s £ S , P roof . Given X =^ Y , put t = Y —h- Y <£= X. Q ( s ) ' t = l y : Y - i - ^ Y X Y <== Y N Y ^ - i f Q ( s ) = l v X ==$> Y <£== Y Y ^ = X Y ^ l [8]. ( C „ , Q ) i s u n i v e r s a l . —S Proof . Given G : C > JD w i th G(s) i s o f o r a l l s e S . ————— )^ def ine H: (} » D by H(X) = G(X). On quasi-morphism - 49 -s G ( s > ~ 1  X > Y = X — • Y'<£= Y, H(X > Y) = G(X) — y G(Y') > G(Y) . H i s we l l-def ined s ince equiva lent quasi-arrows can be f i t t e d in to Z whose outer edges are equa l i sed . Suppose H' : .C • D. i s another functor s a t i s f y i n g H ' .Q = G, then H and H' agree on objects of C^  . To see that they agree on o f J s quasi-morphisms X •> Y = X —> Y' Y , we have H(X Y'<=rS Y) = G ( s ) - 1 -G ( f ) = [H '-Q(s ) ] " 1 .H 'Q ( f ) = H ' (Q (s ) " 1 )-H ' (Q ( f ) ) = H ' (Q (s ) - 1 -Q( f ) ) = H' (X Y' Y) . Thus the ex is tence of the l o c a l i s a t i o n (C C ,Q) i s e s t a b l i s h e d . As f o r i t s uniqueness, i f (C ! ,Q ' ) i s another l o c a l i s a t i o n then by the u n i v e r s a l i t y of (C „ ,Q ) there i s a unique H : C„ * C' s a t i s f y i n g —a —b —S H*Q = Q' ; s i m i l a r l y , there i s a unique H' : C_' ^ C_ such that H'-Q' = Q . Hence H'-H = 1 and H-H' = 1 , . -S ^5 Remark. Dual ly us ing the ca l cu l as of r i gh t f r a c t i ons and FRO 0 , F R 1 ° , F R 2 ° , one can def ine quasi-morphisms X *{ Y' > Y and show that i n case S i s a l e f t m u l t i p l i c a t i v e system, the l o c a l i s a t i o n (C ,Q) a lso e x i s t s . - 50 -D e f i n i t i o n . A category I i s s a id to be f i l t e r e d i f : L I ) . Every p a i r of objects of I can be embedded i n a diagram X Y L2 ) . Given X i n I, there ex i s t s Z such V Y* Y ^ that the square X _ Z commutes. ^ Y ' L3 ) . Given a diagram X > Y , there ex i s t s a map Y > Z such that the two maps obtained by composit ion are the same. I f JE i s f i l t e r e d , then 1 behaves as w e l l as an induc t i ve system fo r taking l i m i t s ( Grothendieck Topo log ies , Chapter I ). [9] . For each object Y of C_, we def ine a category 1^: s Objects of I are morphisms Y ""' ^ X with s £ S . s A morphism i n 1^  between two objects Y = ^ X and t X >^ x' Y =4>X' i s a morphism f : X —>• X' such that the diagram ^ commutes. We c l a im: (1) . I i s f i l t e r e d . (2) . For objects X,Y in C, Horn (X,Y) = lira Horn (X,Y ' ) _ c ->- }± Y'eOb I y - 51 -(3) . I f C i s a d d i t i v e , so i s C„ , — —S Proof . X' (L I ) . For two objects Y = ^ X and Y =T> X' apply (FR1) to ft Y =$> X X' — » Z to get the commutative diagram *fr "ft . Y X 5** V S ' (L2). Given tff f ^ with fs = gs = s 1 by (FR2) there ex i s t s x • » X' X' ===? Z such that 8 ^ Y .S 1 hf = h g Jt-jJS h x ; x == there fore (L2) i s s a t i s f i e d . (L3). Given Y = T ' X' with t r i ang l e s (1) (2) commuting. Complete X ' * - X x' X _ vc- -• Z- Y to a k i t e X ^ <• ^ Y ==^ W and note that the morphism Y =^ > X equa l izes the p a i r of composites X X , ,^_J^ W. Therefore by (FR2) there i s W Z such that the top and bottom rows of X -^.x'D*W=^Z a r e e c l u a l - F i n a l l y the diagram * X' Y J = ^ X _ ^ X " shows that (L3) i s s a t i s f i e d . - 52 -(2) i s c l ea r from the d e f i n i t i o n of l i m (3) Because Horn (X,Y' ) i s an abe l i an group fo r each Y ' , Horn (X,Y) = l im Horn (X,Y' ) i s a l so an abe l i an group, -s \ , ~ Remark. I f S i s a l e f t m u l t i p l i c a t i v e system i n C_, one can a l so c a l cu l a t e Horn (X,Y) as l im Horn (X ' ,Y) where J i s the -s x' e ob J ~ X f i l t e r e d category o f objects X 1 — r x and morphisms x' • X " . 2. A NOTE ON F-ACYCLIC SUBCATEGORIES The not ion of a F-acyc l i c subcategory appears to leave much freedom of cho ice . In genera l , i t i s not necessa r i l y unique. However, we show that there i s always a maximal such subcategory. D e f i n i t i o n . Let F : C D and (RF ,£ ) the r i g h t approximate extens ion of F. An object of X of C i s s a i d to be r i g h t F-acyc l i c i f and only i f p : F(X) > RF(X) i s an isomorphism. ^ X * Lemma. The f u l l subcategory _I of C cons i s t i ng of a l l r i g h t F-acyc l i c objects i s a r i g h t F-acyc l i c subcategory of C_ conta in ing I. Proof . (RAC1) Set F* = F|I*, S* = SOARI* , F Q = F | l * s. * For objects X, Y i n I and a morphism X 7 Y i n S . We have X ==^ r(X) F(X) F(rX) = RF(X) SJJ v ^ Q ' F * ( s ) | I F(o0 = EF Q (a ) Y r(Y) F(Y) —»• F(rY) = RF(Y) - 54 -EF n (a ) i s i n v e r t i b l e i n D „ , s ince a i s i n v e r t i b l e i n C „ . — U —b —o Hence Q F (s) i s i n v e r t i b l e i n ^ , , so F i s (S , S) exact . (RAC2) In order to show each X i n C admits X * * * with s e S and 1 6 ob I_ , i t s u f f i c e s to show I_^ I_ s ince I has th i s proper ty . But f o r X i n J., and X .,, »^ r(X) wi th v Y £ S = S f lAR I , F_ i s ( S „ , S ' ) exact . There fore , - 55 -3. QUASI-SPLIT SEQUENCES A short exact sequence * i * n * Z : 0 —> X C Y —> 0 i n C(A) i s c a l l e d pseudo or quasi s p l i t i f each of the short exact sequences 0 —>• X n —> C n — * Y n — * 0 s p l i t s i . e . , C n - X n ® Y n , * [1] . Let us see how the complex C i n th i s case may be descr ibed i n terms of X and Y . The maps i and p can be wr i t t en as mat r i ces : * and the d i f f e r e n t i a l operator 3 of C may be represented by 3 c a * ) : X n O Y n • X n + 1 » Y n + 1 n .n \Y « , n n ^ „n+ l n v n v n + l where a : X ——*• X 3 : Y ^ X yn : Xn > Y n + 1 6 n : Y n > Y n + 1 - 56 -From 9 c i = i 3 x , fo. 3 \ / 1 a Y, From P 3 C = 3 y p , (0,l y) (" \ \ = 3V(0,1V) we get (Y,6) = (0,3y) There fore , 3 has the form 3 3 X 3 o v Moreover from 3 = 0, 8 x P 3 x & 0 3 ) l o 3 V /o 0 0 0 We get 3 3 + 33„ = 0. x Y Thus 3 i s a complex map Y > X [1]. 