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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Derived categories and functors Loo, Donald Doo Fuey 1971
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Title | Derived categories and functors |
Creator |
Loo, Donald Doo Fuey |
Publisher | University of British Columbia |
Date Issued | 1971 |
Description | For each abelian category A, there is a category D(A), called the derived category of A, whose objects are complexes of objects of A, and whose morphisms are formal fractions of homotopy classes of complex morphisms having as denominators homotopy classes inducing isomorphisms in cohomology. If F : A →B is an additive functor between abelian categories, then under suitable conditions on A, there is a functor RF : D(A) → D(B) with the property that if objects X of A are considered as complexes concentrated at degree 0, then there are isomorphisms [formula omitted] for all n, where [formula omitted] is the ordinary [formula omitted] right derived functor of F. RF is called the derived functor of F, and one may look upon it as a kind of extension of F. |
Subject |
Functor theory. Categories (Mathematics) |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-04-07 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080155 |
URI | http://hdl.handle.net/2429/33402 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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