Derived Limits in QuasiAbelian Categories

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1 Prépublcatos Mathématques de l Uversté ParsNord Derved Lmts QuasAbela Categores by Fabee Prosmas March 98 Laboratore Aalyse, Géométre et Applcatos, UMR 7539 sttut Gallée, Uversté ParsNord Vlletaeuse (Frace)
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3 Derved Lmts QuasAbela Categores Fabee Prosmas Aprl 1, 1998 Abstract ths paper, we study the derved fuctors of projectve lmt fuctors quasabela categores. Frst, we show that f s a quasabela category wth exact products, projectve lmt fuctors are rght dervable ad ther derved fuctors are computable usg a geeralzato of a costructo of Roos. Next, we study dex restrcto ad exteso fuctors ad lk them trough the symbolc Hom fuctor. f : s a fuctor betwee small categores ad f s a projectve system dexed by, ths allows us to gve a codto for the derved projectve lmts of ad to be somorphc. Note that ths codto holds, f ad are flterg ad s cofal. Usg the precedg results, we establsh that the th left cohomologcal fuctor of the derved projectve lmt of a projectve system dexed by vashes for k, f the cofalty of s strctly lower tha the kth fte cardal umber. Fally, we cosder the lmts of proobjects of a quasabela category. From our study, t follows, partcular, that the derved projectve lmt of a flterg projectve system depeds oly o the assocated proobject. Cotets 0 troducto 2 1 Quasabela homologcal algebra Quasabela categores Dervato of a quasabela category tstructure ad heart of the derved category Dervato of fuctors betwee quasabela categores AMS Mathematcs Subject Classfcato. 18G50, 18A30, 46M20. Key words ad phrases. Noabela homologcal algebra, quasabela categores, derved projectve lmts, homologcal methods for fuctoal aalyss. 1
4 2 Fabee Prosmas 2 Projectve systems quasabela categores Categores of projectve systems Projectve systems of product ad croduct type jectve ad projectve objects dex restrcto ad exteso Dervato of the projectve lmt fuctor The case where has eough jectve objects Roos complexes The case where has exact products Derved projectve lmt fuctor ad dex restrcto Dual results for the ductve lmt fuctor Relatos betwee RHom ad derved lmts Derved lmts ad the symbolchom fuctor The symbolchom fuctor Dervato of the symbolchom fuctor Lks wth the derved projectve lmt fuctor dex restrcto Derved projectve lmts ad cofalty Cofal dex restrcto Cofalty ad ampltude of derved projectve lmts Proobjects Categores of proobjects Prorepresetable fuctors Represetato of dagrams of proobjects Lmts categores of proobjects Derved projectve lmts ad proobjects Proobjects of a quasabela category The fuctor L Dervato of L troducto t s wellkow that the projectve lmt of a short exact sequece of projectve systems of abela groups s ot always a exact sequece. Ths pheomeo ofte explas the problems oe meets the globalzato of local results algebra or
5 Derved Lmts QuasAbela Categores 3 aalyss. To uderstad ths loss of exactess, t s atural to study the derved fuctors of the projectve lmt fuctor. Ths was doe the sxtes by varous authors ([3, 4, 6, 8], etc.) ad led to a rather good uderstadg of the homologcal algebra of projectve lmts abela categores. However, for varous applcatos to algebrac aalyss, t would be very useful to exted these results to oabela categores such as the category of tologcal abela groups or the category of locally covex tologcal vector spaces. Ths s what we do ths paper the more geeral framework of quasabela categores. We start wth a summary of the facts about the homologcal algebra of quasabela categores whch are eeded the other sectos (see [9] ad [7] for more detaled expostos). Ths should help the reader whch has a good kowledge of the laguage of homologcal algebra ad derved categores the abela case (as exposed e.g. [5]) to uderstad the rest of the paper. Frst, we recall the codtos a category has to satsfy to be quasabela. Next, we expla brefly the costructo of the derved category D( ) ad we gve the ma results about the two caocal tstructures o D( ) ad ther correspodg hearts LH( ) ad RH( ). We ed ths secto by recallg how to derve a addtve fuctor betwee two quasabela categores. Secto 2 s devoted to the study of the category of projectve systems a quasabela category. We show frst that they form a quasabela category. The, usg projectve systems of product type, we prove that ths category has eough jectve objects whe tself has eough jectve objects. We coclude by defg the dex restrcto ad exteso fuctors. Secto 3, we expla how to derve the projectve lmt fuctor lm : where s a quasabela category ad a small category. Frst, we cosder the easy case where has eough jectve objects. Next, we treat the case where has exact products. ths case, we show that the derved projectve lmt of a projectve system of s somorphc to ts Roos complex. Moreover, f : s a fuctor betwee two small categores ad s a projectve system of dexed by, we show how to compute the caocal morphsm R lm ( ) R lm j ( ( )) by meas of Roos complexes. By dualty, we get correspodg results for the ductve lmts. At the ed of ths secto, we establsh commutato formulas for derved lmts ad the derved Hom fuctor. j
6 4 Fabee Prosmas Secto 4, we recall a few prertes of symbolchom fuctors ad show how to derve them. The, we prove that derved projectve lmts may be computed usg sutable derved symbolchom fuctors. Ths allows us to gve a codto for the caocal morphsm R lm ( ) R lm j ( ( )) to be a somorphsm. the frst part of Secto 5, usg the precedg result, we show that f a fuctor : betwee small flterg categores s cofal, the j R lm ( ) R lm j ( ( )) D ( ). the secod part, we establsh that, f the cofalty of the small flterg category s strctly lower tha the kth fte cardal umber, j LH (R lm ( )) = 0 k for ay projectve system of dexed by. Secto 6 s devoted to a revew of the results about proobjects we eed Secto 7. We refer the reader to [2] for detals. the frst three parts of ths secto, we recall basc results about proobjects, represetable fuctors ad represetato of dagrams of proobjects. the last part, we show that the category of proobjects of a arbtrary category has flterg projectve lmts ad we establsh some prertes of these lmts. Secto 7, we prove that the category of proobjects of a quasabela category s also quasabela ad has exact products. Next, we troduce the fuctor L: Pro( ) ad we establsh that f the category the fuctor L s rght dervable ad s quasabela ad has exact products, the RL lm ( ) R lm ( ) for ay flterg projectve system dexed by. Ths shows partcular that the derved projectve lmt of a flterg projectve system depeds oly o the assocated proobject. Note that the theory develed ths paper may be appled to the category of tologcal abela groups or the category of locally covex tologcal vector