3 depends not only on the given q u a s i - s p l i t sequence Z , but a lso on the sec t ions S n : Y ° —>• C n used to decompose „n . . , , „ n /0\ „ n „n C . In our matrxx notatxon, we have used S = ^ \ : Y —> C and I a 3 \ / 0 V / 0 Y I & \ 10 \ / 3 0 Now, any other s ec t i on must be of the form S' = !M where h n : Y n • X n - 57 -* * and induces g' : Y >• X [1] a l so s a t i s f y i n g Hence g - g ' = h3 - 3 h X A. Taking i n to account of the d i f f e r e n t s igns of the d i f f e r e n t i a l s of .* * X and X [1], we see that g : Y* > X*[ l ] i s determined up to homotopy by the given q u a s i - s p l i t sequence. D e f i n i t i o n . The homotopy c lass of maps g : Y X [1] assoc ia ted wi th a q u a s i - s p l i t sequence ^ ^ ^ ^ * ^ *^ *^ ^ i n the manner descr ibed i s c a l l e d the twis t of the sequence. , , * i * n * [2]. I f Z : 0 f X • C — Y • 0 q u a s i - s p l i t s w i th twis t g : Y > X [1] , then apply ing 0-cohomology to the sequence * i * D * 6 R , T ( i ) * T (D ) * T(6) * 0 x — ^ C -A Y -Ax* [ l ] — [ 1 ] iiPJ+ Y [1] X [2] -» - 58 -and noting that the diagram n, *\ ^ ( i ) n * H n(p) n * 6 r n+l * H n + 1 ( i ) n+i * H n(X ) — V H n ( C ) -——> H (Y ) ~^->H n + 1CX ) ¥JEn l(C ) H°(X*[n]) •* R°(C*[n]) •* H°CY*[n]) •* H°(X*[n+l]) H°(C*[n+l]) H°(T ni) H ° ( T n P ) H°(T n3) H ° ( T n + 1 i ) i s commutative f o r every integer n, and since a l l the isomorphisms are nat u r a l , we conclude that the connecting morphism 6^ i n cohomology i s induced by the twist. [3] Given £ : 0 —> X* C* -A- Y* —>- 0 q u a s i - s p l i t with twist 3, then: (a) 3 i s homotopic to 0 i f f the sequence s p l i t s , i n t h i s case, the se c t i o n i s ( ^) where h : g~ 0. (b) I f X i s c o n t r a c t i b l e with h : 1~0 then p has a homotopy inverse given by p -1 " (11 Proof. (a) Let h : 3-0, - ^ n + h n + 1 ay = $ n f o r a l l n. Define S n =/ ^ l y 1 1 Yn > cn - 59 -Then ( p . S ) " . ^ ^ g ; y * s ince c i s a chain - h g Y map , n+1 -h i a n Y . Conversely i f Z q u a s i - s p l i t s with sec t i on S = C 1 ) , then from X P 0 3 -h 1 n' -h 1 n+1 3" Y we get - 0 n = h n + 1 3 n - 3 n h n Y X x. e. h : 6 * 0 . (b). From the homotopy commutative diagram X 0 X1 * p X1 * Y we see that p i s an isomorphism i n K(A). Because h : 1 « 0 , - l v n = 3* h n + 1 + h" 3 n + 1 A A A X - e n - £ h n + V + h naf 1 e n - 60 -Subs t i t u t i ng 9 £ + 1 3 n = -Bn+19J we get - 3 ; h n + 1 3 n + h n + 2 g n + 1 j £ = B n Hence h ' 3 i s a homotopy from 3 to 0, and by (a) , p _ 1 = ( ~ h B ) 1 it [4]• For any complex X , the mapping cone + i d x * i s c o n t r a c t i b l e with