UBC Theses and Dissertations

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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.Sc,  University  A THESIS SUBMITTED  o f B r i t i s h C o l u m b i a , 1970  IN PARTIAL FULFILMENT OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  i n the Department of Mathematics  We a c c e p t t h i s t h e s i s as conforming to required standard  THE UNIVERSITY OF BRITISH COLUMBIA September,  1971  the  In p r e s e n t i n g t h i s  thesis  an advanced degree at the L i b r a r y I  in p a r t i a l  the U n i v e r s i t y  s h a l l make i t  freely  f u l f i l m e n t o f the of B r i t i s h  available  for  requirements f o r  Columbia, I agree  that  reference and study.  f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s  thesis  f o r s c h o l a r l y purposes may be granted by the Head o f my Department o r by h i s of  this  representatives.  It  thesis for financial  i s understood that copying o r p u b l i c a t i o n gain shall  written permission.  Department The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  not be allowed without my  ii.  ABSTRACT  F o r each a b e l i a n c a t e g o r y A , t h e r e i s a c a t e g o r y D ( A ) , c a l l e d the d e r i v e d c a t e g o r y o f A , whose o b j e c t s complexes o f o b j e c t s fractions  are  o f A , and whose morphisms a r e f o r m a l  o f homotopy c l a s s e s o f complex morphisms h a v i n g  as denominators homotopy c l a s s e s i n d u c i n g isomorphisms i n cohomology.  If  F : A  > IS i s an a d d i t i v e  abelian categories,  between  then under s u i t a b l e c o n d i t i o n s on A,  t h e r e i s a f u n c t o r RF : D(A) that i f objects  functor  >• D(B) w i t h the p r o p e r t y  X o f A a r e c o n s i d e r e d as complexes  c o n c e n t r a t e d a t degree 0, then t h e r e a r e isomorphisms H R F ( X ) ~ R F ( X ) f o r a l l n, where R F i s the o r d i n a r y n n  n  U  r i g h t d e r i v e d f u n c t o r o f F. d e r i v e d f u n c t o r o f F,  RF i s c a l l e d the  and one may l o o k upon i t as a  k i n d o f e x t e n s i o n o f F.  t  h  iii.  CONTENTS  Page C h a p t e r I.  Introduction  C h a p t e r II.  Calculas  of L e f t  1 Fractions  §1.  Definitions  §2.  E x i s t e n c e and d e s c r i p t i o n  C h a p t e r III. Chapter IV.  of l o c a l i s a t i o n s  7 10  Pseudo S p l i t Sequences and Mapping Cones The D e r i v e d C a t e g o r y  Comparison w i t h c l a s s i c a l Chapter V.  and e x t e n s i o n s  7  19 28  theory  Ext  36 38  Appendix 1.  C a l c u l a s of l e f t - f r a c t i o n s  41  2.  A n o t e on F - a c y c l i c s u b c a t e g o r i e s  53  3.  Quasi-split  55  4.  Quisos and the d e r i v e d c a t e g o r y  63  5.  Ext  70  References  sequences  72  Acknowledgements  I would l i k e  to thank D r . K. Hoechsmann f o r  h i s g u i d a n c e and a s s i s t a n c e d u r i n g the w r i t i n g o f thesis. In a d d i t i o n , I  am g r a t e f u l to the N a t i o n a l  Research C o u n c i l f o r t h e i r f i n a n c i a l support.  this  - 1 CHAPTER  I  INTRODUCTION  L e t F: A  > 15 be an a d d i t i v e f u n c t o r between  c a t e g o r i e s where A has enough i n j e c t i v e s .  abelian  The " r i g h t  derived  functor"  o f F i s an e x a c t 6 f u n c t o r o r an e x a c t connected sequence  {R F,6 :  n>0} o f f u n c t o r s from A t o B t o g e t h e r w i t h a n a t u r a l map  n: F  >R°F.  n  n  I t s d e f i n i t i o n i s as f o l l o w s : F o r X £ O b j e c t A , take an i n j e c t i v e I*: and s e t  >X  0  R F(X) n  n : F  0  —  =  ^  1°  "  > I  resolution  "  1  >  I  >  < ••  H F(I*). n  > R ° F i s i n d u c e d by the u n i v e r s a l i t y  > F(X)  £>F(I°)  ^  V'F(dO)  F  d  )  1  o f Ker d ° = R ° F .  >  = R°F(X)  G i v e n a map X — - — > Y the n i c e mapping p r o p e r t i e s o f y i e l d a morphism  0  0  a:I*  >X i*  > Y  ^ J  > 1° 1«°  > J°  >I  |-  > J  >  1  1  ]  —  >  injectives  - 2 -  o f r e s o l u t i o n s , which i s unique up to homotopy o f complexes. the cohomology f u n c t o r s H  Since  do n o t d i s t i n g u i s h homotopic morphisms,  n  t h i s produces a w e l l - d e f i n e d map  R F(f)  : R F(X)  n  The c o n n e c t i n g morphisms 6 from the f a c t  n  a r e d e f i n e d v i a the " Snake Lemma "  n  t h a t e v e r y s h o r t e x a c t sequence i n A can be i n j e c t e d  i n t o a short exact  sequence  & o  &  4e  >  > i  of i n j e c t i v e  *- R F ( Y )  n  y  J  r e s o l u t i o n s and the l a t t e r  >o  K  stays  exact a f t e r an  a p p l i c a t i o n of F . If  F is left  exact,  n  i s an isomorphism and  c o i n c i d e s w i t h the sequence o f " r i g h t s a t e l l i t e s "  { R F, 8 n  of F —  n  }  i.e.  an e x a c t connected sequence o f f u n c t o r s whose 0 ^ term i s i s o m o r p h i c to F. The d i s a d v a n t a g e o f the theory o u t l i n e d above i s i t s heavy r e l i a n c e on i n j e c t i v e s .  But i n case F i s l e f t  computed from any F - a c y c l i c R F(X) n  = 0 f o r n>_l ) f o r i n t h i s  X is  resolution  *  :  an F - a c y c l i c  0  rX  n  n  > X.  1  = H F(X*) n  if  if  r e s o l u t i o n o f A , then  R F(A)  R F can be  ( X£0b A i s F-acyclic  situation, 0  >A  exact,  2 > X  ->  • • •  - 3 -  and even i f  injectives  are n o t a v a i l a b l e ,  o f F can be d e f i n e d i n is  the r i g h t  terras of F - a c y c l i c r e s o l u t i o n s i f  a f u n c t o r i a l way of a s s i g n i n g such r e s o l u t i o n s  A and F i s  e x a c t on these r e s o l u t i o n s .  sheaf-cohomology w i t h f l a b b y all  to o b j e c t s  of  ordinary  sheaves.  cases the f o l l o w i n g p a t t e r n emerges:  We have a c l a s s (AC1)  there  T h i s i s u s u a l l y done i n  group-cohomology w i t h c o i n d u c e d modules and i n  In  satellites  1^ o f o b j e c t s  Each o b j e c t  o f A w i t h the f o l l o w i n g p r o p e r t i e s :  of A admits a monomorphism i n t o  an  o b j e c t o f JE. (AC2) .1 i s  c l o s e d under f i n i t e  d i r e c t sums and c o k e r n e l s  of  monomorphisms. (AC3) after  Short  e x a c t sequences of o b j e c t s  an a p p l i c a t i o n o f 1 is  i n I remain  F.  c a l l e d a c l a s s of F - a c y c l i c o b j e c t s  unique i n g e n e r a l but t h e r e i s  i n A.  a "right  derived  we seem t o need a f u n c t o r i a l way o f s t a r t i n g w i t h  it  One o f  It  is  not  always a maximal o n e .  In o r d e r t o d e f i n e something l i k e  resolutions.  exact  functor"  F-acyclic  the p o i n t s to be made i n the s e q u e l i s  i s not n e c e s s a r y .  The e x i s t e n c e o f any c l a s s  AC1 - AC3 w i l l enable us to c o n s t r u c t a d e r i v e d  that  1^ w i t h p r o p e r t i e s functor.  Outline: Let  a quiso (quasi-isomorphism),  be a morphism X*  f  s  arrow,  Y* of c o - c h a i n complexes i n d u c i n g  isomorphisms i n cohomology. inverse.  denoted by a double  Thus,  i n cohomology, a q u i s o has  an  - 4-  S t a r t i n g w i t h the c a t e g o r y K(A) o f c o - c h a i n complexes and homotopy c l a s s e s whose o b j e c t s  o f c o - c h a i n maps o f A , we form a c a t e g o r y D ( A ) ,  are objects  o f A b u t whose morphisms,  called  " q u a s i - m o r p h i s m s " and denoted by broken arrows X are f o r m a l f r a c t i o n s denominators.  >• Y ,  o f morphisms o f K(A) w i t h q u i s o s as  Every quasi-morphism from X to Y i n d u c e s a  w e l l - d e f i n e d map i n cohomology. An F - a c y c l i c r e s o l u t i o n can be viewed as a q u i s o o f complexes.  A  :  > 0  :  \  >0  >» A  >0  > 1°  1  I  I* : ' * '  >  0  i  >0  >  >0  I  I  > "'  I •  1  G i v e n a map X —^->Y and r e s o l u t i o n s X  * I  >  2  I*,  > • • •  Y ~  v J* ,  we o b t a i n the diagram X  >  I*  4 which e s t a b l i s h e s  a quasi-morphism I  As a consequence o f the e x a c t n e s s  >  J  from I  to J  .  o f F on I_, we know t h a t F  turns quasi-morphisms between 1 - complexes i n t o quasi-morphisms between t h e i r the  images.  quasi-morphism  H F(I*) n  T a k i n g the map i n d u c e d i n cohomology by F(I  )  > F(J  >  H F(J*) n  ) , we h a v e :  Procuring  the l o n g e x a c t sequence from t h i s  f u n c t o r i s a more s u b t l e We note t h a t t h i s make h a r d l y X  task,  but i t  t o o , can be done.  p r o c e d u r e o f d e f i n i n g the  any d i s t i n c t i o n between o b j e c t s  o f such o b j e c t s .  