7 a a { { Derved Lmts QuasAbela Categores 5 spaces sce these categores are quasabela ad have exact products. these cases, more specfc results may be obtaed. Work ths drecto s progress ad wll appear elsewhere. To coclude ths troducto, wat to thak.p. Scheders for the useful dscussos we had durg the preparato of ths paper. 1 Quasabela homologcal algebra 1.1 Quasabela categores To avod cofusos, let us frst recall a few basc deftos. Defto Let Abe a addtve category ad let f : A B be a morphsm of A. () A kerel of f s the data of a par (ker f,) where ker f b( A) ad Hom A(ker f,a) are such that f = 0 ad for ay g Hom A( C,A) verfyg f g = 0, there s a uque g Hom ( C, ker f) makg the dagram A ker g A?? f A B D DD D DD g D D commutatve. () A cokerel of f s the data of a par (coker f,q) where coker f b( A) ad q Hom A( B, coker f) are such that q f = 0 ad for ay g Hom A( B,C) verfyg g f = 0, there s a uque g Hom (coker f,c) makg the dagram C f 0 0 f A B coker f???????? C g commutatve. e ca check that : ker f A s moomorphc ad that q : B coker f s epmorphc. Moreover, two kerels (resp. two cokerels) of f are caocally somorphc. () f the morphsm : ker f A has a cokerel, t s called the comage of f ad deoted by com f. (v) f the morphsm q : B coker f has a kerel, t s called the mage of f ad deoted by m f. q w ww g w ww w ww
8 d d " " { { Fabee Prosmas Remark There s a caocal morphsm com f m f. As a matter of fact, sce com f s the cokerel of : ker f A, there s a uque morphsm f : com f B makg the dagram 0 D D D D D D D D f commutatve. We have q f q = q f = 0. The morphsm q beg epmorphc, t follows that q f = 0. Sce m f s the kerel of q : B coker f, there s a uque morphsm com f m f makg the dagram commutatve. m H q ker f A com f B x xx f x xx q x xx f B coker f H s H s s H f s s H s s com f s s s 0 Defto A category s quasabela f () t s addtve, () ay morphsm has a kerel ad a cokerel, () a cartesa square X f Y f s a strct epmorphsm, the f X f Y s a strct epmorphsm, (v) a cocartesa square X f Y f s a strct moomorphsm, the f s a strct moomorphsm. X f Y
9 Derved Lmts QuasAbela Categores Dervato of a quasabela category ths secto, wll be a quasabela category. We deote by C( ) the category of complexes of ad by K( ) the category defed by b( K( )) = b( C( )) ad where K( ) C( ) Hom ( X,Y ) = Hom ( X,Y )Ht( X,Y ) Ht( X,Y ) = { f : X Y : f s homotc to zero}. Recall that K( ) has a caocal structure of tragulated category. Defto A sequece u v A B C of such that v u= 0 s strctly exact f u s strct ad f the caocal morphsm m u ker v s somorphc. Remark Note that a sequece of such that F G Hom ( X, ) Hom ( X, F ) Hom ( X, G) s exact for ay X () A of s f the se Defto quece b( ) s strctly exact. complex X k 1 k k 1 d k d k1 X X X strctly exact degree k s strctly exact. () A complex of s strctly exact f t s strctly exact every degree. () We deote by N( ) the full subcategory of K( ) whose objects are the strctly exact complexes of.
10 8 Fabee Prosmas (v) A morphsm f : X Y of K( ) s called a strct quassomorphsm f there s a dstgushed tragle N of K( ) such that Z b( ( )). X Y Z X [1] Prosto The subcategory N( ) of K( ) s a ull system,.e. t verfes the followg codtos: () 0 N( ), () ( ) f ad oly f [1] ( ), X () f N X N X Y Z X [1] N s a dstgushed tragle of K( ) where X, Y b( ( )), the Z b( N( )). Defto The derved category of deoted by D( ) s the localzato of the tragulated category K( ) by N( ). The, D( ) = K ( ) N( ). Remark Note that as the abela case, a strctly exact sequece 0 X Y Z 0 of C( ) gves rse to a dstgushed tragle X Y Z X [1] of D( ). 1.3 tstructure ad heart of the derved category Frst, let us recall some usual results about tstructures o a tragulated category. Defto T T T strctly full subcategores of T. We set 0 0 Let be a tragulated category ad let ad be two 0 0 T = T [ ] ad T = T [ ]. 0 0 T T T The, the pars (, ) forms a tstructure o f t verfes the followg codtos:
11 Derved Lmts QuasAbela Categores () ad, T T T T 0 1 () Hom T ( X, Y ) = 0 f X b( T ) ad Y b( T ), () for ay X b( T ), there s a dstgushed tragle T T such that X b( ) ad b( ) T X1 T 0 0 The heart of the tstructure ( T, T ), deoted by H, s the full subcategory of T defed by 0 0 H = T T. Theorem The heart of ay tstructure s a abela category. 0 0 Prosto Let ( T, T ) be a tstructure o a tragulated category T. () There s a fuctor τ : T T such that X X X X [1] Hom ( X, τ ( Y )) Hom ( X, Y ) for ay X b( T ) ad ay Y b( T ). the same way, there s a fuctor such that τ for ay X b( T ) ad Y b( T ). These fuctors τ ad τ are the 0 0 trucato fuctors assocated to the tstructure ( T, T ). () For ay Z, the fuctor defed by Hom ( τ ( X ),Y) Hom ( X,Y) H : T T : T H H ( X) = ( τ τ ( X))[ ] s a cohomologcal fuctor,.e. ay dstgushed tragle X Y Z X[1] T T of T gves rse to the log exact sequece H ( X) H ( Y) H ( Z) D BC H ( X) H ( Y) H ( Z)
12 10 Fabee Prosmas Defto Let be a quasabela category. We deote by D 0 0 ( ) (resp. D ( )) the full subcategory of D( ) whose objects are the complexes whch are strctly exact degree k> 0 (resp. k< 0). Prosto a tstructure o D( ). We call t the left tstructure of D( ). ( D,D ) 0 0 Let be a quasabela category. The par forms Remark The heart of the left tstructure s deoted by 0 0 We call t the left heart of D( ). f course, the objects of LH( ) are the complexes whch are strctly exact every degree but zero. The cohomologcal fuctors are deoted by where s degree ad k where s degree. Hece, the cohomologcal fuctors are gve by where s degree. LH( ) = D ( ) D ( ). LH : D ( ) LH( ). Prosto D( ). The trucato fuctors are gve by ker d X Let be a quasabela category. Let be a object of τ ( X ) : X ker d 0 τ ( X ) : 0 com d X X ker d 0 1 LH ( X ) : 0 com d ker d 0 X Prosto Let be a quasabela category. The fuctor : LH ( ) whch assocates to ay object of the complex 0 0 where s degree 0 s fully fathful.
