Derived Limits in QuasiAbelian Categories


 Janel Owens
 1 years ago
 Views:
Transcription
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)
2
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 (, ( ))
CHAPTER 4 RADICAL EXPRESSIONS
6 CHAPTER RADICAL EXPRESSIONS. The th Root of a Real Number A real umber a s called the th root of a real umber b f Thus, for example: s a square root of sce. s also a square root of sce ( ). s a cube
More informationPROJECTION PROBLEM FOR REGULAR POLYGONS
Joural of Mathematcal Sceces: Advaces ad Applcatos Volume, Number, 008, Pages 9550 PROJECTION PROBLEM FOR REGULAR POLYGONS College of Scece Bejg Forestry Uversty Bejg 0008 P. R. Cha emal: sl@bjfu.edu.c
More informationMATH 247/Winter Notes on the adjoint and on normal operators.
MATH 47/Wter 00 Notes o the adjot ad o ormal operators I these otes, V s a fte dmesoal er product space over, wth gve er * product uv, T, S, T, are lear operators o V U, W are subspaces of V Whe we say
More informationarxiv:math/ v1 [math.rt] 17 Jul 2006
ON THE STRUCTURE OF CALABIYAU CATEGORIES WITH A CLUSTER TILTING SUBCATEGORY arxv:math/0607394v1 [math.rt] 17 Jul 2006 GONÇALO TABUADA Abstract. We prove that for d 2, a algebrac dcalabyau tragulated
More information1 Onto functions and bijections Applications to Counting
1 Oto fuctos ad bectos Applcatos to Coutg Now we move o to a ew topc. Defto 1.1 (Surecto. A fucto f : A B s sad to be surectve or oto f for each b B there s some a A so that f(a B. What are examples of
More informationUnit 9. The Tangent Bundle
Ut 9. The Taget Budle ==========================================================================================  The taget sace of a submafold of R, detfcato of taget vectors wth dervatos at
More informationarxiv: v2 [math.ag] 9 Jun 2015
THE EULER CHARATERISTIC OF THE GENERALIZED KUMMER SCHEME OF AN ABELIAN THREEFOLD Mart G. Gulbradse Adrea T. Rcolf arxv:1506.01229v2 [math.ag] 9 Ju 2015 Abstract Let X be a Abela threefold. We prove a formula,
More informationEntropy ISSN by MDPI
Etropy 2003, 5, 233238 Etropy ISSN 10994300 2003 by MDPI www.mdp.org/etropy O the Measure Etropy of Addtve Cellular Automata Hasa Aı Arts ad Sceces Faculty, Departmet of Mathematcs, Harra Uversty; 63100,
More informationOn the Primitive Classes of K * KHALED S. FELALI Department of Mathematical Sciences, Umm AlQura University, Makkah AlMukarramah, Saudi Arabia
JKAU: Sc., O vol. the Prmtve, pp. 5562 Classes (49 of A.H. K (BU) / 999 A.D.) * 55 O the Prmtve Classes of K * (BU) KHALED S. FELALI Departmet of Mathematcal Sceces, Umm AlQura Uversty, Makkah AlMukarramah,
More informationarxiv: v2 [math.rt] 28 Jan 2013
Fuso procedure for wreath products of fte groups by the symmetrc group L. Poula d Adecy arxv:30.4399v2 [math.rt] 28 Ja 203 Mathematcs Laboratory of Versalles (CNRS UMR 800, VersallesSatQuet Uversty,
More informationEconometric Methods. Review of Estimation
Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators
More informationDIFFERENTIAL GEOMETRIC APPROACH TO HAMILTONIAN MECHANICS
DIFFERENTIAL GEOMETRIC APPROACH TO HAMILTONIAN MECHANICS Course Project: Classcal Mechacs (PHY 40) Suja Dabholkar (Y430) Sul Yeshwath (Y444). Itroducto Hamltoa mechacs s geometry phase space. It deals
More informationDiscrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b
CS 70 Dscrete Mathematcs ad Probablty Theory Fall 206 Sesha ad Walrad DIS 0b. Wll I Get My Package? Seaky delvery guy of some compay s out delverg packages to customers. Not oly does he had a radom package
More informationArithmetic Mean and Geometric Mean
Acta Mathematca Ntresa Vol, No, p 43 48 ISSN 4536083 Arthmetc Mea ad Geometrc Mea Mare Varga a * Peter Mchalča b a Departmet of Mathematcs, Faculty of Natural Sceces, Costate the Phlosopher Uversty Ntra,
More informationON THE LOGARITHMIC INTEGRAL
Hacettepe Joural of Mathematcs ad Statstcs Volume 39(3) (21), 393 41 ON THE LOGARITHMIC INTEGRAL Bra Fsher ad Bljaa JolevskaTueska Receved 29:9 :29 : Accepted 2 :3 :21 Abstract The logarthmc tegral l(x)
More informationFunctions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More informationOn L Fuzzy Sets. T. Rama Rao, Ch. Prabhakara Rao, Dawit Solomon And Derso Abeje.