kind of " d e r i v e d "  "R F's" n  X of A and complexes  In g e n e r a l we w i l l work on the l e v e l o f  the complexes. Evidently X f  rr?  I  any complex X  into  : X  ^ Y  "F-acyclic"  a complex I  of objects  o f A admits a quxso  of o b j e c t s  o f I , and any morphism  i n d u c e s a quasi-morphism o f t h e i r  respective  resolutions X*  = >  I*  i The f u n c t o r RF : D(A) RF(X*)  =  > D(B),  i s d e f i n e d by s e t t i n g  F(I*).  The R F can then be r e c o v e r e d from RF by t a k i n g n ^ n  cohomology.  Thus RF w i l l be r e f e r r e d  1  t o as the " d e r i v e d "  functor  o f F and we may l o o k upon i t as a k i n d o f e x t e n s i o n o f F from K(A) has  to D ( A ) .  T h i s p o l i c y of s t a y i n g on the l e v e l  the advantage  of circumventing  o f the complexes  the u s u a l s p e c t r a l sequences F  the s t u d y o f composite f u n c t o r s . we o b t a i n a n a t u r a l map y : R(G*F)  Indeed,  given A  in  G > IS  > RG»'RF which i s an  > C.,  - 6 -  isomorphism under f a v o u r a b l e  conditions.  D u a l l y , what has been s a i d about r i g h t a p p l i e s e q u a l l y w e l l to l e f t the a p p r o p r i a t e a r r o w s , "right", In of t h i s  "projective"  functors  derived functors a f t e r  reversing  i n t e r c h a n g i n g the terms " l e f t "  and " i n j e c t i v e " ,  and  etc.  c o n c l u s i o n we sum up some o f the s t r e n g t h s  and weaknesses  approach:  (1) class  derived  We can d e f i n e d e r i v e d f u n c t o r s whenever we have a  1^ o f F - a c y c l i c o b j e c t s  dualized for left-derived application is have p l e n t y (2)  of f l a t s  but hardly  Derived functors  An example o f a  as e a s i l y  any  non-trivial  projectives.  f o r complexes — hypercohomology, as those f o r  objects.  The s p e c t r a l sequence r e l a t i n g  o f a composite i s functors.  functors.  appropriately  the problem of d e f i n i n g TOR f o r sheaves where we  are h a n d l e d j u s t (3)  as d e s c r i b e d above;  to the d e r i v e d  s u p p l a n t e d by a s i m p l e c o m p o s i t i o n o f  However,  f o r the more d e l i c a t e  from a s p e c t r a l sequence, i t  information  seems t h a t t h i s v e r s i o n  functors derived  extractable will  u s u a l l y be too c r u d e , though the u s u a l s p e c t r a l sequences always be s e t up.  can  - 7CHAPTER CALCULAS OF LEFT  §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 Let  like  II FRACTIONS  and E x t e n s i o n s  S be a c l a s s of morphisms i n a c a t e g o r y JC.  to " i n v e r t "  "smallest"  the elements  of S.  In  o t h e r words,  c a t e g o r y such t h a t every morphism of S i s  The f a m i l i a r  s i t u a t i o n o f a r i n g l o c a l i s e d at a  system g i v e s  us a c l u e as how to d e f i n e such a  Definition. s h o u l d be a p a i r Q(s)  (Cg,Q) where Q : C  to t h i s p r o p e r t y ,  (D, G) , t h e r e i s  find  the  invertible.  multiplicative category.  The l o c a l i s a t i o n o f C: w i t h r e s p e c t to S  b e i n g an isomorphism  respect  We would  for a l l i.e.  > Cg i s s€S  and i s  a functor with universal with  g i v e n any o t h e r s i m i l a r  a unique f u n c t o r H : C_  pair  > JJ such t h a t  H-Q = G.  G \  • H D  As i n the case of e x t e n d i n g a homomorphism between r i n g s to t h e i r  l o c a l i s a t i o n s , we w i l l study the c o n d i t i o n s under  which a g i v e n f u n c t o r F : £_ sations diagram  (Cg,Q)  two  and ( D g , , Q ' ) .  > D w i l l extend to the  locali-  That i s , when we can complete  the  - 8 -  C  —-—>  D  Q  £s  £s  O b v i o u s l y , we h a v e : P r o p o s i t i o n 1.  A necessary  F t o extend t o Cg i s F w i l l be c a l l e d In  and s u f f i c i e n t  t h a t Q ' F ( s ) be an isomorphism i n Dg' (S,S')  Definition.  A right  approximate e x t e n s i o n of F i s  a pair  > Dg r and £ : Q ' F  natural  the u n i v e r s a l  property  > RFQ a  : f o r any o t h e r  (G,<j>) w i t h G : Cg  > D»  unique map n : RF  > G such t h a t HQ* 5 = <> j-  F  > D  weaker  extension.  ( R F , £ ) , where RF : Cg map s a t i s f y i n g  g  and < > J : Q'F  Q'F  pair  > G«Q t h e r e i s  5  RFQ  Q  In  \/sfeS.  exact.  case an e x t e n s i o n does not e x i s t we c o n s i d e r the  n o t i o n of an approximate  C  condition for  c o n j u n c t i o n w i t h e x t e n s i o n s we c o n s i d e r  composite  a  - 9F functors A (RF,?),  H3  (RG,?')  of t h e p a i r (unique)  G and r i g h t approximate e x t e n s i o n s  and ( R ( G F ) , £ " ) .  (R(GF),  A p p l y i n g the u n i v e r s a l i t y  t o ( RG-RF,RG(5_) . £  C")  n a t u r a l map y : R(GF)  J  F  ),  we o b t a i n a  > RG-RF s a t i s f y i n g t h e r e l a t i o n  Yx' 5'x =RG( C X )' C' ( x ) .  A —E-v  In  B  —  c  case y i s an i s o m o r p h i s m , t h e r e i s a n a t u r a 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 d e f i n e d by ex the n a t u r a l map.  Yx** J^G(?x) where E, : F  >- RF-Q i s  - 10 E x i s t e n c e and D e s c r i p t i o n  §2.  We assume S c o n t a i n s i d e n t i t i e s composition.  For b r e v i t y  arrows  ^  .  and i s  c l o s e d under  elements of S w i l l be denoted by double  ObCg  =  ObC.  Arrows of Cg are  c l a s s e s of r e d u c e d words fs'-'-gt "'"  that i s ,  -  f  s  diagrams o f the form X —  .  8  equivalence  finite  t  .  .  Y  with  the  equivalence  relation: X  ===£> Y  <4=== X r e p l a c e d by 1„ A.  Y 4*  Y  s  —  Y  r e p l a c e d by  l  y  The c o m p o s i t i o n of any two reduced words Xand Y - X ^ : - ^ . . Z  is  X-+<=-*^... Y-»-^:-*<= . . . Z . f  f u n c t o r Q : C —»> C£ i s  -1 Q(s)  d e f i n e d by Q(X — > • Y)  The f  «  X  Y with  s  = X-4= Y f o r s i n S.  the p r o p e r t y t h a t G(s)  is  If  G:C_ —>• D i s  any f u n c t o r w i t h  an isomorphism i n B f o r every  s in  S,  t h e n the f u n c t o r H : _Cg is  > 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 G f s ) ^ G f t') "'" u n i q u e l y determined by H(X—><= -><== . . . Y) =? G(X) —> -+ -*»".. .G(Y) GCf) G(g) -  Remark.  This i s  too academic and cumbersome to be o f  much u s e . We would l i k e f g X—>^= — s  to s i m p l i f y  the sequence o f  arrows  fc  Y by s u c c e s s i v e " p u s h - o u t s " of  pair  and thus r e d u c i n g the words to the form X — * » ^ = Y.  each  -  - 11 f  In o r d e r to ensure t h a t p a i r s  s  and  g  t  r e s u l t i n g from the same " p u s h o u t " diagram a r e i d e n t i f i e d we r e q u i r e the a d d i t i o n a l h y p o t h e s i s on S t h a t i f arrows  >  p a i r s of  are e q u a l i z e d by an element s 6 S,  then they  are  6 c o e q u a l i z e d by some t e S .  More g e n e r a l l y  c o n s i d e r an a x i o m i z a t i o n of Definition. system  this  l e a d s us t o  S.  The c l a s s S i s  called a right  multiplicative  if:  (FRO).  S contains a l l  identities  and i s  c l o s e d under  composition. (FR1).  Any diagram  >.  w i t h s € S can be  •If X*  > Y  1  X  8  ;> Z  completed to a commutative square  w i t h t 6 S. X'  (FR2) such t h a t  G i v e n a p a i r o f maps X as  =  a  »• Y  * Y if  there i s  3s, then t h e r e i s a t e S such t h a t t a  D u a l l y we can d e f i n e a l e f t - m u l t i p l i c a t i v e FRO", F R 1 ° , F R 2 ° by r e v e r s i n g a l l morphisms which i s b o t h a l e f t in A is  a s£ S  called a multiplicative  the arrows.  tB.  system  A c l a s s of  and r i g h t m u l t i p l i c a t i v e system i n A.  =  system  - 12 P r o p o s i t i o n 2.  L e t £ be a c a t e g o r y and S a r i g h t  m u l t i p l i c a t i v e system i n C_, then the l o c a l i s a t i o n o f £ w i t h r e s p e c t to S e x i s t s and i s unique up t o isomorphism. Proof.  Appendix.  