13 Derved Lmts QuasAbela Categores 11 Remark wrte X Let be a object of ( ). By a abuse of otatos, we wll X X LH f s somorphc to ( ) for some object of. Prosto Let be a quasabela category. (a) Ay object of LH( ) s somorphc to a complex u 0 A B 0 where B s degree 0 ad us a moomorphsm. Moreover, such a object s the essetal mage of f ad oly f u s strct. (b) A sequece F G of s strctly exact f ad oly f the sequece ( ) F ( ) G ( ) of LH( ) s exact. Corollary D( ). The, Let be a quasabela category ad let be a object of X k LH X X k () ( )=0 s strctly exact degree, () k k 1 LH ( X ) dx s strct. Remark Replacg the oto of strctly exact sequece by the oto of costrctly exact sequece, we may defe a secod tstructure o D( ). We call t the rght tstructure ad ts assocated heart (the rght heart) s deoted by RH( ). 1.4 Dervato of fuctors betwee quasabela categores F ths secto, : wll deote a fuctor betwee quasabela categores. Defto Let Q : K ( ) D ( ) ad Q : K ( ) D ( ) be the caocal fuctors. A rght derved fuctor of F s the data of a par ( T,s) where T : D ( ) D ( )
14 & & 12 Fabee Prosmas s a fuctor of tragulated categores ad s M M M M t M M M M M α dq k k k k such that, for ay k, b( ) ad u : X s a strct moomorphsm. M T,t there s a uque morphsm α : T T of fuctors makg the dagram commutatve. Defto A full subcategory of s Fjectve f () 0 0 s a strctly exact sequece of such that, Prosto f s a Fjectve subcategory of, the for ay object X of C ( ), there s a strct quassomorphsm Prosto s : Q K ( F) T Q s a morphsm of fuctors such that for ay par ( T : D ( ) D ( ) t : Q K ( F) T Q, Q K ( F) T Q T Q () for ay b( ), there s a strct moomorphsm where b( ), b( ), the (a) b( ), (b) 0 F( ) F( ) F( ) 0 s strctly exact. u : X We call a Fjectve resoluto of X. f has a Fjectve subcategory, the fuctor ) where F : s rght dervable ad ts derved fuctor s gve by F X where s a jectve resoluto of. RF : D ( ) D ( ) RF ( X ) = F ( )
15 Derved Lmts QuasAbela Categores 13 Defto () A object of s jectve f for ay strct moomorphsm u : F ad ay morphsm v :, there s a morphsm v : F makg the dagram v ~ ~~ u v ~ ~~ commutatve. () The category has eough jectve objects f for ay object of, there s a strct moomorphsm wth jectve. ~ ~ F Prosto f has eough jectve objects, the the full subcategory of formed by the jectve objects s Fjectve for ay fuctor F :. partcular, ay fuctor F : s rght dervable. Now, let us expla how to derve a bfuctor. Prosto Let F (, ): be a bfuctor betwee quasabela categores. Assume that there are full subcategores ad of ad respectvely such that () for ay b( ), s F (,)jectve, () for ay b( ), s F(, )jectve. The, the fuctor F (, ) s rght dervable ad ts derved fuctor RF : D ( ) D ( ) D ( ) s gve by RF ( X, Y ) = sf (, ) where (resp. ) s a jectve resoluto of X (resp. Y ) ad sf (, ) s the smple complex assocated to the double complex F(, ). Remark Dually, t s possble to derve fuctors o the left by cosderg Fprojectve subcategores.
16 14 Fabee Prosmas 2 Projectve systems quasabela categores 2.1 Categores of projectve systems Defto Let C be a category ad let be a small category. We deote by C the category of fuctors from to C. The objects of C wll be called projectve systems of C dexed by. Prosto Let be a small category. Assume s a quasabela category. The, s a quasabela category. Proof. We kow that the category s addtve. Cosder a morphsm f : F of. The kerel of f s gve by a object K of ad a morphsm u : K of such that for ay, the object K ( ) of ad the morphsm u ( ) : K ( ) ( ) form a kerel of f ( ). The cokerel of f s defed smlarly. t follows that a morphsm f : F of s strct f ad oly f f ( ) : ( ) F ( ) s strct for ay. Cosder a cartesa square f F f F of, where f s a strct epmorphsm. Sce for ay, the square f() () F () f () () F () of s cartesa ad sce f ( ) s a strct epmorphsm, f( ) s a strct epmorphsm. t follows that f s a strct epmorphsm of. Usg the same kd of argumets, a cocartesa square f F f F of, f f s a strct moomorphsm, the f s also a strct moomorphsm.
17 Derved Lmts QuasAbela Categores Projectve systems of product ad croduct type Remark Hereafter, by a abuse of otatos, we wll deote by the same symbol a set ad ts assocated dscrete category. Defto Let be a small category ad let be a addtve category wth products. We defe the fuctor Π: b( ) the followg way. At the level of objects, for ay fuctor S : b( ) we defe the fuctor Π( S) : by settg Π( S)( ) = S( j) α j for ay. Let be a object of. For ay morphsm α : j of, we deote by p α S S j j :Π( )() ( ) the caocal projecto. The, f f : s a morphsm of, we defe by settg Π( S)( f) : Π( S)( ) Π( S)( ) p S f p Π( )( ) = α j α f α j j for ay morphsm :. b( ) At the level of morphsms, for ay morphsm s : S S of, we defe Π( s) : Π( S) Π( S ) by settg p Π( s)( ) = s( j) p for ay object of ad ay morphsm α : j of. α α j j Defto Let be a small category ad let be a addtve category wth croducts. Applyg the precedg defto to ad, we get a fuctor Through the caocal somorphsm b( ) ( ) ( ) ( ) ( D) D ( C ) ( C ),.
18 16 Fabee Prosmas ths gves us a fuctor Note that ( S)( ) = S( j ). α j b( ) b( ) b( ) Defto Let be a small category ad let be a addtve category wth products (resp. wth croducts). A projectve system product type : : croduct type b( ) s of (resp. of ) f there s a object of such that b( ) wth products (resp. wth croducts). For ay object S of ad ay object Π( S) (resp. ( S)). Defto Let be a small category ad let be a arbtrary category. We defe the fuctor b( ) : by ( )( ) = ( ) for ay object of. f f : s a morphsm of, we defe ( f) : ( ) ( ) by settg ( f)( ) = f( ). e checks easly that we have: Prosto Let be a small category ad let be a addtve category of, we have Hom ((,S ) ) Hom (, Π( S)) ( resp. Hom ( S, ( )) Hom ( ( S ), )). Prosto Let be a small category ad let be a addtve category b( ) wth products (resp. wth croducts). For ay object S of, we have the somorphsm lm Π( S)( ) S( ) ( resp. lm ( S)( ) S( )).. S
19 Derved Lmts QuasAbela Categores 17 Proof. lmts. Ths follows drectly from the defto of the projectve ad the ductve 2.3 jectve ad projectve objects Prosto S of. Let be a small category ad let be a quasabela category S b( ) wth products. f s a jectve object of, the Π( ) s a jectve object Proof. S b( ) Let be a jectve object of. Cosder a strct moomorphsm f : of. Sce for ay, f( ) : ( ) ( ) s a strct moomorphsm ad sce S ( ) s jectve, the sequece Hom ( f( ),S( )) Hom ( ( ),S( )) Hom ( ( ),S( )) 0 s exact. t follows that the sequece Hom (( f ),S) b( ) b( ) Hom (( ),S) Hom (( ),S) 0 s exact. By Prosto 2.2.6, the sequece Hom ( f, Π( S)) Hom (, Π( S)) Hom (, Π( S)) 0 s also exact ad the cocluso follows. Prosto Let be a small category ad let be a quasabela category wth products (resp. wth croducts). For ay object of, there s a strct moomorphsm (resp. strct epmorphsm) f : Π(( )) ( resp. g : (( )) ) of. Proof. Let be a object of. We defe the morphsm f : Π(( )) by settg p α j f()= ( α)
20 18 Fabee Prosmas for ay object of ad ay morphsm α : j of. Sce for ay object of, we have p d f, ( ) = (d) = d p d f f ( ) s a strct moomorphsm. Cosequetly, for ay, ( ) s a strct moomorphsm of ad f s a strct moomorphsm of. Prosto Let be a small category ad let be a quasabela category wth products. f has eough jectve objects the the category has eough jectve objects. Proof. Let be a object of. We kow that there s a strct moomorphsm f : Π(( )) of. Moreover, sce has eough jectve objects, for ay, there s a strct moomorphsm s (): () () of, where ( ) s a jectve object of. These morphsms defe a morphsm s :( ) b( ) b( ) of where s a jectve object of. Now, cosder the morphsm Π( s) f : Π( ) of. Sce the product of strct moomorphsms s a strct moomorphsm, for ay, Π( s)( ) s a strct moomorphsm of. Cosequetly, Π( s) s a strct moomorphsm of ad Π( s) fs a strct moomorphsm of. Fally, by Prosto 2.3.1, the object Π( ) s jectve. Ad the cocluso follows. Dually, we have: Prosto Let be a small category ad let be a quasabela category wth croducts. f has eough projectve objects, the the category has eough projectve objects.