homotopy h = ^ ^ ^ ^ ) " Proof . o o ) / y ± i x * \ +f 3 / ± i x * W o 0 l x * 0/0 o / v 0 ^ X ° / V x V/ 0 v I H . I f S : 0 — » X * - L * X * X * > 0 quas i-s p l i t s wi th twist 3 and sec t i on S = ( ° ) then * * q ^ : c_i r X 2 has a homotopy inverse given by p = ( ) . P roof . - 61 -1 C o ; ' xo x: x£ © x!J © x£ + 1 Set a : X 2 [ - l ] * J - i d * X 0 n ,n- l a Then C" - X? © x" + 1 9 X n 8 C * "0 ~ 0 /'S -id n+1 X 0 -8 .n+1 0 0 3 0 n n and the map g : C C def ined by £ = -1 i s an isomorphism with inverse £ ~ = £ . We have a commutative diagram - l (P,0) L X 2 * ^ 0) where W i s def ined by - 62 --id * X0 a X 2 [ - l ] ( [ l ] ) C _ 1 i s con t r a c t i b l e with homotopy h =| 0 0  X0 I - 1 0 - 63 -4 QUISOS AND THE DERIVED CATEGORY A quiso or quasi-isomorphism i s a co-chain map or homotopy c l ass of one , which induces an isomorphism i n cohomology. A [1] . The c lass of quisos i n K (A) form a r i gh t and l e f t mu l t i -p l i c a t i v e system. Proof . (FRO) i s c l e a r . * (FR1): Given X , make Y - + Z A A * C. - X [1] c 1 ^ T ( s ) t t 8 A \ ; T ( f ) t A Y [1] ' > Z [1] 1 t 1 T * T < f > cs cs T * X i it "k By p ropos i t i on 7(or TR3) there ex i s t s a morphism h : X [1] » C * , h * complet ing the square X [1] • T(s) f f t * T ( f ) * Y [1] L L * Z [1] From the long exact sequence - 64 -n , * H " ( g ) n * H n ( t ) n * n * H n ( C _ ) — ^ H n ( Z [1]) » H n ( C ) > H n ( C c [ l ] ) y .. and the f a c t t h a t H ( C g ) = 0, f o r a l l n , we see t h a t t i s a q u i s o . * s A h v A (FR2): G i v e n X Y ZZ£ Z w i t h hs = gs . To p r o v e g FR2 we c o n s i d e r the morph ism f = h - g and r e d u c e to show ing t h a t A 3 A f A f o r any X Y > X f s = 0, t h e r e i s a q u i s o t s u c h t h a t A t f = 0. Because H o m ^ ^ C - , Z ) i s e x a c t on the sequence A s . A A A X Y » C g > X [1] y . . A A and f s = 0, t h e r e e x i s t s a morph ism g : C > Z c o m p l e t i n g it S & f & (1) o f the d i a g r a m X Y > Z A A Now t ake t : Z > C^ . Then f r om the e x a c t sequence H N ( C * > » V > ^ V « £ > — , H"(C*[1J ) — * . . . i t f o l l o w s t h a t t i s a q u i s o . D u a l l y we can p r o v e FRO 0 - FR2° . We d e f i n e the d e r i v e d c a t e g o r y D(A) o f A as the l o c a l i s a t i o n o f K(A) w i t h r e s p e c t to q u i s o s . S i m i l a r l y , D (A ) , D (A) , D~(A) , D b ( A ) a r e the l o c a l i s a t i o n s o f K*(A) , K + ( A ) , K~(A) and K b ( A ) - 65 -r e s p e c t i v e l y . [1] gives us a good ho ld on D (A) i n the sense that i t can be handled as a category of l e f t or r i g h t f r a c t i ons ( us ing FR0-FR2 or FR0 °-FR2 ° ). [2] . (a) . Let S be a m u l t i p l i c a t i v e system i n a category (2. Let D be a f u l l subcategory of C_ and assume that SHMorD i s a