Granted the e x i s t e n c e of a r i g h t m u l t i p l i c a t i v e S i n C we c o n s t r u c t the l o c a l i s a t i o n ObCg  =  (Cg,Q)  by s e t t i n g :  ObC  F o r any X, Y £ ObCg, a morphism from X to Y, by b r o k e n arrows X  •> Y ,  system  denoted  and c a l l e d a quasi-morphism from X f  to Y,  is  an e q u i v a l e n c e c l a s s o f morphisms X  s  v Y ' <"  Y  f where the e q u i v a l e n c e r e l a t i o n between two p a i r s X and X  ~f  -  y  »• Y S  ~s  Y holds i f  The c o m p o s i t i o n o f  f  Y  + Z = Y  X  > Y = X  s  • Y  1  <^  Y and  u  > Z <  Y' <  Z is  d e f i n e d by the diagram  Y —2—* Z' < (1)  W  —  are commutative and the  o u t e r edges form a commutative s q u a r e .  h  >• Y '  t h e r e i s a diagram  such t h a t the two i n n e r t r i a n g l e s  quasi-morphisms X  s  Z  two  - 13 v/ith (1)  o b t a i n e d by a p p l y i n g  The f u n c t o r Q : C Q(X  Y)  for s  (FR1)  > Cg i s  = X -J—>  Y  to Y '  <^=  d e f i n e d by Q(X)  <—^ Y with Q ( s )  Y •  h  • > Z'.  = X and = Y < ~  - 1  X  e S. Alternately  one can a l s o d e s c r i b e the c a t e g o r y C„ b y :  P r o p o s i t i o n 2' . where Jy i s  HomrjgCX.Y)  the f i l t e r e d  = lira  category  HomrjCXjY')  Y ' 6 0bJ of diagrams Y Y  s  :—> Y '  and  s whose morphisms are commutative  diagrams  Y  Y  1  Y" We now go on w i t h the problem of c o n s t r u c t i n g approximate e x t e n s i o n s . sucy  Let .1  be a f u l l  right  subcategory  that: (RAC1).  F|JE  (RAC2).  is  (S >S')  e x a c t where S  0  F o r every  = ARIHS.  0  o b j e c t X of C, there  is  a s £ S and an  g object  I  of I  such t h a t X :—>  (RAC2) immediately  implies  system i n I, and moreover, Cg.  By P r o p o s i t i o n 1,  I. that S  that Ij  0  is  is  a right  a full  subcategory  F |_I has an e x t e n s i o n E:(F  We n e x t note t h a t the i n c l u s i o n f u n c t o r i  multiplicative  : I_  to  i s not only f u l l y categories.  faithful,  but a l s o y i e l d s  :  x  "  )> r ( X ) ,  r(X)£obJE.  So  c  —1>  an e q u i v a l e n c e  To see t h i s , we a s s i g n t o each o b j e c t  a morphism Vx  l .  > C  —1> o  of  X o f Cg  This defines  a  of  - 14 functor r  :  —» I_g  and an isomorphism  v : Id^  =  i'r  —S which i s n a t u r a l  s i n c e any morphism X X  = >  r(X)  Y  =r>  r(Y)  I  representing a well-defined  > Y s e t s up the  quasi-morphism r(X)  »  diagram  r(Y).  Upon c a l l i n g I_ a r i g h t F - a c y c l i c s u b c a t e g o r y of C., we have P r o p o s i t i o n 3. and S,S'  Let  F : £  right multiplicative  > D be a c o v a r i a n t  functor  system i n (2,1) r e s p e c t i v e l y .  I be a r i g h t F - a c y c l i c s u b c a t e g o r y  of C, then the  functor  RF  t o g e t h e r w i t h the n a t u r a l map F(v ) y  £ is a right  x  : F(X)  d e r i v e d f u n c t o r of F.  up to isomorphism of Proof.  =  RF(X)  Moreover,  (RF,£)  is  unique  functors.  We must prove u n i v e r s a l i t y . G : Co  Want:  £-> F(rX)  > Dgi  a unique t, : RF  n : Q'F  Let  (G,n)  be g i v e n .  *• GQ  > G such that  C^'S = n-  Let  - 15 Q ' F — £ - > RFQ  G.Q  First,  to see the uniqueness o f £, f o r any o b j e c t  X o f C_,  p i c k X ==£>• r ( X ) .  F(v ) x  F(X)  ->  r(x)  >• RF(rX)  RF(X) /  F ( r X ) = RF(X)  RF(v ) x  N  > r(x)  'r(x)  C  4, G(X)  G(rX)  G(v ) x  Note t h a t G ( v )  i s an isomorphism s i n c e v  x  because £ ( ) i r  x  s  choice f o r C ( ) r  But is  the lower  x  a  i s one i n Cg and  x  isomorphism t h e r e i s o n l y one p o s s i b l e  n  as shown (namely,  C ( ) r  x  =  n  r  ( ) x  t r a p e z o i d has i s o s on top and b a s e ,  o n l y one p o s s i b l e c h o i c e f o r £ x C  =  x  G(v ) x  _ 1  -C  r ( x )  -RF(v )  o r s i n c e F ( r X ) = R F ( X ) , we a l s o have  C  x  = G(v ) x  -n y  1  r(x  x  *? (x) r  thus  ^' there  - 16 Natural!ty: C is  clearly  x  : F(rX)  n / s r  >  W  b i j e c t i o n between  - 1  » G(X) • Y i n C_.  formally  there i s a  the s e t s  Horn (RFQ,G'Q)  « Hom(RF,G).  Hence  £ i s n a t u r a l with respect  X  > Y' <  to quasi-morphisms  Y i n Cg.  Finally  t o show t h a t  another p a i r  (RF',?')  universality  o f (RF,?)  the p a i r  (RF,5)  t>..''n' 'K = Q Q Remark.  : RF'  i s u n i q u e , assume  h a v i n g the same p r o p e r t i e s , guarantees  then the  the e x i s t e n c e o f a unique  • RF' s a t i s f y i n g H Q • ? =  a l s o a unique n '  LF : Cg  )  n a t u r a l w i t h r e s p e c t t o morphisms X  Next by e x t e n s i o n , we observe t h a t  n : RF  G(v  G(rX)  > RF w i t h  £'.  Similarly,  n'  =  £.  there  is  But  £; n'n 'V - VT h e r e f o r e n -n* = I d ; n' -1 Q Q Q Q Q Q A l e f t approximate e x t e n s i o n o f F i s a f u n c t o r • Dg t o g e t h e r w i t h a n a t u r a l map a : LF-Q  w i t h the p r o p e r t y unique to : G  that  f o r any s i m i l a r p a i r  > L.F s a t i s f y i n g a ' u q = T .  > Q'F  (G,x) t h e r e i s a By a p p r o p r i a t e l y  r e v e r s i n g t h e arrows p r o p o s i t i o n 3 can a l s o be proved f o r l e f t F-acyclic  s u b c a t e g o r i e s and l e f t  a l t h o u g h the n o t i o n o f an a c y c l i c  approximate e x t e n s i o n s . s u b c a t e g o r y do l e a v e  freedom o f c h o i c e , we w i l l show i n the appendix t h a t in fact,  always a maximal such s u b c a t e g o r y .  And  us much  there i s ,  = Id.  - 17 Next suppose we have two F  : C  • D  G  : D  »•  and r i g h t m u l t i p l i c a t i v e categories. category  E  systems S , S ' , S ' '  Suppose f u r t h e r  there i s  I o f C and a r i g h t G - a c y c l i c  We then have r i g h t  Y  i n these  If :  'F(1)QI , ,  R(GF)  respective  a right F-acyclic  sub-  subcategory V  J).  approximate e x t e n s i o n s RF,  P r o p o s i t i o n 4.  is  functors  of  RG, R(GF)  then the c a n o n i c a l morphism  — — > RG.RF  an i s o m o r p h i s m . Proof.  a p p l y G to I'  F o r X i n ObCg, i n the  to o b t a i n RG-RF we have  situation 5x  F(X)  —>• RF(X)  I  =  F(rX)  V  This  gives GF(X)  v GF(rX)  G(I')  Because F ( r X ) £ I_',  and  = R(GF)  (X)  = RG-RF(X).  y i s now an isomorphism i n Eg i i  to  - 18 Finally,  under the c o n d i t i o n s of P r o p o s i t i o n 4, we  can d e f i n e an edge morphism e : RG-F commutative  diagram  RG-F  5£i£=)  ^  RG-RF  R(GF)  R.(GF) by  the  - 19 CHAPTER  III  PSEUDO SPLIT SEQUENCES AND MAPPING CONES  L e t A be an a b e l i a n c a t e g o r y .  We denote by C(A) the  a b e l i a n c a t e g o r y of a l l c o - c h a i n complexes over A and by K(A)  the homotopy c a t e g o r y t h e r e o f .  C(A)  ( K(A) ),  d  T(X*)  is  *  "  D  X *  F o r each o b j e c t X o f  d e f i n e the complex T ( X * ) by T ( X * ) T  H  E  N  T  :  C  (  - ^  *"  C  (  -  )  (  R  E  S  P  *  K  (  N  =x  n  +  1  - ^  *  , K  ^  )  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 t e n w r i t e X * [ l ] i n s t e a d o f T ( X * ) and X * [ n ] i n s t e a d o f T (X*). N  To every  Z :  s h o r t exact sequence  0  > XQ — i - y  X*  P  > X*.  •  0  i n C(A),  t h e r e c o r r e s p o n d s a l o n g exact cohomology sequence ...  H(xJ)-^2 n  nH  H  n  (  X  * )  n H  6 — H * n  - ^  H  n( |) X  +  1  (J) X  w i t h the c o n n e c t i n g morphism 6^ f u r n i s h e d by the Snake Lemma.  We c o n s i d e r the f o l l o w i n g c o n d i t i o n s on s h o r t sequences under which more r e a d i l y of  accessible descriptions  the c o n n e c t i n g morphisms a r e a v a i l a b l e :  Definition.  Z : 0  We c a l l a s h o r t e x a c t sequence  *-X* — i - v X * Q  exact  P  >X*  • 0  — v  - 20 i n C(A)  q u a s i o r pseudo s p l i t  i.e.  *  0  splits  Xjj  if  it  splits  > X"  •  i n each d i m e n s i o n ,  V 0  f o r each n.  t P r o p o s i t i o n 5.  If  Z quasi-splits,  from the d i f f e r e n t i a l o p e r a t o r  [ "  ^\  [6 X  1  has  , a morphism 3: X  c  2  then one can  Y  a  l  l  j  = [ !?x* ^  d  I  of  d *j  \0 e  extract  x  the t w i s t of E.  the most i m p o r t a n t p r o p e r t y t h a t i n cohomology the  morphism 6 £ : H ( X ) n  Lemma i s exact  >- H  2  n + 1  i n d u c e d by the t w i s t .  (X ) Q  In  cohomology sequence o f Z i s  3 connecting  one o b t a i n s from the Snake o t h e r words,  the  long  o b t a i n e d from the sequence  o f complexes  •x^tn] ^ ^ [ n + l ] m . X *  ... — « • [ „ ]  [ n + 1 1  — *  •••  by t a k i n g 0-cohomology. Furthermore 3 measures the degree o f s p l i t n e s s of i n the sense t h a t i t  i s homotopic to zero i f  and o n l y i f  sequence s p l i t s . Proof.  