21 Derved Lmts QuasAbela Categores dex restrcto ad exteso To fx the otatos, let us recall a few deftos of the theory of categores. Defto Let F : A C ad G : B C be two fuctors betwee arbtrary categores. We deote by F G the category whose objects are the trples ( a, f, b) where a s a object of A, b s a object of B ad f : F( a) G( b) s a morphsm of C. f ( a,f,b) ad ( a,f,b) are two objects of F G, a morphsm f ( u, v): ( a, f, b) ( a, f, b ) A B of F G s the data of a morphsm u : a a of ad a morphsm v : b b of such that the dagram F( a) G( b) F( u) G( v) s commutatve. Remark f G b F( a) ( ) Let 1 deote the category wth oe object. () f the fuctor F : 1 Cassocates to the object of 1, the object of C, the category F G wll smply be deoted G. f, moreover, G= d, the the category G wll be deoted C. C () Smlarly, f the fuctor G : 1 Cassocates to the object of 1, the object of C, the category F G wll be deoted F. Moreover, f F = d, the the category F wll be deoted C. C Prosto Let be a arbtrary category. For ay object of, (d,) ( resp. (, d )) s a tal object (resp. a termal object) of (resp. ).
22 20 Fabee Prosmas Defto Let : be a fuctor betwee two arbtrary categores ad let be a object of. () We defe the fuctor : by settg for ay object ( f,j) of ad by settg for ay morphsm β : ( f,j) ( f,j ) of. () We may defe the fuctor the same way. α α α α : Prosto Let : be a fuctor betwee arbtrary categores ad let α : be a morphsm of. () There s a fuctor α : such that () There s a fuctor such that Proof. defed by The fuctor for ay object ( f,j) of ad by ( f,j) = j ( β) = β α =. α : α =. : ( f,j) = ( f α,j) ( u) = u for ay morphsm u : ( f 1,j1) ( f 2,j2) of, solves the problem. The fuctor s defed smlarly.
23 Derved Lmts QuasAbela Categores 21 Remark f = d the the fuctors wll be deoted respectvely by α,, ad :, :, α α : ad :. Defto Let C be a arbtrary category ad let : be a fuctor betwee two small categores. We defe the fuctor α : C C by settg ( C) = C for ay object C of C. f f : C C s a morphsm of C, we defe by settg for ay j. ( f) : ( C) ( C ) ( f)( j) = f( ( j)) Defto Let C be a cocomplete category ad let : be a fuctor betwee two small categores. We defe the fuctor : C C the followg way. At the level of objects, for ay fuctor :, we defe the fuctor by settg ( f,j) ( G) :, ( G)( ) = lm ( G )( f,j) = lm G( j) ( f,j ) α C ( f,j) ( f,j) G ( f,j) for ay. Let be a object of. For ay object ( f,j) of, deote by r : G( j) ( G)( ) ( G)( α) : lm G( j) lm G( j ) C the caocal morphsm. The, f : s a morphsm of, we defe
24 22 Fabee Prosmas by settg ( )( ) ( f,j ) = ( f α,j ) G α r r for ay object ( f,j) of. At the level of morphsms, for ay morphsm g : G G of C, we defe ( g) : ( G) ( G) by settg ( f,j ) ( f,j ) ( g)( ) r = r g( j) for ay object of ad ay object ( f,j) of. e ca check easly that we have: Prosto Let C be a cocomplete category ad let : beafuctor betwee small categores. For ay object C of C ad ay object G of C, we have Hom ( ( G ), C) Hom ( G, ( C )). C C 3 Dervato of the projectve lmt fuctor 3.1 The case where has eough jectve objects Prosto Let be a small category ad let be a quasabela category wth products. f has eough jectve objects, the the fuctor s rght dervable. lm : Proof. Ths follows drectly from Prosto Dually, we have: Prosto Let be a small category ad let be a quasabela category wth croducts. f has eough projectve objects, the the fuctor s left dervable. lm :
25 Derved Lmts QuasAbela Categores Roos complexes ths secto, wll deote a small category ad a quasabela category wth products. Defto We defe the fuctor the followg way. At the level of objects, for ay fuctor :, we defe by settg ad where R (,) = ( ) 0, R (,) α1 0 0 α1 α 0 α1 α p α1 α 1 d ( ) 1 α2 α R, α p 1 0 = ( ) l ( 1) p α1 1 α 1 l=1 1 R (,) : C ( ) R (,) C ( ) R (,)=0 < α l α l 0 l 1 l1 1 α1 0 At the level of morphsms, for ay morphsm : of, we defe α s a cha of morphsms of. Deotg by p : R (,) ( 0) the caocal projecto, we defe the dfferetal by settg d : R (,) R (,) ( 1) p α. f R (,f) : R (,) R (, ) by settg α1 α 0 α1 α 0 ( ) = ( ) 0 p R,f f p.
26 24 Fabee Prosmas Notato Let be a object of. For ay, we deote by q :lm () () the caocal morphsm. Prosto For ay object of, there s a caocal somorphsm whch duces a caocal morphsm 0 0 ( ) : R lm ( ) ker (, ) ɛ, d ɛ (,) : lm ( ) R (, ). Proof. We defe the morphsm 0 0 ɛ (,) : lm ( ) R (,) = ( ) by settg 0 p ɛ (,) = q for ay. Sce d ɛ (,) = 0, ɛ (,) duces a morphsm R (, ) ɛ (,) : lm ( ) R (,) of ( ). t follows drectly from the deftos that (lm ( ) ( )) s a kerel 0 C,ɛ, 0 dr (, ) of. Defto Let : beafuctor betwee small categores. We defe the morphsm of fuctors by settg R (, ) : R (,) R (, ()) β1 β ( β 1 ) ( β) j0 j ( j0) ( j) p R (,) = p 0 for ay object of.
27 Derved Lmts QuasAbela Categores 25 Defto We defe the fuctor R (): C ( ) the followg way. At the level of objects, for ay fuctor :, we defe by settg α α R ( ) R ( ) ()= R (, ( )) At the level of morphsms, f : s a morphsm of fuctors, we defe 0 0 t follows that for ay 0 ad ay R ( )( ) = ( ) 0 R( ) C ( ) α1 α1 α α α R ( )( ) = R (, ( )) for ay. f α : s a morphsm of, we defe by settg R( )( α) = R (, ( )) f α1 α f 0 0 ad usg the fact that = ( ) =. For ay 0, we defe the dfferetal d 1 : R ( ) R ( ) by settg for ay object of. by settg R ( )( α) : R (, ( )) R (, ( )) d d f R( f) : R( ) R( ) R ( f)( ) = R (, ( f)) for ay object of. Remark Let us otce that to gve a cha of morphsms (,f) (,f) of s equvalet to gve a cha of morphsms of of the form.