m u l t i p l i c a t i v e system i n D, assume fu r the r that one of the f o l l o w -ing two cond i t ions h o l d : s i ) . Whenever X ' = ^ X with s <L S and X € Ob D, there e x i s t s X " X' such that s f £ S and X ' 6 0b D. s i i ) . Whenever X' = ^ X with s e S and X' € Ob D, there e x i s t s X X " such that f s 6 S and X " € Ob D. then P_g|-^Mor D c a n be i d e n t i f i e d with a f u l l subcategory of C_g. (b) . Each of the functors D +(A) ^ •» D b (A) D(A) ^ D*(A) i s a f u l l embedding. Proof . (a) : We prove only f o r r i g h t m u l t i p l i c a t i v e systems and i i ) , The other cases are s i m i l a r . - 66 -f s Given a quasi-morphism X Y ' <£= Y with X, Y6 Ob D. By hypothes i s , there ex i s t s t : Y ' — > D such that ts eS and DeOb D. From the k i t e We conclude that (X — Y ' Y ) £ HOITL ( X , Y ) . - S f l M o r D (b) ( i ) D (A) — 1 > D(A) i s a f u l l embedding. Let X * s Y be a quiso with X £K (A) and m e Set s a t i s f y i n g X = 0 f o r a l l n<m. ,m m+1 -7* r A 4 J*-*^ v m Y d Y m+2 Z = ( -s>- 0 —• imcL. —> Y — Y —* Y A A n ^ f : Y — * Z f = | L^n n>m ,n-l -d l n = m-1 .) 0 n<m I ' 0 • 0 -Jo Jo vm-2 _^ v m - l ,m ,m+l v m X v m+l X vm+2 A >• A * X I m I m+1 I m+2 |s \s 4 s ,m v m+l _ ^ vm+2 J 0 I 0 -> imi m-1 m-1 „m+l K"1 i1 i1 Tm+2 - 67 Thus H (fs) : H (X ) .= H (Z ) f o r a l l n; fs £ S , and by (a) , we are done. ( i i ) A > D + (A) f A , S * Let A —> X B be a quasi-morphism where A, B £ ObA , * b * X € Ob D (A). Since s i s a quiso we may assume X i s truncated -1 "X 0 "X 1 and looks l i k e . . • 0 1 X y X ? X —* . . with H ° ( s ) : B = Ker d ° . Now from f A s f ,0 = we see that X > X <( B can be replaced by A > Ker d •< B, which i s a morphism i n A . [3] The fo l l ow ing i s a subs t i tu te fo r "Car tan-E i l enberg" r e so l u t i ons : Let I be a f u l l subcategory of A such that every object of A admits a monomorphism in to an object of 1^ , then every object of C + (A) admits a quiso i n to an object of C+(_I). Proof . A + We may assume X €: C (A) i s t r i v i a l i n negat ive dimensions. * 0 1 X = . . — * 0 —> X —> X - 68 -Def ine I n = 0, s n = 0, and = 0 V n < 0 > and 0 s ° o n imbed X > > I wi th I e o b i . A T n n . Assume I , s ex i s t wi th d ^ " 1 = 0 V n f k - l and s ^ " 1 = d ^ 1 s n - 1 \ / ^ k . d k - l d k . . - * o - , x 0 ^ . . . . ^ x k - 2 - , x k - 1 i x k J l x k + 1 k s k+1 1c V+1 For n = k+1 l e t (P x , g K, a K x) b e def ined by the push-out diagram ^ x k ^ x k + l _ ^ xk+2 "1 7* T~ / VI 1 J k - 1 P „ T,k+1 1 4m d k _ 1 = C ° k d I > P k+1 Now imbed P k + 1 > I — ¥ I k + 1 a n d set s k + 1 - Y k + 1 a k + \ . d k = Y k + 1 B k . 