Remark: "track"  Appendix  As to t h e i r  down a l l  availability,  we would l i k e  the q u a s i - s p l i t sequences and w i l l  to show  Z the  that up to homotopy isomorphism—an isomorphism o f  short  e x a c t sequences i n K(A) , e v e r y q u a s i - s p l i t sequence i s  a  " t r u n c a t e d mapping c o n e " sequence. We n e x t c o n s t r u c t is  a q u a s i - s p l i t sequence * whose  the homotopy c l a s s of a g i v e n map f  the mapping cone o f f i s  c  n  =  Y  n  : X  > Y .  the c o - c h a i n complex  e  twist First,  d e f i n e d by  x  n+1  or  Then the  sequence  * :0  Y*  1  > C*  P  quasi-splits with twist T ( f ) , the homotopy c a t e g o r y . mean a sequence of the  > X*[l]  >0  which i s  equivalent  to f  By a mapping cone sequence we form  in shall  - 22 * f  *  ** : X  *  Y  *  -—> C  *  f  Y  squares  f  i n d u c e s isomorphisms i n A  is  trivial.  1  -V C*  a  2  ^C  diagram  *  » 2  y*  g  > Y  2  • C  f  \2 •k  *,  |  *  a : C..  X*  [2]  X  in  Y  Y  the cohomology of  l  I a map  *  : X • >  *  *  have  7. Given any homotopy commutative  X  there i s  T(r>)  yc [l]^h  to * * we immediately  and o n l y i f  Proposition  *  cohomology sequence d e r i v e d from  o r a p p l y i n g 0-cohomology  cohomology i f  *  >X [1] ±^±*Y [1]  From the l o n g exact  P r o p o s i t i o n 6.  T(f)  g  *  g  the  • X*[l]  \ I a  such t h a t each of  T( ) a i  *  —*Y [1] ±  commutes.  Proof.  In m a t r i x n o t a t i o n ,  a =( 2 a  0 h  n  : x" ^ +  • Y^  is  ^  \  a,  where  the homotopy of the o r i g i n a l  square.  - 23 Commutativity i s  trivial.  In o r d e r 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 s e q u e n c e s , we f i r s t e x a c t sequence  * > X  E : 0  •;  Q  compare any s h o r t  * > X-  P  X  L  *  ^ 0  2  w i t h a t r u n c a t e d mapping cone sequence * :  Z :  0  * i > X —±->  X  >X  X  Q  * :0  By P r o p o s i t i o n  i>  Q  *  p  1  >X  *  •  2  >•( X Q [1]  * C_.  1  0  7 t h e r e i s a morphism w: C  > Xn  rendering  —i the  diagram  X  I 0  —i-^X  p  -»XQ[1]  I I "" ' I <p 0)  -l  x commutative.  * -> C .  x  >• x  2  -*-0  2  A p p l y i n g cohomology and the 5 Lemma, H (p)  H (i)  n  n  H (xj) — * H n  n  al) — * H  f  n  n  ( X*)  5>H  (X*)  (C_.)  ,n, * » H " ( X [ 1])  n+1  H  n + 1  (X*)  H (tr)  -1  n  H (X )  *H"(X )  Q  H (-i) n  r  o  > H " ( X [ 1 ])• 1  - 24 we conclude t h a t -n induces an isomorphism i n cohomology. Hence up to homotopy and cohomology, every quence Z  : 0  &  > X  i  * X  Q  it  a t r u n c a t e d mapping cone sequence.  X  * 0  i  *" l  p X  \  i * x  represents  *  X  -*-  o — >  )• 0  That i s ,  the  is  diagram  • *  i  x  A 2  se-  * 2  t  *  D  ——> X  ±  s h o r t exact  •  C  -i  an isomorphism of complexes i n some  suitable  homotopy c a t e g o r y where morphisms i n d u c i n g isomorphisms i n cohomology are i n v e r t i b l e .  Proposition.If  * ' s,  then ir : C .  Moreover:  Z q u a s i - s p l i t s w i t h t w i s t 6 and s e c t i o n  * y X^  has a homotopy i n v e r s e  g i v e n by  C ) . Thus we have the v e r y homotopy c a t e g o r y K(A)  important r e s u l t  every  quasi-split  that i n  sequence i s  morphic to a t r u n c a t e d mapping cone sequence. TT a s s o c i a t e d w i t h  homology.  and i s  iso-  The morphism  Z measures the d e v i a t i o n of the  from b e i n g q u a s i - s p l i t  the  sequence  always an isomorphism i n  co-  - 25 Remark. we are g i v e n  There i s  a dual s t o r y .  _  E : 0  >• X  A  i  Then t h e r e e x i s t s a morphism  *  •  Q  Suppose once p  A  *- 0.  > X^  A  t : X [1] 0  again  A  >» C p  (  again  e v o k i n g p r o p o s i t i o n 7) such t h a t  x*  > . o  »• x * [ i ]  0  •  >'  1  0 P  A  x  A  •  By the  A  yx  c  2  commutes.  0  A  x 1  X*[l]  p  [l] l  5 lemma i a l s o i n d u c e s an isomorphism i n  cohomology.  Proposition. commutative  ir and i a r e r e l a t e d by the homotopy  diagram A  c . -1  x*  x *[i] 0  A  C P  Proof. appears  is  as  A  The maps C . -  1  the r e q u i r e d homotopy.  ».  +  A  C  P  in matrix  notation  Now i t  is  c l e a r that:  Proposition.  The f u n c t o r s H o m ^ ^ ( - , Z  ) and Horn^.^  when a p p l i e d to any mapping cone sequence  X  —i+ Y  C  y i e l d long exact  Hom  H  o  m  K ( A )  (Z  K ( A )  *  (  *  £JI  ,X ) — H o m ^  X  * '  Z  *  c a t e g o r y K(A)  is  )  ^  a  (Z  *  i n g e n e r a l not a b e l i a n .  category i s  K ( A )  *  * ,C ) f  (C*. Z*) «  i n view all  a triple  c a t e g o r y , where a  (C, A, T)  an automorphism of (2.  (2)  A is  a c o l l e c t i o n of s e x t u p l e s of C^, where i n each — T ( X )  A morphism o f t r i a n g l e s  However,  the  enough i n f o r m a t i o n f o r our p u r -  T is  Z  Hom  (Z  c h a p t e r , we can axiomize  (1)  X —±+ Y —  [1]  assumed t o be an a b e l i a n c a t e g o r y ,  that carry  called triangles  C  * J# ,Y ) — H o r n ^  poses by the n o t i o n o f a t r i a n g u l a t e d triangulated  f  [1]  H o n ^ ^ C Y * ^ * ) * ^  the developments i n t h i s  the s t r u c t u r e s  Y  [1]  sequences  Although A i s  of  • X  f  is  such  (X,Y,Z,  triangle  .  a commutative  diagram  that:  u,v,w)  •  - 27 -  x  —±L+ Y  f|  -^U  gj  X'  h{  Y'  The t r i p l e  —  (_C, A,T)  (TR1).  A is  T(f)  W  |  -^-*- T(X')  satisfies  .  the axioms:  c l o s e d under i s o m o r p h i s m s .  Every morphism in a triangle X —  T(X)  —2-*-  W  Y  u : X  > Z  V  *- Y can be embedded )  W  T(X).  F o r any o b j e c t X i n <C, the s e x t u p l e is  (X,X,0,id  x >  0,0)  a triangle. (TR2).  is  (X,Y,Z,u,v,w)  is  a triangle i f f  (Y,Z,T(X),v,w,-T(u) )  a triangle. (TR3).  G i v e n a diagram —  x  X*  where the f i r s t triangles, is  —2-+.  Y'  —  square i s  —2-*  Z  Z  '  —2-»  *• Z'  the case of K ( A ) ,  T : K(A)  f  y Y  *  such that  y K(A) is  to a mapping cone sequence o f the form *  T(X')  are  (f,g,h)  triangles.  automorphism and a t r i a n g l e i n K(A)  X  T(X)  commutative and the rows  there e x i s t s h : Z  a mapping o f  In  —2-*  y  i  * C  * > X [1] .  is  the  translation  any s e x t u p l e i s o m o r p h i c  - 28 CHAPTER  IV  THE DERIVED CATEGORY  We w i l l now f o l l o w f o r the c o n s t r u c t i o n o f F  : A  the programme o u t l i n e d i n Chapter the d e r i v e d f u n c t o r RF o f any  II  functor  B. A  Let A be C(A)  abelian.  o r any one o f i t s  and C^(A)  whose o b j e c t s  bounded below, similarly  We denote C (A) full  subcategories  c o r r e s p o n d i n g homotopy  category  C (A), +  C  (A),  are complexes of A bounded above,  and bounded on b o t h s i d e s  denote K ( A ) ,  f o r the  K (A),  K~(A)  respectively,  and K (A)  for  and  their  categories. A  P r o p o s i t i o n 9.  In K (A)  the c l a s s of morphisms which  i n d u c e s isomorphisms i n cohomology, c a l l e d the  class  q u a s i - i s o m o r p h i s m s o r q u i s o s , form a m u l t i p l i c a t i v e Proof.  system.  Appendix.  For b r e v i t y , Definition. localisation  of  q u i s o s w i l l be denoted by double arrows The d e r i v e d c a t e g o r y  of K(A)  with respect  D(A)  of A i s  to the c l a s s  = ^  the  of a l l  quisos.  S i m i l a r l y , t h e r e 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 h e i r r e l a t i v e s i z e s we have: +  Proposition:  Each of  the  functors  .  - 29 -  D (A) +  A  D(A)  • D (A) D  is  (A)  a f u l l embedding. Proof.  Appendix.  C o n t i n u i n g w i t h the programme o f Chapter I I , for "suitable" If  I  is  acyclic  any s u b c a t e g o r y of A such t h a t every  i n negative  0 —• X  1  —• X  V  0 -+ I  J -  of I then f o r X  object  in C  (A)  d i m e n s i o n s , we can c o n s t r u c t a q u i s o  —• X 1  J I -.  n-1  ..  n _ 1  I x  s  n-1  2  1  cokdj.  by i n d u c t i o n and the f a c t s  -JM- x  n-1  i jn-2  -  n  n+1  X  pushout  _n  > P  t h a t i n any pushout  diagram  B  1 C  I  g i n d u c e s an epimorphism Kera P  n  look  subcategories.  admits a monic i n t o an o b j e c t trivial  we n e x t  ^Kera'  of A  - 30 -  and $' i n d u c e s a monomorphism cok a > P r o p o s i t i o n 11.  o f A admits a mono i n t o an o b j e c t  o b j e c t o f C ( A ) admits a q u i s o i n t o an o b j e c t +  Proof.  be an a d d i t i v e  functor.  and F : A  F i n d u c e s f u n c t o r s C (A)  o f C (_I) +  • 13 y C (B)  *> K (B) which w i l l s t i l l be denoted by F.  recall  o f I_,  Appendix.  