28 26 Fabee Prosmas ad that for ay morphsm α : of R ( )( α) : ( ) ( ) s defed by 0 0 α1 α f α1 α f 0 0 For ay α1 α f ( )( ) = α1 α α f 0 0 p R α p. 0, the dfferetal R ( ) 1 d : R ( ) R ( ) s gve by p α1 α 1 f dr ( ) α1 p α2 α 1 f 0 1 ()= ( ) 1 1 l ( 1) p α1 α l 1 α l α 1 f 0 l 1 l1 1 l=1 1 ( 1) p α1 α f α 1 for ay. Fally, for ay morphsm f : of s gve by for ay 0 ad ay. R( f) : R( ) R( ) α1 α g 0 α1 α g 0 ( )( ) = ( ) 0 p R f f p 0 Lemma f the category has a termal object, the for ay object of, there s a caocal homoty equvalece ( ) R (,) Proof. For ay 0, defe 1 h : R (,) R (,) by settg h 0 = 0 ad α1 α 1 α1 α α 1 0, p h = ( 1) p 1
29 Derved Lmts QuasAbela Categores 27 where α, 1 s the uque morphsm of Hom ( 1, ). Defe the caocal morphsm f : ( ) R (,) by settg k 0 0 0, p f = ( α ) Sce dr (, ) f = 0, f duces a caocal morphsm We wll also cosder the morphsm defed by settg Clearly, g f : ( ) R (, ). g : R (,) ( ) { p f k = 0 = 0 f k 1. g f =d. Moreover, f g s homotc to the detty map sce we have R (, ) =d h d f g ad 1 1 R (,) R (, ) = d R (, ) d h h d. Prosto For ay object of, there s a somorphsm ɛ( ) : R ( ) ( ). D Proof. Let be a object of. Sce (, d ) s a termal object of, by Lemma f ( ) : ( )(, d ) = ( ) R(, ( )) = R( )( ) K ( ), where 0 0 f (): () R ( )()= ( ) α 0 0
30 28 Fabee Prosmas s defed by So, for ay, we defe α 0 0 p f ()= ( α ). 0 0 ɛ ( )( ) : ( ) R ( )( ) by settg 0 0 ɛ ( )( ) = f ( ) Sce dr ( ) ɛ ( ) = 0, ɛ ( ) duces a morphsm ɛ( ) : R( ) of ( ). By costructo, for ay, we have the somorphsm C ɛ( )( ) : ( ) R( )( ) D ( ). t follows that ɛ( ) : R ( ) ( ). D Lemma For ay object of ad ay 0, there s a object S ( ) b( ) of such that R ( ) Π( S ( )). Proof. e checks easly that the fuctor S ( ) : b( ) defed by for ay solves the problem. S ( )( ) = ( ) α1 α 0 0 Prosto For ay object of, there s a somorphsm lm R( )( ) R (,) ( ) of C.
31 Derved Lmts QuasAbela Categores 29 Proof. Ths follows from the cha of somorphsms lm R ( )( ) lm Π( S ( ))( ) S ( )( ) α1 0 α1 0 R (, ). α α ( ) ( ) The case where has exact products ths secto, wll deote a small category ad a quasabela category wth products. Defto A object of s a Roosacyclc projectve system f the coaugmeted complex j k lm ( ) R (,) R (,) s strctly exact. other words, s Roosacyclc f ad oly f LH ( R (, )) = 0 for ay k> 0. b( ) Prosto For ay object Sof, there s a caocal homoty equvalece Sj () R (, Π( S )). partcular, Π( S) s a Roosacyclc projectve system. Proof. For ay 0, defe 1 h : R (, Π( S)) R (, Π( S)) by settg h 0 = 0 ad β α1 α 1 d β α1 α 1 j 0 0 = 1 j j j 0 1 p p h p p
32 30 Fabee Prosmas for 1. Defe the caocal morphsm 0 0 u : S( j) R (, Π( S)) by settg j p p u = p. β 0 j Sce dr (, Π( S )) u = 0, u duces a caocal morphsm u : S( j) R (, Π( S )). j We wll also cosder the morphsm 0 0 v : R (, Π( S)) S( j) 0 j j defed by settg ad the duced morphsm Clearly, 0 pj v = p p d j j v : R (, Π( S)) S( j ). v u =d. j j Moreover, u v s homotc to the detty map sce we have R (, Π( S )) =d h d u v ad for R (, Π( S)) R (, Π( S)) = dr (, Π( S)) d h h d Prosto Assume has exact products. The, the fuctor s rght dervable. lm :
33 Derved Lmts QuasAbela Categores 31 Proof. t s suffcet to show that the famly s lmjectve. F = { b( ) : s Roosacyclc} () Let be a object of. By Prosto 2.3.2, there s a strct moomorphsm Π(( )) ad by the precedg prosto, Π(( )) belogs to F. () Cosder a strctly exact sequece 0 0 of where ad belog to F. Sce has exact products, the sequece 0 R (, ) R (,) R (, ) 0 s strctly exact ad gves rse to the log exact sequece LH R, LH R, LH R, 0 ( ( )) ( ( )) ( ( )) D BC LH ( R (, )) LH ( R (, )) LH ( R (, )) D BC LH ( R (, )) LH ( R (, )) LH ( R (, )) Sce ad are objects of, t follows that ad that belogs to. k k k F LH ( R (, )) = LH ( R (, )) = 0 k > 0. 0 LH ( R (, ))=0 k > 0 F Moreover, by Prosto 3.2.3, for ay object of, LH ( R (, )) lm ( ) ad the precedg log exact sequece shows that the sequece s exact. 0 lm ( ) lm ( ) lm ( ) 0
34 o o 32 Fabee Prosmas Prosto Assume has exact products. The, for ay object of, we have a caocal somorphsm R lm ( ) R (,. ) Proof. By Prosto ad Lemma 3.2.9, R( ) s a Roosacyclc resoluto of. The, R lm ( ) lm R( )( ) R (,) where the last somorphsm follows from Prosto Derved projectve lmt fuctor ad dex restrcto ths secto, wll deote a quasabela category wth products ad : a fuctor betwee small categores. Defto We defe the morphsm of fuctors by settg ρ(,. ): ( R()) R( ()) β1 β g ( β 1 ) ( β) ( g) j0 j ( )( ) = j ( j0) ( j) ( j) p ρ, j p for ay object of, ay j ad ay 0. Prosto The caocal morphsm of fuctors characterzed by the fact that q j () j for ay object of ad ay duces a caocal morphsm () : lm lm j q q ( ) = q j j R lm R lm. Moreover, f has exact products, the dagram R lm ( ) R (,) R (,) j s commutatve for ay object of. R lm ( )( j) R (, ( ))
35 o Derved Lmts QuasAbela Categores 33 Proof. By a wellkow procedure of homologcal algebra, the caocal morphsm q( ) : lm lm j duces a caocal morphsm Sce the fuctor j j R lm R lm R. s exact, we get the caocal morphsm R lm R lm. Assumg has exact products, we may vsualze the costructo of ths morphsm the followg way. Cosder a object of. We kow that s a lmacyclc resoluto of. The fuctor beg exact, we have the somor phsm D ( ). Sce s a lmacyclc resoluto of ( ), j ɛ( ) : R ( ) ( ɛ( )) : ( ) ( R( )) ɛ( ( )) : ( ) R( ( )) 1 ɛ( ( )) ( ( ɛ( ))) : ( R( )) R ( ( )) s a lmacyclc resoluto of ( ( )). Moreover, the dagram j R ( ɛ ( )) ρ (,) ( R( )) R ( ( )) k 5 k k k k k 5u k kk k k k k k k ɛ ( ( )) k ( ) of D ( ) beg commutatve, we have 1 ɛ( ( )) ( ( ɛ( ))) = ρ(, ). Hece, the caocal morphsm R lm ( ) R lm ( )( j) j
36 34 Fabee Prosmas s gve by the commutatve dagram j R lm ( ) R lm ( )( ) j q ( R ( )) j lm R( )( ) lm ( R ( ))( j) lm R( ( ))( j) lm ρ (,)( j) j j Sce a drect computato shows that the dagram q ( R ( )) lm ρ (,)( j) j lm R( )( ) lm ( R ( ))( j) lm R( ( ))( j) j j R (,) R (,) R (, ( )) s commutatve, the cocluso follows. 3.5 Dual results for the ductve lmt fuctor ths secto wll deote a small category ad a quasabela category wth croducts. By dualty, the results ad costructos the precedg sectos ca be easly adapted to derve the fuctor lm :. We wll ot do ths explctly here. However, the rest of ths paper, we wll eed to work wth the derved fuctor of lm :. To avod cofusos, we wll fx below the otatos used ths case. The fuctor R (,) : C ( ) s defed by ad the dfferetal s gve by R (,)=( R (, )) R (,) 1 =( R (,) ) d d.