1 r k . Then, d ^ " 1 - 0, s k + 1 d k « y k + 1 . a k + 1 d k - Y k + 1 . B k . , k . s k - l-s^. -L X A 1 Hence, by induc t ion we obta in s : X > I . - 69 -To see that s i s a qu iso , we r e c a l l that i f : a A • B C > D' i s a pushout i n A, then 1) 3 induces an ep i : ker a ker a ' 2) 3' induces a mono : "cok a > > cok a ' In our s i t u a t i o n we have fo r each k k k 1) IT s induces e p i ' s Z k (X*) = ker d k — » ker3 k - k e r Y k + 13 k= ker d k / = H1 / • A ^ 1 ' lm dj. k+1 2) a induces monos: , ,k % ^ . Q k „ . _k k ^_ v , . k+ l „k k. , ,k cok d > > cok 3 - cok3 n > •-) cok(y 3 TT ) = cok d_. A I From the commutative diagram with exact rows „ k , * . ^ . ,k ^ . ,k+l H (X ) > cok d >• coim d X X 1 1 „ u k , T * . ^ , ,k ^ . ,k+l ^ 0 f H (I ) > cok d^ . > coxm d^ . • k * k * We see that H (X ) > * H (I ) i s monic. Hence, H k(X*) > H k( I*) i s i s o f o r a l l k. - 70 -5. EXT [1] Let A be an abe l i an category and f : X — * I be a morphism i n C(A) . Assume: 1) X* i s a c y c l i c i . e . H n(X*) = o V n , 2) each i n i s i n f e c t i v e , * 3) I i s bounded below, then f i s homotopic to zero . Proof . Wel l known, (by i nduc t i on ) . [2] Let A be an abe l i an category and s : I —> Y a morphism in C(A). Assume: 1) s i s a quasi-isomorphism, 2) each I n i s i n f e c t i v e , 3) I i s bounded below, then s has a homotopy i nve r se . P roof . (from R. Har tshorne 's Residues and Dua l i t y ) * The mapping cone C i s a c y c l i c . The morphism s * * v = ( ld ,0 ) : C — • I [1] s a t i s f i e s [1] and i s therefore homotopic to s o. Let H = (k , t ) : I [1] ® Y —> I be the homotopy operator . v = ( ld ,0) = (k , t )3 * + 3 T ( k , t ) - 71 -Separat ing components, we have l d j * ^ = T O ] . ) k + k T O ^ + t-T(s) and 3 yt - t9 v = 0. * * Thus t : Y y I i s a morphism of complexes and Id* i s homotopic to t « s , so t i s a homotopy inverse of s, - 72 -References A r t i n , M. Grothendieck Topo log ies , mimeographed seminar notes , Harvard Un i ve r s i t y , 1962. Bredon, G . E . , Sheaf Theory, McGraw-Hil l , New York, 1967. Car tan , H. and S. E i l enbe rg , Homological A lgebra , Pr inceton Un i ve r s i t y Press , P r ince ton , N. J . , 1956. G a b r i e l , P., and M. Zisman, Ca lcu las of F rac t ions and Homotopy Theory, Spr inger-Ver lag New York Inc . , 1967. Gamst, J . and K. Hoechsmann, Products i n Sheaf Cohomology I, II. Hartshorne, R., Residues and Dua l i t y , Lecture notes i n mathematics, no. 20, Spr inger-Ver lag , 1966. Hoechsmann, K., Notes on Der ived Categories and Functors , (unpubl ished. ) MacLane, S., Homology, Academic P ress , New York, 1963. M i t c h e l l , B., Theory of Categor ies , Academic P ress , New York, 1965. Spanier , E .H . , A lgebra i c Topology, McGraw-Hil l , New York, 1966. Swan, R., The Theory of Sheaves, The Un i ve r s i t y of Chicago Press , Chicago, 1964. 

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