Now l e t J3 be another a b e l i a n c a t e g o r y ,  K (A)  Thus:  L e t I be a f u l l - s u b c a t e g o r y o f A  such t h a t every o b j e c t then every  > cok a ' .  We  t h a t i f F p r e s e r v e s s h o r t e x a c t sequences o f o b j e c t s o f  1 and _I i s c l o s e d under c o k e r n e l s o f monomorphisms, then F takes s h o r t exact sequences o f C ( I )  i n t o exact  +  of C ( B ) .  Moreover,  +  * f o r a q u i s o 1^ " F(C*)  s  > /  i f 1^ i s c l o s e d under d i r e c t sums, *  F  = F ( I * [ 1 ] © I*)  ( ) s  -* \>- F ( I ) 2  = F(I*[1]) © F(I*)  i s a quiso,  and i s i n p a r t i c u l a r ,  P r o p o s i t i o n 12. is  then  + of objects o f C (I),  and the e x a c t n e s s o f F on o b j e c t s *• F(I^)  sequences  of C (I) +  = C*  (  implies  s  )  that  so F maps q u i s o s i n t o q u i s o s ,  (quiso,quiso)  exact.  Summarizing, we have  L e t I be a f u l l s u b c a t e g o r y o f A which  c l o s e d under d i r e c t sums and c o k e r n e l s o f monomorphisms, f u r t h e r ,  assume F p r e s e r v e s  s h o r t e x a c t sequences o f o b j e c t s  then the r e s t r i c t i o n o f F to K (I)  o f 1^,  maps q u i s o s i n t o q u i s o s .  - 31 -  In view of P r o p o s i t i o n s 11 and 12, we say category  I of A i s  right F-acyclic  that a f u l l  sub-  if  (AC1).  Every o b j e c t X i n A admits a mono i n t o an o b j e c t  (AC2).  I is  of I. c l o s e d under  (finite)  direct  sums and co-  k e r n e l s o f monos. (AC3) .  F preserves  Then, K (I) +  i n the sense of Chapter II  Theorem 1; I_,  If  (RAC1, RAC2).  of F to  D (A) +  g i v e n by  D (A)  D (I)  +  -  +  as i n P r o p o s i t i o n  (RF,£)  Hence  A contains a r i g h t F - a c y c l i c subcategory  then the r i g h t approximate e x t e n s i o n ( R F , £ )  e x i s t s and i s  I_.  w i l l be a r i g h t F - a c y c l i c s u b c a t e g o r y o f  +  K (A)  s h o r t e x a c t sequences of o b j e c t s o f  is  ^  2  *  D(B)  3.  c a l l e d the r i g h t  derived functor of  F.  Granted the presence of a r i g h t F - a c y c l i c s u b c a t e g o r y JE of A, RF(X  A  )  is  taking a quiso X  *  found on any o b j e c t '.  X  v  ^  *  I  x  =  F  (  A  i n t o an o b j e c t  RF(X*) = F(I*) ?  X  V  and '  -f-  of K (A) + of C (I)  by  first  and s e t t i n g  - 32 -  Dually  the p r e s e n c e o f a l e f t  of A guarantees w i t h LF  the e x i s t e n c e of a l e f t  : D (A)  to speak of the l e f t  F - a c y c l i c s u b c a t e g o r y P_  • D(B)  the r i g h t  .  In  derived  principle it  functor  (LF,cr )  i s of course p o s s i b l e  approximate e x t e n s i o n of F to D(A)  approximate e x t e n s i o n o f F to D ( A ) .  For i t s  or  existence  one would n e e d : (a) . S t r o n g e r  c o n d i t i o n s on I_ as to o b t a i n the  appropriate  g e n e r a l i z a t i o n o f P r o p o s i t i o n 1 1 , and (b) .  Stronger  e x a c t n e s s c o n d i t i o n s o f F on  For our purposes we w i l l  I.  f o c u s our a t t e n t i o n o n l y on the  right  + derived  -  f u n c t o r of F on D , and the l e f t  F o r the r e s t , o f an a d d i t i v e  functor,  this chapter, I  F:  d e r i v e d f u n c t o r on D .  A — * 13 w i l l always  denote  a r i g h t F - a c y c l i c s u b c a t e g o r y o f A.  The h i g h e r o r d e r cohomology f u n c t o r s can be o b t a i n e d from RF by s e t t i n g R. F(X*) = R^RFCX*) and t h e r e a r e n a t u r a l maps n  £  N  : H F  £  N  : H (£ )  We next c o n s t r u c t functors.  >R F  n  N  n  : H F(X*) n  the l o n g exact  d e f i n e d by  • HV(X*)  sequence of  = R F (X*)  the  n  derived  .  - 33 -  Let 1  :  > X*  0  > X*  1  > X*  P  » 0  be any  short  * e x a c t sequence i n C (A) is  essentially  .  Up to isomorphisms i n cohomology, E  a t r u n c a t e d mapping cone sequence  * C . * i  *p  A  t^x0-x1-2»x  2  ^  *  "  ,  A  *o -+h [1]  c  A  [1]  * ? \  p A  More p r e c i s e l y , if and i  i n , the d e r i v e d c a t e g o r y D ( A ) ,  the morphisms  a r e i s o m o r p h i s m s , and the diagrams A  i  A  • x  xQ  A  n  A  »  x2  ±  4  4 A  xn  i  0  A  n  • x. — x _ 1  X [1] Q  Jh A  A  > c  2  p  and  A  —i  xQ A  xQ  A  • xx i  A  * c_  ±  » x0[i]  t  A  * x  A  Z  A  A  >  > x2  X [1] Q  A  a r e isomorphisms of sequences i n D (A) morphisms are n a t u r a l Lemma 1.  i n the  .  Moreover, these i s o -  sense o f :  For a g i v e n morphism of s h o r t exact  sequences  - 34 *  A  -J  xQ  l o  E  : 0  • Y  *  4  I  1  Q  *  > x2  lA  f  1  D  >x  1  > Y  > 0  i  f  p'  1  2  f  A  >• Y  I  2  • 0  the diagram  *  X  i  >X  Q  l 0 Y  is  p  — X  x  *  l 2  f  i  0  A  1  >  f  p'  l  Y  E  d  * , *»X [1] r  2  | l  f  A  *  1  d  E  1  A Y  2  A  *  Y  0  [  1  ]  commutative.  Proof.  A  By P r o p o s i t i o n 7 t h e r e i s a morphism a : C P  r e n d e r i n g square  (1) o f the  \  X  diagram  *  2  |f2  p ^ =  C  commutative.  !  T  X  0  [  (1) | a (2)  A  ^  2  Square (2) i s  A  > C , P  A  ^  C  ]  {  ,•  A  p ' ^ =  clearly  1  Y  0  [  1  ]  commutative.  Now the q u i s o r e s o l u t i o n of the mapping cone sequence (*) be chosen a g a i n to be a mapping cone sequence (**), under the isomorphism of c a t e g o r i e s D ( l )  = D(A),  or  can  equivalently,  mapping cone sequences  - 35 are preserved.  *  d-  * P * — • x — • x2  i. • *  1  xQ  Q  1  1  . I  A  1  p  A  x Q —> x 1  4  i  ^  A  *  x2  > cp  A  A  U 4  a  p  A  * x2[i]  1  I *  *  • x1[i] —• x2[i]  U  fl  A  * —* x1[i]  > X [1]  U  4  A  A  (**) i Q —+ i 1 —*• i 2 )• c g • I.J1] —> i 2 [ i ] S i n c e F p r e s e r v e s 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  F (  A TT (Q I ) -JL±H  the l o n g mapping cone sequence  ^  A  A  y  i  F  (  C  g  F i n a l l y , t a k i n g 0-cohomology y i e l d s  0  A  +  A  )  » FC^tl]) — •  the l o n g exact sequence  0  H°F(I*)  A  H°F(I*[1])  H F(C ) U  g  which i s p r e c i s e l y  O A  O  R F(X ) U  A  • R F(X ) U  1  2  1  A  1  •R -F(X ) J  A  * R F(X >  ()  1  and we have  Theorem 2.  {R F n  A  R F(d >. n  n  • X  I : 0  where <$£ =  , S }  z  Q  l  i s e x a c t on s h o r t exact sequences -v  *  x  ±  p  A  >X  2  -> 0  o f C (A)  - 36 Comparison w i t h C l a s s i c a l  Classically,  Theory.  one i s i n t e r e s t e d  i n R F(X) f o r X i n A. n  By c o n s i d e r i n g X as a complex c o n c e n t r a t e d at degree 0 the quiso r e s o l u t i o n X  •' •  •••  )> I  >0  >• X  1  I  >0  + 1°  can always be chosen to be the form  ——> 0  >  \ >I  •  b •  1  0  I  >  2  0 w i t h the sequence 0  >X  >I  1 y I  > exact.  We always have R F ( X ) = 0 f o r n<0. n  The l o n g e x a c t takes  cohomology sequence f o r 0 —> X —»• Y — Z  — 0  the form 0 —> R ° F ( X ) —»• R ° F ( Y ) — > R ° F ( Z ) — > R ^ C X ) — . . .  Hence: (1) F is left  The map £ ° : F exact.  >• R^F i s an isomorphism i f and o n l y  In that case,  the _R F ' s a r e the r i g h t  F o r any X i n ob(I),  R F ( X ) = 0 f o r n>0.  (3)  If  i s an isomorphism)  X is acyclic  (£  satellites  n  (2)  if  n  then R F ( X ) = 0 n  f o r a l l n>0. And i n case A has enough i n j e c t i v e s , to the c l a s s i c a l  derived  the R F ' s are i s o m o r p h i c n  functors. F  G  We c o n c l u d e w i t h a note on composite f u n c t o r s A — • B^ If  F sends the r i g h t F - a c y c l i c s u b c a t e g o r y  G-acyclic  s u b c a t e g o r y J_ o f  I of A i n t o a r i g h t  then we have an isomorphism  > C_.  of F.  Y :  which g i v e s  e  R(G-F)|  rise  :  *>  to an  edge  (RG.RF)|  morphism  R(G-F) |  RG| .F B  d e f i n e d by the commutative  A  diagram  -y R(G.F) |  RG| .F B  RG(g)  (RG-RF) | . A  and i s o b t a i n e d by a p p l y i n g G to  F(X )  F(v )  F(I  )  yielding GF(X*)  GF(I*)  RG.F(X*) = This replaces  =  R(G-F)(X*)  G(J*)  the u s u a l s p e c t r a l sequence  c o n c e r n i n g composite  functors.  arguments  - 38 -  CHAPTER V EXT  We c o n c l u d e w i t h an example o f how the E x t - f u n c t o r s out i n the language o f d e r i v e d f u n c t o r s . *  Lemma a. Any q u i s o s : I i n C ( A ) has a homotopy +  work  F i r s t we note  A  A  = ^ Y  where 1  i s an  infective  inverse. A  Hence every morphism i n D(A) o f a complex X of i n j e c t i v e s  to a complex  bounded below i s r e p r e s e n t e d by an a c t u a l morphism  o f complexes, and i f A has enough i n j e c t i v e s , cal equivalence  of  there i s a c a n o n i -  categories: K (I)  = D (A)  +  .  +  N e x t , we o b s e r v e t h a t the Hom-functors of A can be extended to a b i f u n c t o r Horn* : C ( A ) °  by s e t t i n g  x C(A)  P P  Hom (X*,Y*) = n  II  7/  Hom.( X ,  P €™ d  Under t h i s A  A  n  definition,  =  > C(Ab)  P  Y  P + n  )  _A  n ( d p €2 x  P  _  1  + (-l)  n + 1  d  P + n  y  )  the n - c y c l e s o f the complex  A  Horn ( X ,Y ) a r e i n a one-to-one correspondence w i t h morphisms of X  A  A  to Y [n] and the n-boundaries  corresponds to those mor-  - 39 -  phisms which are homotopic to z e r o .  Thus one has a  natural  isomorphism (*)  H (Hom*(X*,Y*))  *  n  which,  t o g e t h e r w i t h Lemma a  Hom  F o r each i n j e c t i v e  A  (X*,Y*[n])  gives  * Lemma b.  K ( A )  I  + i n C (A)  the  X  > Horn (X  ,1 ) p r e s e r v e s  Therefore, defines  quisos i n  C(A).  assuming t h a t A has enough i n f e c t i v e s ,  n  (X  *  *  ,Y )  n  =  A A A  H (Horn (X  ,1 ))  A  for X  A  i n C(A)  and an i n j e c t i v e  A  resolution Y  F o r o b j e c t s X, Y o f A and an i n j e c t i v e  > I  n  Ext (X,Y)  n  defined i s  n  the u s u a l  f o r any q u i s o s : Y  of i n j e c t i v e s , Hom  A  (X*,I*[n])  e  :—^  I  of Y  H o m ^ ^ (X*, I*[n])  (*)  H o m  K(A)  ( X  *  , ] :  *  [ n ] )  =  the  A  by Lemma a  D ( A )  Thus  ^  Ext. A  Finally  .  resolution Y  o f Y, E x t ( X , Y ) = H ( H o m * ( X , I * ) ) = H ( H o m ( X , I * ) ) . n  one  the Ext-groups by Ext  and  functor  A A A  ( *( V*))  Hn Hom  x  i n t o a complex  I  - 40 one g e t s n a t u r a l isomorphisms  Ext (X*,Y*) n  e  Hom  f o r X* i n C ( A ) , Y* i n C ( A ) . +  D ( A )  (X*,Y*[n])  A P P E N D I X  - 41 -  1.  CALCULAS OF LEFT - FRACTIONS  We use the axioms: (FRO)  S contains a l l  identities  and i s  c l o s e d under  composition. (FR1)  Any  X  w i t h s G S can be completed  4  X'  • Y X  to a commutative square  >» Y '  i  *fr X'  (FR2) as  = 3s,  Given  X  > Y,  then t h e r e i s  Definition.  t€S  w i t h t 6 S.  n >> Y  suppose  such t h a t  3  s £ S with  tot =  tg .  A q u a s i - a r r o w from X to Y i s  a diagram  X — > Z •£= Y. Definition.  A kite  in C is  a diagram o f the form  W  X  with triangles  (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 o f FR1 to  Y''  ft Y yields  Y'  Y"  a commutative square  Z  it Y Add on to  :  Y'  (*) Y' X  Y Y' '  Definition. if  they f i t  [2]  Two q u a s i - a r r o w s a r e s a i d to be " ~ "  i n t o a k i t e whose edges form a commutative " s q u a r e " .  "~" is  an e q u i v a l e n c e r e l a t i o n on the s e t o f  all  q u a s i - a r r o w s from X t o Y.  Proof.  Symmetry i s  trivial;  reflexivity  X' ^  X f S  , Y'  |  Y ^  s.  Y'  follows  from  - 43 -  Transitivity:  Assume (X and  (X  Y' —•  ^ = Y) - (X - » Y "  <*= Y)  Y' ' ^ = Y) ~ (X ->• Y " »  Y)  then t h e r e are k i t e s w i t h e q u a l e d g e s :  Y' Z'  X  By FR1)  3  X  a commutative square  Z' Y  —> W Z"  But s i n c e the maps Y = ^ Z' — ^ Z " = ^ It  follows  Z'  Y'  z"^  W = ^ Z are a l s o e q u a l ( f o r  t h a t the o u t e r edges o f  ^ X  Y'  w  a r e  et  *  ua1  some Z by  ' FR2)  Y => Z  are equal, Definition. Y is  An e q u i v a l e n c e c l a s s o f quasi-arrows f o r X to  c a l l e d a quasi-morphism from X to Y. We w i l l denote quasi-morphisms by b r o k e n arrows X -> Y.  - 44 -  [3] a  Equality  kite  can be t e s t e d on any k i t e .  That i s ,  given  Y'  y  %  X  w i t h (X -> Y  Y  Y" there i s Z = ^ Proof.  W  equalising its  < = Y)  1  - (X  Y)  o u t e r edges.  By d e f i n i t i o n o f e q u i v a l e n c e ,  there i s a k i t e  Y' X  Y  Z'  whose edges are e q u a l , and we have  w i t h the square on the l e f t  commutative.  By FR1 we get a commutative diagram Z'  4=  Y  V S i n c e the morphism Y W =^ W "  Y'  equalizes  r e n d e r i n g the rows o f Y  1  1  Y' Z  l  ^  W' , t h e r e i s  • Z w  '  ^  w  "  C o r r e s p o n d i n g l y the e q u a l i s a t i o n o f Y = ^ Y " on the Y'  1  ^  Z ^  a  equal. rows  1  W  i n d u c e s a morphism W =^ W ' " c o e q u a l i s i n g them;  and an a p p l i c a t i o n o f FR1 to W"'<£= W*=$> W' '  yields  - 45 -  W " ' £ = W =^ W "  .  Finally  from  W  ^  where (1)  (2)  X  and (3)  W  a r e commutative, i t  Y =^ Z =^ W  follows  t h a t the  kite  has e q u a l i s e d o u t e r e d g e s .  Y" Note.  [3] works f o r p s e u d o - k i t e s as w e l l .  That i s ,  kite-like  Y' diagrams  [4].  X  >*.  . Y —V W  Y' '  & ——'  i n which Y -> W i s  Any two q u a s i - a r r o w s making a f i x e d  i n t o commutative s q u a r e s ( b y FR1 ) are  Proof.  we use  [1]  Assuming  to f i t  X  X — • Y' 4b "rt" X' —> Y X' —• Y  and  "ft X'  arbitrary.  —y  Y  equivalent.  X —*• Y ' -fr ^ x —• Y  1  !  into a kite  commute,  - 46 -  whose o u t e r edges are e q u a l i s e d by X* =^ X ; hence a l s o by some Z =^ W. [5].  are  If  X'  the quasi-arrows  X ^ J" >*• Y '  Y  Y  X  f  Y  are e q u i v a l e n t ,  so  "  1  ^ Proof.  Any k i t e X  Y' ^  ^ Y  ^ '  whose o u t e r edges  are  Y  e q u a l i s e d produces by c o m p o s i t i o n a k i t e  whose edges are t r i v i a l l y  [6],  equalised.  C o m p o s i t i o n o f quasi-morphisms.  Given X  Y'  by a p p l y i n g FR1 to  Y , Y - * Z' 4 = Z , we d e f i n e t h e i r Y  1  ^ = Y -> Z '  Y  »- Z'  W By  [4],  [5]  this  does n o t depend on W.  Dependence on Y : 1  Suppose X —> Y  are equivalent with compositions  1  Y and X  composite  - 47 -  y  >  x  Then any k i t e  on  W"  *  X  ^  z' 4 = z  — >  Y  Y "  w'  Z  gives  a p s e u d o - k i t e on  W"  ^  ^  Y  X  -  Dependence on Z'  Y .  The l a t t e r  can be e q u a l i z e d by some s € S.  can be a n a l o g o u s l y p r o v e d .  Hence  c o m p o s i t i o n of  quasi-morphisms i s w e l l d e f i n e d . Associativity  follows  3=  X —> Y '  from the  Y  diagrams  Z'  Z — » W' <£= W ->  4>d B  X —> Y ' < $ = Y —>Z'-4= Z —»• W \ .  C  a  ^ ^ £ b  B  The quasi-morphism X  by  X  # d B  B  » X = X ^ X ^ = X i s s  Note that f o r s : X =^Y  W  , X  the  identity.  s Y •<= X i s  a l s o the  identity  - 48 -  Definition.  We form the c a t e g o r y C^:  Ob  =  Ob C —  —D  Horn  (X,Y)  = set  of a l l  quasi-morphisms X  > Y.  -S and d e f i n e Q : C Q(l  x  ) = l, Q(x)  and the  x  n  • C  by Q(X  g  Proof.  is  4=  Y  shows t h a t Q(g^f) = Q ( g ) . Q ( f ) Q(s)  Y)  =  Y  -  8  Z  — •  ^=  Z  G i v e n X =^ Y , put t = Y —h- Y <£= X.  fQ(s)  :  Y - i - ^Y  X  = l  Y <== Y Y  ( C„, Q ) is —S  Y ^=  X  ^ l  universal.  Given G : C  —————  ^ - i  X ==$> Y <£== Y  v  Y  d e f i n e H: (}  1 Y 4= Y . Then  s£S,  N  Proof.  f  .  an isomorphism f o r a l l  Q(s)'t = ly  [8].  X  diagram  X  [7].  f  > JD  w i t h G(s)  iso for a l l  ^) » D  by H(X)  = G(X).  On quasi-morphism  seS.  - 49 X  s > Y = X — • Y'<£= Y, H(X  > Y) = G(X) — y G ( Y ' )  H i s w e l l - d e f i n e d s i n c e e q u i v a l e n t quasi-arrows  Z  Suppose H'  : .C  G ( s  >~ > G(Y) 1  can be f i t t e d  .  into  whose o u t e r edges a r e e q u a l i s e d .  • D. i s another f u n c t o r s a t i s f y i n g H ' . Q = G,  then H and H' agree on o b j e c t s o f C^ .  To see t h a t they agree on  o  J  f quasi-morphisms H(X  X  •> Y = X — > Y '  Y'<=r Y) = G ( s ) S  =  s  - 1  Y , we have  -G(f) = [H'-Q(s)]" .H'Q(f) 1  H'(Q(s)" )-H'(Q(f)) 1  = H'(Q(s) -Q(f)) - 1  = H' (X  Y'  Y)  Thus the e x i s t e n c e o f the l o c a l i s a t i o n  . (C ,Q) C  is established.  f o r i t s uniqueness, i f  (C!,Q')  universality  t h e r e i s a unique H : C „  * C'  —b  —S  of (C„,Q)  i s another l o c a l i s a t i o n then by the  —a  H*Q = Q' ; s i m i l a r l y , H'-Q'  = Q .  t h e r e i s a unique H'  Hence H'-H = 1 -S  Remark. FR1°,FR2°  As  and  : C_'  satisfying  ^ C_ such t h a t  H-H' = 1 , .  ^5  D u a l l y u s i n g the c a l c u l a s o f r i g h t f r a c t i o n s and F R O ,  , one can d e f i n e quasi-morphisms X *{  0  Y'  > Y and  show t h a t 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 l s o e x i s t s .  - 50 -  Definition. LI).  A c a t e g o r y I i s s a i d to be f i l t e r e d i f  Every p a i r o f o b j e c t s  :  o f I can be embedded i n a diagram  X  Y L2).  Given  X  i n I,  V t h a t the square  Y  X  there e x i s t s  Z  such  Y*  ^ _  Z  commutes.  ^ Y ' L3). such that  G i v e n a diagram X  >Z  the two maps o b t a i n e d by c o m p o s i t i o n a r e the same.  I f JE i s f i l t e r e d , for  > Y , t h e r e e x i s t s a map Y  taking l i m i t s  [9].  then 1 behaves as w e l l as an i n d u c t i v e  ( G r o t h e n d i e c k T o p o l o g i e s , Chapter I  system  ).  F o r each o b j e c t Y o f C_, we d e f i n e a c a t e g o r y 1^: s Objects of I  a r e morphisms Y  ""' ^ X w i t h s £ S . s  A morphism i n 1^ between two o b j e c t s Y = ^ X and t Y =4>X' i s a morphism f commutes. (1) . I  : X —>• X' such t h a t the diagram  X  We c l a i m : is  filtered.  (2) . F o r o b j e c t s X,Y i n C, Horn _c  (X,Y)  = l i r a Horn ( X , Y ' ) ->-  Y'eOb  }±  I  y  ^> x' ^  - 51 (3).  If  C is —  additive,  so i s C „ , —S  Proof. (LI).  F o r two o b j e c t s Y = ^ X and Y =T> X'  X' *fr Y  t o get the commutative diagram  5** (L2).  X'  Given  tff  ===? Z such  therefore (L3).  •  x  that  (L2)  Given  is  V  '  Z "ft X  — »  Jt-jJS 8 ^  g  x  Y  X' ft Y =$> X  to  .  w i t h f s = gs = s  X'  hf = h  .S  1  by  (FR2)  there  exists  h  1  x ==  ;  satisfied.  Y = T ' X' X'*-  S  f ^ »  a p p l y (FR1)  with t r i a n g l e s  (1)  (2)  commuting.  Complete  X  x' X _  - • Z- Y to a k i t e  c-  v  X ^ <•  ^  Y ==^ W  morphism Y =^> X e q u a l i z e s the p a i r o f composites X T h e r e f o r e by rows o f  X  (FR2)  there i s W  -^.x'D* ^ W=  Z  a  r  e  Z such that e c  l *  Y  (L3)  is  satisfied.  l  -  X'  J=^X_ ^  shows t h a t  u a  X"  Finally  and note t h a t  X  , ,^_J^  the top and bottom the diagram  the  W.  - 52 (2) i s c l e a r  (3)  from the d e f i n i t i o n o f  Because Horn ( X , Y ' )  Horn  -s  (X,Y) =  i s an a b e l i a n group f o r each Y ' ,  l i m Horn ( X , Y ' )  \ ,  lim  ~  i s a l s o an a b e l i a n group,  Remark. If  S is a left  calculate  Horn  multiplicative  (X,Y)  as  filtered  x'  lim  category o f objects  •  Horn ( X ' , Y ) where J  x' e ob J  -s  X"  .  X  system i n C_, one can a l s o  1  i s the  ~ X — r  x  and morphisms  A NOTE ON F-ACYCLIC SUBCATEGORIES  2.  The n o t i o n of a F - a c y c l i c freedom of c h o i c e .  s u b c a t e g o r y appears  In g e n e r a l , i t  However, we show that  there i s  to l e a v e much  i s not n e c e s s a r i l y  unique.  always a maximal such s u b c a t e g o r y .  Definition. Let  F : C  D and ( R F , £ )  of F.  An o b j e c t of X o f C i s  if  : F(X)  p  > RF(X)  is  the r i g h t approximate e x t e n s i o n  s a i d t o be r i g h t F - a c y c l i c  if  and o n l y  an i s o m o r p h i s m .  ^X * Lemma.  The f u l l s u b c a t e g o r y _I  right F-acyclic  objects  is  o f C c o n s i s t i n g of  a right F-acyclic  all  s u b c a t e g o r y of C_ c o n t a i n i n g  I. Proof. (RAC1)  Set F* = F|I*,  S* = S O A R I * ,  F o r o b j e c t s X, Y i n I  *  F  Q  =  F|l  and a morphism X  s. 7  * Y in S .  We have  X S  JJ Y  = = ^ r(X) v  ^ r(Y)  F(X) Q'F*(s)| F(Y)  F(rX)  I —»• F ( r Y )  = F(o0 =  RF(X) = EF (a) Q  RF(Y)  - 54 -  EF (a) — U n  is  invertible  Hence Q F  (s)  (RAC2) with s e S  *  is  invertible  v £ S Y  in ^ ,  , so F  is  (S  *  and 1 6 ob I_ , i t  suffices  to show I_^I_  But f o r X i n J.,  = SflARI,  F_  is  (S„,S')  and X .,, exact.  in C „ . —o , S)  In o r d e r to show each X i n C admits  has t h i s p r o p e r t y . with  in D„, since a is invertible —b  exact.  X  *  since I  ^» r ( X ) Therefore,  - 55 QUASI-SPLIT  3.  A s h o r t exact  sequence *  is  i  : 0 —> X  Z i n C(A)  SEQUENCES  * C  n  c a l l e d pseudo o r q u a s i s p l i t  s h o r t e x a c t sequences 0 —>• X  n  splits  Y ,  i.e.,  C  - X  n  n  ®  —> C  Y if  —>  0  each of  —* Y  n  *  n  the  —* 0  n  * [1] .  Let  us see how the complex C  d e s c r i b e d i n terms o f X The maps i  in this  case may be  and Y .  and p can be w r i t t e n as m a t r i c e s :  * and the d i f f e r e n t i a l o p e r a t o r 3  3  c  a  \Y , where  n  n  a  y  : X  n  * n  :  X  n  .n «  )  : X  n  O  of C  Y  •  n  ^ „n+l ——*• X > Y  n + 1  may be r e p r e s e n t e d by  n 3 6  n  v  n  : Y : Y  n  X  n + 1  v  »  n+l  ^ X > Y  n + 1  Y  n + 1  - 56 -  From 9 i  = i 3  c  x  ,  o. 3 \ / 1  f  a Y,  From  P  3  C  =  Therefore,  3 p,  (0,l ) (" \ \ = 3 (0,1 ) we get ( ,6) = (0,3 )  y  3  V  =  x Thus 3 i s  3  8  0  x  Y  y  3  X  o  0,  3 3 + 33„ =  V  3  has the form  Moreover from 3  We g e t  y  v  P  3  3 )  lo  x  &  3  /o  0  0  0  V  0.  Y  a complex map Y  >  X [1].  3 depends n o t o n l y on the g i v e n q u a s i - s p l i t sequence Z b u t a l s o on the s e c t i o n s S „n C .  : Y ° —>• C  n  n  used to  decompose  . . , , „n /0\ In our matrxx n o t a t x o n , we have used S = ^ \ :  Ia 3\ /0V  and  I& \ 1 \ 0  / Y 0  Now, any o t h e r s e c t i o n must be o f the  S'  =  !M  where  h  n  :  Y  n  form  • X  n  ,  „n „n Y —> C  /3  0  - 57 -  and i n d u c e s g'  : Y  *  Hence g - g ' = h3  *  >• X [1] a l s o  - 3 h X  A.  T a k i n g i n t o account o f the d i f f e r e n t .* * X and X [1], we see  >  The homotopy c l a s s  associated with a quasi-split  ^  ^ ^*  ^  , If  [2]. with twist  0  Z : 0  i  f X  g : Y  of maps g : Y  ^*  called *  sequence. X  > X [1] ,  the t w i s t of the *  • C  ^  n —  sequence.  *  Y  • 0  quasi-splits  then a p p l y i n g 0-cohomology  sequence *  x  i  *  —^ C  D  *  -A Y  6  ,  T(i)  *  -Ax*[l] — [ 1 ] R  [1]  sequence  ^ ^*  i n the manner d e s c r i b e d i s ,  of  X*[l]  determined up to homotopy by the g i v e n q u a s i - s p l i t Definition.  the  s i g n s of the d i f f e r e n t i a l s  that  g : Y* is  satisfying  T(D)  *  iiPJ+ Y [1]  to  T(6)  *  X [2] -»  - 58 and n o t i n g  that  the diagram  n, *\ ^ ( i ) n * H (p) * r n+l * H H ( X ) — V H ( C ) -——> H (Y ) ~ ^ - > H C X ) n  n  6  n + 1  n  n  n + 1  ( i ) +i * ¥E (C ) J  n  H°(X*[n]) •* R°(C*[n]) •* H°CY*[n]) •* H°(X*[n+l]) H°(T i) n  is  H°(T ) n  H°(T 3)  n + 1  i)  commutative f o r e v e r y i n t e g e r n, and s i n c e a l l the  isomorphisms a r e n a t u r a l , we 6^  l  H°(C*[n+l])  H°(T  n  P  n  i n cohomology  [3]  £ : 0 —>  C* -A- Y* —>-  X*  3  0 quasi-split  i s homotopic t o 0 i f f the sequence  i n t h i s case, the s e c t i o n i s (b)  If X  (  S  n  =/  - ^  3-0,  ^ ly  n  n  +  Y c >  1 1  p  n n + h  splits,  where h : g~ 0.  g i v e n by  Proof. Let h :  ^)  i s contractible with h  then p has a homotopy i n v e r s e  Define  morphism  3, then: (a)  (a)  the c o n n e c t i n g  i s i n d u c e d by the t w i s t .  Given  with twist  conclude t h a t  1  :  1~0  - " (11 1  a  y  =  $  n  f o r a l l n.  - 59 Then (  p  .  )"  S  .  ^  ^  g  ;  *  y  is a  c  c h a i n map  since  - h  g Y  , n+1 -h i n a  Y  Conversely i f  quasi-splits with section S =  Z  X  from  (b).  -0  n  -h n'  P  0  we get  3  =  .  h  n  -h  1  +  1  3  n  Y  n  h  X  n  x. e.  h : 6 *  diagram  X 1  *  0  X  p  Y  we see t h a t p i s an isomorphism i n Because h : 1  « 0 , -l n v  A  A  = 3* h  1  *  K(A). +  n + 1  A n +  h"  3 X  - e - £ h V + h af n  n  )  3" Y  From the homotopy commutative  X  1  n+1  1  - 3  C  1  e  n  n + 1  0 .  ,  then  - 60 Substituting  we get Hence  9  £  - ; h 3  h'3  is  3  + 1  =  n  3  n + 1  n  -B J n+1  9  +  h  g  n + 2  n  +  1  j£ =  B  n  a homotopy from 3 to 0, and by ( a ) ,  p  _ 1  = ( ~  h B  1  it  [4]•  F o r any complex X ,  c o n t r a c t i b l e w i t h homotopy  the mapping cone + i d * x  h = ^ ^  ^  ^ )  is  "  Proof.  o  o ) / y  ±i *\  l *  0/0  o  °/Vx  V/  x  ^X  IH.  If  S  x  f  +  3/  /  : 0 — » X* - L *  * :  _  c  i  0  x  v  0  v  0  s p l i t s w i t h t w i s t 3 and s e c t i o n ^  ±i *Wo  X* S = ( °  X  )  *  > 0 then  * r 2 X  Proof.  quasi-  q has a homotopy i n v e r s e g i v e n by p = (  )  .  )  - 61 -  1  o  C  ;  x:  'o x  x£ © x!J © x £  Set  :  a  + 1  *  X [-l] 2  J  a  -id * 0  ,n-l  n  X  Then  C"  - X? © x" "0 ~ 0  /'S 8  C*  9  +1  X  n  -id n+1 0 .n+1 -8 0  3n  X  0  and the map g : C  C  0 n  d e f i n e d by £ =  -1 isomorphism w i t h i n v e r s e £ ~ = £ .  We have a commutative  diagram -l  (P,0)  L  * ^ 0)  where  W is  d e f i n e d by  is  X  2  an  - 62 -  -id * X  C_  0  a  i s c o n t r a c t i b l e w i t h homotopy  1  X  0  X [-l]([l]) 2  h =|  I  0  0  - 1 0  - 63 -  4 QUISOS AND THE DERIVED CATEGORY  A quiso or quasi-isomorphism i s  a co-chain map o r homotopy  c l a s s o f one , which i n d u c e s an isomorphism i n cohomology. A  [1]. plicative  The c l a s s  o f q u i s o s i n K (A)  form a r i g h t  and l e f t  multi-  system.  