37 Derved Lmts QuasAbela Categores 35 f f : s a morphsm of, the R (,) 0 b( ) R (,f)=( R (,f )). As Prosto 3.2.3, there s a caocal somorphsm The fuctor s defed by ad the dfferetal s gve by ɛ 0(,) : coker d lm ( ). R ( ) 1 =( R ( ) ) R(): C ( ) R( ) = ( R ( )) d d. f f : s a morphsm of, the R( f) = ( R ( f )). As Prosto 3.2.8, there s a caocal somorphsm ɛ( ) : R( ) D ( ). For ay 0, there s a object S ( ) of such that R ( ) ( S ( )). Moreover, there s a caocal somorphsm R (,) lm R ( )( ) C ( ). Therefore, as Prostos ad 3.3.4, f fuctor lm : s left dervable ad we have L lm ( ) R (,. ) has exact croducts, the
38 36 Fabee Prosmas f : s a fuctor betwee small categores, the we defe the morphsm of fuctors ρ(, ): R( ()) ( R()) of as the mage of ρ(,) by the adjucto by settg ρ (,) = ( ρ(, )). For ay 0, we defe the morphsm ρ (,) : ( R ( ( ))) R ( ) Hom ( R ( ( )), ( R ( ))) Hom ( ( R ( ( ))),R ( )) The morphsms ρ (,) duce a morphsm of fuctors ρ (, ) : ( R( ())) R (). 3.6 Relatos betwee RHom ad derved lmts Defto Let be a small category ad let be a quasabela category. Cosder a object Xof. We defe the fuctor the followg way: at the level of objects, f s a object of, we set Hom ( X, ) : Ab Hom ( X, Y )( ) = Hom ( X, Y ( )) for ay object of. f α : j s a morphsm of, Y s defed by Hom ( X, Y )( α) : Hom ( X, Y ( )) Hom ( X, Y ( j)) for ay f Hom ( X,Y( )). Hom ( X, Y )( α)( f) = Hom ( X, Y ( α))( f) = Y ( α) f At the level of morphsms, f : s a morphsm of, F Y Y Hom ( X, F ) : Hom ( X, Y ) Hom ( X, Y ) s defed by Hom ( X, F )( ) = Hom ( X, F ( )) for ay. Dually, we defe also the fuctor Hom (,X) : ( ) Ab.
39 Derved Lmts QuasAbela Categores 37 Lemma Let be a small category ad let be a quasabela category. () For ay object X of ad ay object of, we have Hom ( X, R (, )) R (, Hom ( X, )). () For ay object of ad ay object of, we have X Hom ( R (, ),X) R (, Hom (,X )). Proof. Ths follows drectly from the deftos. Prosto Let be a small category ad let be a quasabela category X wth eough jectve objects. For ay object of ad ay object of, we have ad RHom (L lm,x ( ) ) R lm(rhom (,X))( ) 0 1 RHom ( X, R lm ( )) R lm(rhom ( X, ))( ). Proof. Frst, recall that sce has eough jectve objects, croducts are exact. Hece, the ductve lmt fuctor s left dervable. Let 0 be a jectve resoluto of X. oe had, RHom (L lm ( ),X) s gve by the smple complex assocated to Hom ( R (, ), ) Hom ( R (, ), ) Ths complex s somorphc to the smple complex assocated to R (, Hom (, )) R (, Hom (, )) the other had, RHom (,X) s gve by the complex Hom (, ) Hom (, ) Therefore, R lm(rhom (,X))( ) s somorphc to the smple complex assocated to R lm Hom (, )( ) R lm Hom (, )( )
40 38 Fabee Prosmas Sce for ay l l b( ) of such that for l 0, S s a jectve object of. oe had, sce lm Hom ( X, Π( S ))( ) Hom ( X, lm Π( S )( )) l Hom ( X, S ( )) Hom ( X, S ( )) Hom ( X, S ( )) Ths complex s somorphc to the complex S ( ) S ( ) l R lm Hom (, )( ) R (, Hom (, )), the frst somorphsm s establshed. Next, we kow that has a jectve resoluto of the form l 0 Π( S ) Π( S ) Hom ( X, Π( S )) Π Hom ( X, S ) ad sce projectve systems of product type are lmacyclc, by composto of the derved fuctors, we have R lm(rhom ( X, ))( ) R(lm Hom ( X, )( )). Hece, R lm(rhom ( X, ))( ) s somorphc to the complex lm Hom ( X, Π( S ))( ) lm Hom ( X, Π( S ))( ) Moreover, for ay l 0, we get l 0 1 Hece, R lm(rhom ( X, ))( ) s gve by the complex the other had, R lm ( ) s gve by the complex 0 lm Π( S )( ) lm Π( S )( ) l l l
41 Derved Lmts QuasAbela Categores 39 Sce the product of jectve objects s a jectve object, the last complex s a jectve resoluto of R lm ( ). Therefore, RHom ( X, R lm ( )) s somorphc to the complex ad the cocluso follows Hom ( X, S ( )) Hom ( X, S ( )) D R, Prosto Let be a small category ad let be a quasabela category wth exact products. Cosder a object of. f for ay object X of, Hom ( X, ) s lmacyclc, the s lmacyclc. Proof. Cosder X b( ). We kow that R lm Hom ( X, )( ) R (, Hom ( X, )) Hom ( X, R (, )). Sce Hom ( X, ) s lmacyclc, we have Hom ( X, R (, )) lm Hom ( X, )( ) Hom ( X, lm ( )). Therefore, the complex Hom ( X, R (, )) s exact degree k = 0 for ay X b( ). Hece, Remark shows that R (,) s strctly exact degree k = 0. t follows that R (,) lm ( ) ( ). Sce R lm ( ) ( ), we get R lm ( ) lm. ( ) 4 Derved lmts ad the symbolchom fuctor 4.1 The symbolchom fuctor ths secto, wll deote a small category ad a complete addtve category.
42 40 Fabee Prosmas Defto We deote by b A the symbolchom fuctor. For ay object M of Ab ad ay object of, the object [ M, ] of s characterzed by Ab [, ] : ( A b ) X Hom ( X, [ M, ] ) Hom ( M, h ) X b( ). f s the oe pot category, the [, ] wll be deoted by [, ] : A b. ths case, for ay abela group M ad ay object of, the object [ M,] of s characterzed by Hom ( X, [ M, ]) Hom ( M, Hom ( X, )) X b( ). Let us recall the followg easy formulas: Prosto () For ay object of, we have [ Z,]. () For ay object M of Ab ad ay object of, we have [lm M, ( ) ] lm[ M,. ( ) ] () For ay abela group M ad ay object of, we have [ M, lm ( )] lm[ M, ( )]. 4.2 Dervato of the symbolchom fuctor ths secto, wll deote a small category. Lemma Let be a quasabela category wth exact products. f P s a projectve abela group, the the fuctor [ P, ] : s exact.
43 Derved Lmts QuasAbela Categores 41 Proof. Frst, let us prove the result whe P s a free abela group,.e., ( ) P = Z = Z. Cosder a strctly exact sequece of. Sce for ay object of, ( ) [ Z,] [ Z,] 0 [ Z, ] [ Z,] [ Z, ] 0 ( ) [ Z, ] [(ker q) P, ] [ker q, ] [ P, ]. ( ) The fuctor [ Z,] beg exact, the fuctor [ P, ] s also exact. Lemma Let be a complete addtve category. b( ) () For ay object S of Ab ad ay object of,we have [ ( S, ) ] [ S, ( )] [ S, () ()]. ( ) ( ) ( ) b( ) b( ) () For ay object M of Ab ad ay object S of, we have [ M, Π( S)] [( M ),S] [ M( ),S( )]. 0 0 ad sce products are exact, the sequece s exact. Next, cosder a projectve abela group P. We kow that there s a exact sequece ( ) q 0 ker q Z P 0. Sce P s projectve, ths sequece splts ad we have b( ) Proof. () Let X be a object of. Frst, we have Hom ( X, [ ( S ), ] ) Hom Ab ( ( S ), hx ) Hom Abb( ) ( S, ( hx )) (*) Hom Abb( ) ( S, hx ( )) Hom ( X, [ S, ( )] ), b( )
44 42 Fabee Prosmas where the somorphsm (*) follows from Prosto Next, we get Hom ( X, [ S, ( )] b( ) ) Hom Abb( ) ( S, hx ( )) Hom Ab( S, ( ) Hom ( X, ( ))) Hom ( X, [ S( ), ( )]) Hom ( X, [ S( ), ( )]). () Let X be a object of. oe had, we have successvely the other had, we get Hom ( X, [ M, Π( S)] ) Hom Ab ( M, hx Π( S)) Hom b A ( M, Π( hx S)) Hom Abb( ) (( M,h ) X S) Hom ( X, [( M ), S ] ). b( ) Hom ( X, [( M ), S] b( ) ) Hom Abb( ) (( M ), hx S) Hom ( M, ( ) Hom ( X,S ( ))) Ab Hom ( X, [ M( ), S( )]) Hom ( X, [ M( ), S( )]). Lemma Let be a quasabela category wth exact products. f P s a projectve object of Ab, the the fuctor [ P, ] : s exact. Proof. Frst, let us prove the result whe P = ( S) where S s a projectve object b( ) of Ab. Cosder a strctly exact sequece 0 0
45 Derved Lmts QuasAbela Categores 43 of. Sce for ay, S( ) s projectve, by Lemma 4.2.1, the fuctor s exact. Moreover, sce for ay object of, [ ( S, ) ] [ S, ( ) ( )] b( ) S b P Prosto Let be a quasabela category wth exact products. The fuctor [, ] : ( A b ) q [ S, ( ) ] : ad sce products are exact, the sequece 0 [ ( S, ) ] [ ( S, ) ] [ ( S, ) ] 0 s exact. Next, cosder a projectve object P of Ab. We kow that there s a epmorphsm q : ( S) P where s a projectve object of A. Sce s projectve, the exact sequece splts ad we have 0 ker q ( S) P 0 [ ( S, ) ] [(ker q) P, ] [ker q, ] [ P, ]. The fuctor [ ( S, ) ] beg exact, the fuctor [ P, ] s also exact. has a rght derved fuctor R[, ] :( D ( A b )) D ( ) D ( ). Proof. Let us show that f P s the full subcategory of projectve objects of Ab, the the par ( P, ) s [,] acyclc. Frst, cosder a object P of P. Sce the fuctor [ P, ] : s exact, the category s [ P, ] jectve. Next, cosder a object of b( ) ad let us show that P s [,]  projectve.
46 44 Fabee Prosmas () For ay object M of Ab, there s a projectve object P of Ab ad a epmorphsm P M. () f 0 P P P 0 s a exact sequece of Ab, where P ad P are two objects of P, the we kow that P s a object of P. () f 0 P P P 0 s a exact sequece of P, the t splts. t follows that the sequece 0 [ P,] [ P,] [ P,] 0 of splts ad that t s exact. Sce the fuctor [, ] s left exact, the cocluso follows from Prosto Lks wth the derved projectve lmt fuctor Lemma Let be a small category ad let be a complete addtve category. f Z : Ab s the costat fuctor whch assocates to ay the abela group Z, the for ay object of, we have Proof. For ay object X of, we have [ Z,] lm ( ). Hom Z b ( X, [, ] ) Hom A ( Z,hX ) lm( hx )( ) lm Hom ( X, ( )) Hom ( X, lm ( )). Prosto Let be a small category ad let be a quasabela category M b S b( ) wth exact products. For ay object of A ad ay object of, we have R [ M, Π( S)] R [( M ), S ]. b( )
47 Derved Lmts QuasAbela Categores 45 Proof. We kow that there s a projectve resoluto P of M Ab such that for ay 0 P = ( S ) b( ) where S s a projectve object of Ab. The, we have R [ M, Π( S)] [ P, Π( S)] [( P ), S] where the secod somorphsm follows from Lemma Sce for ay 0 ad ay, we have ( P )( ) = P ( ) = ( S )( ) = S ( j ), b( ) ( P ) s projectve Ab. The fuctor b( ) α j : b( ) beg exact, ( P) s a projectve resoluto of ( M). t follows that R [( M,S ) ] [( P ),S] b( ) b( ) R[ M, Π( S )]. Corollary Let be a small category ad let be a quasabela category b( ) wth exact products. For ay object S of, we have R [ Z, Π( S )] [ Z, Π( S )]. Proof. b( ) The object ( Z ) of Ab s projectve sce for ay, ( Z )( ) = Z ( ) = Z. t follows that R [ Z, Π( S )] R [( Z ),S] b( ) [( Z ),S] [ Z, Π( S)] b( ) where the last somorphsm follows from Lemma
48 46 Fabee Prosmas Prosto Let be a small category ad let be a quasabela category wth exact products. The, the fuctor lm : s rght dervable ad for ay object of, we have Proof. R lm ( ) R [ Z, ]. Frst, let us remark that, sce the fuctor s left exact, we have for a object of, f ad oly f Next, let us show that the famly s lmjectve. k [, ] : ( A b ) R[ Z,] [ Z,] LH R[ Z, ] 0 k > 0. F = { b( ) : R [ Z,] [ Z,] } () Let be a object of. By Prosto 2.3.2, there s a strct moomorphsm Π(( )) ad by Corollary 4.3.3, Π(( )) belogs to F. () Cosder a strctly exact sequece 0 0 of where ad belog to F. Ths sequece gves rse to the log exact sequece Z, Z, Z, 0 [ ] [ ] [ ] D BC Z Z Z LH R[, ] LH R[, ] LH R[, ] Z Z Z LH R[, ] LH R[, ] LH R[, ] D BC
49 Derved Lmts QuasAbela Categores 47 of C ( ). Sce ad are objects of F, k k LH R[ Z, ] = LH R[ Z, ] =0 k > 0 ad t follows that k LH R[ Z, ] =0 k > 0. Hece, belogs to F. Moreover, by Lemma 4.3.1, for ay object of, we have [ Z,] lm ( ), so the precedg log exact sequece shows that the sequece 0 lm ( ) lm ( ) lm ( ) 0 s exact. Cosequetly, Fs lmjectve ad the fuctor lm : s rght dervable. t follows from Prosto that ay object of has a resoluto by projectve systems of product type. Assume that for ay, s of the form Π( S ). The, for ay 0, we have [ Z, Π( S )] R[ Z, ] R[ Z, Π( S )] [ Z, ]. t follows that Therefore, we get R[ Z, ] [ Z, ]. R lm ( ) lm ( ) [ Z, ] R[ Z, ] R[ Z,] where the frst somorphsm follows from the frst part of the proof ad the secod from Lemma
50 o o 48 Fabee Prosmas Prosto Let be a small category ad let be a quasabela category wth exact products. The, for ay object of, we have a caocal somorphsm R (,) [ R ( Z ),] C ( ) makg the dagram R lm ( ) R [ Z,] R (,) [ R ( Z ),] commutatve. Proof. For ay object of Ab ad ay 0, we have the cha of somorphsms Hom ( R ( Z ),) Hom ( ( S ( Z )),) Ab Ab Hom Abb( ) ( S ( Z), ( )) Hom ( S ( Z )( ), ( )( )) 0 0 α1 0 α1 0 Ab α Ab α Hom ( Z ( ),( )) R (, ). α ( ) α Hom ( Z,( )) Ab 0 0 A drect computato shows that these somorphsms are compatble wth the dfferetals. Hece, we have Hom ( R ( Z ),) R (, ).