Proof. (FRO)  is  clear.  * (FR1):  Given  X  , make - + Z  Y  A  C. 1  A  *  - X [1] ^ T ( s ); A \ Y [1]  c T(f) 1  t  1  '  c  s T *  t  t A Z [1] 8  t  > T  c  *  T  < > i f  s  X  By p r o p o s i t i o n 7(or TR3) c o m p l e t i n g the square  there e x i s t s * , X [1]  *  From the l o n g exact  sequence  *  •  f  T(s) Y  h  a morphism  ft T(f)  [1]  L L *  * Z  [1]  it  h : X [1]  »  "k  C  - 64 -  n, * "(g) H (C_) — ^  n * H (t) * H ( Z [1]) » H (C )  H  n  n  and the  fact  that  FR2  we  consider  tf  any  Given X  = 0.  s .  X  of  A  » C  the  diagram  X  t  is  D u a l l y we  We d e f i n e of  K(A)  D (A) b  the  is  t  is  w i t h hs = gs  and r e d u c e  there  is  exact  to  a  .  quiso. To  showing  prove that  a quiso t  on t h e  such  that  sequence  y ..  [1]  a morphism g : it  S Y  &  f  C  A  > Z  A  completing  & > Z  the  a  Then from the  ^ V « £ >  —  exact  sequence  H"(C*[1J) — *  ,  ...  quiso.  can prove FRO  derived  with respect  are  Z )  > X  g  »V>  that  Z  g  = 0,  A  > C^ .  N  follows  A  v  ZZ£  that  A  Z  H (C*>  it  we s e e  y ..  c  A  exists  A  :  A h  n,  = h - g  fs  A  there  t  Y  all  n  A  Hom^^C-,  = 0,  Now t a k e  s  > X  Y  and f s  (1)  f  Y  Because  A  *  for  the morphism f A  X  = 0,  g  A 3  for  n  H (C )  (FR2):  n * > H (C [l])  n  n  - FR2°  category  to q u i s o s .  localisations  0  of  D(A)  .  o f A as  Similarly, K*(A)  D (A),  , K (A), +  K~(A)  the  localisation  D (A) and  ,  D~(A),  K (A) b  - 65 -  respectively. it  [1]  g i v e s us a good h o l d on D (A)  can be h a n d l e d as a c a t e g o r y o f l e f t  FR0-FR2 o r F R 0 ° - F R 2 °  i n the sense  or r i g h t f r a c t i o n s  that  ( using  ).  [2]. (a) . L e t S be a m u l t i p l i c a t i v e  system i n a c a t e g o r y (2.  L e t D be a f u l l s u b c a t e g o r y o f C_ and assume t h a t S H M o r D multiplicative  system i n D, assume f u r t h e r t h a t one o f the  is  a  follow-  i n g two c o n d i t i o n s h o l d : s i) . there e x i s t s X "  Whenever X ' = ^ X'  such t h a t  X w i t h s <L S and X € Ob D,  s f £ S and X ' 6 0b D. s  ii) . there e x i s t s X then P_g|-^  Mor  D  (b).  X" c  a  n  Whenever X ' such t h a t  = ^ X with s e S  f s 6 S and X " € Ob D.  be i d e n t i f i e d w i t h a f u l l s u b c a t e g o r y o f C_ . g  Each o f the f u n c t o r s  D (A) +  •»  D (A)  ^ D(A)  b  ^ is a f u l l  and X' € Ob D,  D*(A)  embedding.  Proof. (a):  We prove o n l y f o r r i g h t m u l t i p l i c a t i v e systems and  The o t h e r cases a r e  similar.  ii),  - 66 f  s  G i v e n a quasi-morphism X By h y p o t h e s i s , D e O b D.  We  there e x i s t s  From the  conclude  Y ' <£= Y w i t h X, Y 6 Ob D.  that  t  : Y'  — > D such t h a t  ts e S  and  kite  (X  — Y '  Y)  £  HOITL  (X,Y).  -SflMorD  (b)  (i)  Let X  D (A)  *  s  satisfying X Set  — 1 > D(A)  Y  i s a f u l l embedding.  be a q u i s o w i t h X £K (A)  and m e  = 0 f o r a l l n<m. -7*  r  ,m m+1 J*-*^ m Y Y m+2 imcL. —> Y — Y —* Y A A n ^ Y —* Z f = | L^n n>m  Z  = ( -s>- 0 —•  f  :  A  4  d  v  ,n-l dl  n = m-1  0  0  • 0-  Jo  Jo  v  I'  v  A  I m ,m  v  I  0  0  m-1 1  K" -> i m i  v  m+l  >• A  |s  m-2 _ ^ m - l  J  ,m X  m  m-1  n<m ,m+l X m+2 *X v  I m+1  I m+2  4  \s v  s  m + l _ ^ m+2  i  v  1  „m+l  1  i  m+2  T  .)  - 67  Thus H ( f s ) by ( a ) ,  : H (X ) .= H (Z ) f o r a l l n;  fs £ S ,  and  we are done.  (ii)  A A  f  L e t A —> b X € Ob D (A).  , S  *  B be a quasi-morphism  Since  and l o o k s l i k e :  +  X  *  with H ° ( s )  > D (A)  s is  .. B  where A, B £ ObA , * a q u i s o we may assume X i s t r u n c a t e d  • 0  = Ker d °  1 X  -1 "X  y X  0 "X  1  ? X  —* . .  .  Now from  f  we see t h a t X which i s  a morphism i n  s  resolutions  f  B can be r e p l a c e d by A  a substitute for  •<  :  I be a f u l l s u b c a t e g o r y of A such t h a t  of C ( A ) +  every  object  of  every  admits a q u i s o i n t o an o b j e c t o f C (_I). +  Proof. A  +  We may assume X €: C (A)  X  *  =  =  "Cartan-Eilenberg"  A admits a monomorphism i n t o an o b j e c t o f 1^, then object  ,0  > Ker d  A.  The f o l l o w i n g i s  [3]  Let  A  > X <(  .. —*  is  trivial  0 —> X  0  i n negative  —> X  1  dimensions.  B,  - 68 -  Define I 0 imbed X  = 0, s  n  s°  >  A  T  n  =  1  = 0  V  , s  0  >  and  exist with and s ^ "  =d^  1  ^  ....  ^ x  k  -  2  - , x  -  k  1  s n  \/^k.  1  k-l  d  0  <  .  0Vnfk-l  . . - * o - , x  n  n I e obi.  with  n  Assume I d ^ "  o  > I  = 0, and  n  1  k  d  i x  k  J l x  k  +  1  k s  k+1  F o r n = k+1  let  out diagram  7*  1  4m d  _  k ^  "1  VI k  x  , g , a K  K  x  )  b  e  d e f i n e d by t h e push-  ^ x  T~ /  (P  V+1  1c  1  =  C  °  x  J  1 k  d  k  k+l _ ^  -  1  I  P  k+2  x  „ T,k+1 > P  k+1 Now imbed  Then,  P  d ^ " -L  1  k + 1  >I—  - 0, s  ¥  I  k + 1  k+1  d  a n  k  X  « y  d set s  k  +  1  .  a  k  +  1  k + 1  d  k  A  Hence, by i n d u c t i o n we o b t a i n s : X  -  k  Y  -  Y  +  k  +  1  1  >  a  k  \ .  +  d  .B ., . k  I  k  .  =  k  s  k  Y  k + 1  -  B . r . k  1  l-s^. 1  k  - 69 -  To see t h a t s i s a q u i s o , we r e c a l l  that  if:  a  is  A  • B  C  > D'  a pushout i n A , then 1)  3 induces an e p i : k e r a  2)  ker a'  3' induces a mono : "cok a >  >  cok a '  In our s i t u a t i o n we have f o r each k k k IT s i n d u c e s  1)  Z (X*) k  = ker d  2) ,  a ,k  k  k+1  ker3 - k e r Y k  k + 1  3 = ker d / k  k  .  Q  k „  > cok 3  . _k k ^_  , . k + l „ k k.  v  - cok3 n > •-) cok(y  3  TT  „ k , *. H (X )  1  u  k, *. )  diagram w i t h e x a c t  ^ . ,k > cok d  1 ,  H (X*) k  X  ,k  ^  > cok d^.  k * We see t h a t H (X ) >  rows  ^ . ,k+l >• coim d  X  ^  T  f H (I  Hence,  =  ,k  cok d _ . I  From the commutative  0  1  , )  A  „  = H  1  • A ^ '/ lm dj.  i n d u c e s monos:  ^  %  cok d >  — »  epi's  .  ,k+l  > coxm d^.  ^ •  k * * H (I ) i s monic.  > H ( I * ) i s i s o f o r a l l k. k  -  70 -  5.  [1]  EXT  L e t A be an a b e l i a n c a t e g o r y and f  be a morphism i n C ( A ) .  : X  —* I  H (X*) =  oVn,  Assume:  1)  X* i s a c y c l i c i . e .  2)  each i  n  is  n  infective,  * 3)  I  i s bounded below,  then f i s homotopic to Proof.  zero.  W e l l known, (by  induction).  Let A be an a b e l i a n c a t e g o r y and s : I  [2]  a morphism i n C ( A ) .  —>  Y  Assume:  1)  s is  a quasi-isomorphism,  2)  each I  3)  I  n  is  infective,  i s bounded b e l o w ,  then s has a homotopy i n v e r s e . Proof.  (from R.  H a r t s h o r n e ' s Residues and D u a l i t y )  * The mapping cone C  is  acyclic.  The morphism  s  *  v = (ld,0) o.  : C — • s  Let H  = (k,t)  I  *  [1]  satisfies  : I  [1]  (ld,0)  =  ® Y  —>  [1]  and i s  t h e r e f o r e homotopic to  I  be the homotopy  +  3 (k,t)  operator. v  =  (k,t)3 *  T  - 71 -  Separating  components, we have  l d j * ^  =  and  3t y  TO .)k  -  t9  * Thus t  :  Y  + kTO^  ]  v  =  + t-T(s)  0.  * y  I  is  a morphism of complexes  Id* i s homotopic to t « s , so t i s  and  a homotopy i n v e r s e  of  s,  - 72 References Artin,  M.  Bredon,  G r o t h e n d i e c k T o p o l o g i e s , mimeographed seminar Harvard U n i v e r s i t y , 1962.  G.E.,  Sheaf  Theory, McGraw-Hill,  C a r t a n , H. and S. E i l e n b e r g , University Press, G a b r i e l , P.,  Gamst, J .  and K.  1967.  Homological Algebra, Princeton Princeton, N.J., 1956.  and M. Zisman, C a l c u l a s Homotopy T h e o r y ,  New Y o r k ,  notes,  of F r a c t i o n s  and  S p r i n g e r - V e r l a g New York I n c . ,  Hoechsmann, P r o d u c t s  1967.  i n Sheaf Cohomology I,  II.  H a r t s h o r n e , R., Residues and D u a l i t y , L e c t u r e n o t e s i n mathematics, n o . 20, S p r i n g e r - V e r l a g , 1966. Hoechsmann, K., Notes on D e r i v e d C a t e g o r i e s (unpublished.) MacLane, Mitchell, Spanier, Swan, R.,  S.,  Homology, Academic P r e s s ,  B., Theory of C a t e g o r i e s , 1965. E.H.,  and F u n c t o r s ,  New Y o r k ,  Academic P r e s s ,  A l g e b r a i c Topology, McGraw-Hill,  The Theory of Sheaves, P r e s s , C h i c a g o , 1964.  1963.  The U n i v e r s i t y  New Y o r k ,  New Y o r k ,  1966.  of C h i c a g o  

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