51 Derved Lmts QuasAbela Categores 49 The, for ay object of ad ay object X of, we have successvely Hom ( X, [ R ( Z ), ] ) Hom ( R ( Z ), h ) t follows that ad that Sce Ab X R (,hx ) Hom ( X, ( 0)) α1 α 0 Hom ( X, ( 0)) α1 α 0 Hom ( X, R (, )). [ R ( Z ),] R (,) [ R ( Z ),] R (, ). ad S( Z)( 0) Z( ), α1 α 0 t follows from the dual of Prosto that R( Z) s a projectve object of Ab. Together wth the fact that R ( Z) s a resoluto of Z, ths explas the secod vertcal somorphsm of the dagram our statemet. The commutatvty of the dagram follows drectly from the costructo of the varous morphsms. 4.4 dex restrcto R ( Z ) ( S ( Z )) Prosto Let be a complete addtve category ad let : be a fuctor betwee small categores. For ay object M of Ab ad ay object of,wehave [ ( M ),] [ M, ( )]. Proof. Ths somorphsm follows from Prosto As a matter of fact, for ay object X of, we have successvely Hom ( X, [ ( M ),] ) Hom Ab ( ( M ),hx ) Hom Ab ( M, ( hx )) Hom Ab ( M, hx ( )) Hom ( X, [ M, ( )] ).
52 50 Fabee Prosmas Lemma Let be a cocomplete category ad let : be a fuctor betwee two small categores. f P s a projectve object of, the ( P) s a projectve object of. Proof. Cosder a strctly exact sequece of. Sce s exact ad sce s projectve, the sequece Prosto Let be a quasabela category wth exact products ad let : be a fuctor betwee two small categores. For ay object M of Ab ad ay object of, we have [ ] R[ L ( M ), ] R M, ( ). P Hom ( P, ( )) Hom ( P, ( )) Hom ( P, ( )) s exact. The cocluso follows from Prosto Proof. f P s a projectve resoluto of M, the we have successvely [ ] R M, ( ) [ P, ( )] [ ( P ),] R[ ( P ),] R[ L ( M ), ]. Prosto Let be a quasabela category wth exact products ad let : be a fuctor betwee two small categores. The caocal somorphsm commutatve D ( Ab ). L d : Z ( Z ) = Z w : L ( Z ) Z ( Z ) duces by adjucto a caocal morphsm whch makes the dagram w Z Z ρ (, ) Z Z Z ɛ ( ) ( R( ( ))) R( )
53 # # o o Derved Lmts QuasAbela Categores 51 Proof. Recall that w s the composto of ad the morphsm Z Z Z Z Z ( ɛ ( )) u : L ( Z ) ( Z ), v : ( Z ) Z defed as the mage of d : Z ( Z ) = Z by the adjucto Sce R( Z ) s a projectve resoluto of Z, we have D ( Ab ). Cosder the dagram ( R( ( Z ))) ( R( Z )) L ( Z ) Z ρ (, ) Z Z ( Z ) Hom (, ( )) Hom ( ( ), ). ( R( )) R( ) L w Z Z ɛ ( ) u Z [ ] R Z, ( ) R lm ( )( j) Z Z ( ɛ ( )) ɛ ( ) ρ (, ) Z Z s commutatve C ( Ab ) ad the cocluso follows. Prosto Let be a quasabela category wth exact products ad let : be a fuctor betwee two small categores. The, for ay object of, the caocal dagram R [ Z,] R lm ( ) R[ w,] ( Z ) R[ L ( Z ), ] ( Z ) t follows from the costructo of w that the lower square s commutatve. Sce a drect computato shows that the dagram v v Z j Z ( R( )) R( )
54 z o :z d$ $ o o o o o o { ;{ c# # 52 Fabee Prosmas s commutatve. Proof. Cosder the followg dagram: R [ Z,] R lm ( ) (3) (2) [ R ( Z ),] R (,) w ww w ww w ww w ww w ww w ww R [ L ( Z ),] [ ( R ( Z )),] (1) (6) u uu u uu (4) u uu u uu [ R ( Z ), ( )] R (, ( )) u uu u [ uu G G ] R Z, ( ) R lm ( )( j) Lemma Let be a quasabela category wth exact products ad let : be a fuctor betwee two small categores. For ay object of, the caocal dagram s commutatve D ( ). u uu u uu R [ L ( Z ),] [ ( R ( Z )),] (5) Z R[ w,] [ ρ (, ),] R[ Z,] [ R ( Z ),] G G G G G G G G G G G G G G G G G Clearly, the result wll be establshed f we prove that the subdagrams (1) (6) commute. We kow already from Prosto that dagram (6) commutes. Moreover, Prosto shows that dagrams (2) ad (5) are also commutatve. Sce the commutatvty of (1), (3) ad (4) follows from the lemmas below, the proof s complete. G G j Proof. Sce R( Z ) s a projectve resoluto of Z, we have R[ Z,] [ R ( Z ),].
55 o o o o Derved Lmts QuasAbela Categores 53 Moreover, usg Lemma 4.4.2, we see that R[ L ( Z ),] R[ ( R ( Z )),] [ ( R ( Z )),]. Ths explas the horzotal somorphsm. Now, cosder the dagram R[ L ( ),] R[ ( R ( )),] [ ( R ( )),] Z oo Z Z [ ] R Z, ( ) [ R ( Z ), ( )] Z R[ w,] (1) R[ ρ (, ),] (2) [ ρ (, ),] R[ Z,] oo R[ R ( Z ),] [ R ( Z ),] By Prosto 4.4.4, the square (1) s commutatve. The square (2) beg clearly commutatve, the cocluso follows. Lemma Let be a quasabela category wth exact products ad let : be a fuctor betwee two small categores. For ay object of, the caocal dagram Z s commutatve D ( ). R [ L ( Z ),] [ ( R ( Z )),] Proof. Ths follows drectly from the costructo the proof of Prosto f oe keeps md that R( Z ) s a projectve resoluto of Z. Lemma Let be a quasabela category wth exact products ad let : be a fuctor betwee two small categores. For ay object of, the dagram [ R ( Z ),] R (,) Z [ ρ (, ),] [ ( R ( Z )),] R (,) s commutatve C ( ). [ R ( Z ), ( )] R (, ( ))
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