Iteratoal Joural of Fuzzy Mathematcs ad Systems. ISSN 22489940 Volume 3, Number 5 (2013), pp. 375379 Research Ida Publcatos http://www.rpublcato.com O L Fuzzy Sets T. Rama Rao, Ch. Prabhakara Rao, Dawt
More information2. Independence and Bernoulli Trials
. Ideedece ad Beroull Trals Ideedece: Evets ad B are deedet f B B.  It s easy to show that, B deedet mles, B;, B are all deedet ars. For examle, ad so that B or B B B B B φ,.e., ad B are deedet evets.,
More informationA Remark on the Uniform Convergence of Some Sequences of Functions
Advaces Pure Mathematcs 05 5 57533 Publshed Ole July 05 ScRes. http://www.scrp.org/joural/apm http://dx.do.org/0.436/apm.05.59048 A Remark o the Uform Covergece of Some Sequeces of Fuctos Guy Degla Isttut
More informationD KL (P Q) := p i ln p i q i
CheroffBouds 1 The Geeral Boud Let P 1,, m ) ad Q q 1,, q m ) be two dstrbutos o m elemets, e,, q 0, for 1,, m, ad m 1 m 1 q 1 The KullbackLebler dvergece or relatve etroy of P ad Q s defed as m D KL
More information1 Lyapunov Stability Theory
Lyapuov Stablty heory I ths secto we cosder proofs of stablty of equlbra of autoomous systems. hs s stadard theory for olear systems, ad oe of the most mportat tools the aalyss of olear systems. It may
More informationv 1 periodic 2exponents of SU(2 e ) and SU(2 e + 1)
Joural of Pure ad Appled Algebra 216 (2012) 1268 1272 Cotets lsts avalable at ScVerse SceceDrect Joural of Pure ad Appled Algebra joural homepage: www.elsever.com/locate/jpaa v 1 perodc 2expoets of SU(2
More informationAlgorithms Theory, Solution for Assignment 2
JuorProf. Dr. Robert Elsässer, Marco Muñz, Phllp Hedegger WS 2009/200 Algorthms Theory, Soluto for Assgmet 2 http://lak.formatk.ufreburg.de/lak_teachg/ws09_0/algo090.php Exercse 2.  Fast Fourer Trasform
More informationα1 α2 Simplex and Rectangle Elements Multiindex Notation of polynomials of degree Definition: The set P k will be the set of all functions:
Smplex ad Rectagle Elemets Multdex Notato = (,..., ), oegatve tegers = = β = ( β,..., β ) the + β = ( + β,..., + β ) + x = x x x x = x x β β + D = D = D D x x x β β Defto: The set P of polyomals of degree
More informationPTAS for BinPacking
CS 663: Patter Matchg Algorthms Scrbe: Che Jag /9/00. Itroducto PTAS for BPackg The BPackg problem s NPhard. If we use approxmato algorthms, the BPackg problem could be solved polyomal tme. For example,
More informationOrdinary Least Squares Regression. Simple Regression. Algebra and Assumptions.
Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos
More information1 Mixed Quantum State. 2 Density Matrix. CS Density Matrices, von Neumann Entropy 3/7/07 Spring 2007 Lecture 13. ψ = α x x. ρ = p i ψ i ψ i.
CS 94 Desty Matrces, vo Neuma Etropy 3/7/07 Sprg 007 Lecture 3 I ths lecture, we wll dscuss the bascs of quatum formato theory I partcular, we wll dscuss mxed quatum states, desty matrces, vo Neuma etropy
More informationC.11 Bangbang Control
Itroucto to Cotrol heory Iclug Optmal Cotrol Nguye a e .5 C. Bagbag Cotrol. Itroucto hs chapter eals wth the cotrol wth restrctos: s boue a mght well be possble to have scotutes. o llustrate some of
More informationQualifying Exam Statistical Theory Problem Solutions August 2005
Qualfyg Exam Statstcal Theory Problem Solutos August 5. Let X, X,..., X be d uform U(,),
More informationLecture 12 APPROXIMATION OF FIRST ORDER DERIVATIVES
FDM: Appromato of Frst Order Dervatves Lecture APPROXIMATION OF FIRST ORDER DERIVATIVES. INTRODUCTION Covectve term coservato equatos volve frst order dervatves. The smplest possble approach for dscretzato
More information4 Inner Product Spaces
11.MH1 LINEAR ALGEBRA Summary Notes 4 Ier Product Spaces Ier product s the abstracto to geeral vector spaces of the famlar dea of the scalar product of two vectors or 3. I what follows, keep these key
More informationA Characterization of Jacobson Radical in ΓBanach Algebras
Advaces Pure Matheatcs 4348 http://dxdoorg/436/ap66 Publshed Ole Noveber (http://wwwscrporg/joural/ap) A Characterzato of Jacobso Radcal ΓBaach Algebras Nlash Goswa Departet of Matheatcs Gauhat Uversty
More informationPGE 310: Formulation and Solution in Geosystems Engineering. Dr. Balhoff. Interpolation
PGE 30: Formulato ad Soluto Geosystems Egeerg Dr. Balhoff Iterpolato Numercal Methods wth MATLAB, Recktewald, Chapter 0 ad Numercal Methods for Egeers, Chapra ad Caale, 5 th Ed., Part Fve, Chapter 8 ad
More informationMatricial Potentiation
Matrcal Potetato By Ezo March* ad Mart Mates** Abstract I ths short ote we troduce the potetato of matrces of the same sze. We study some smple propertes ad some example. * Emertus Professor UNSL, Sa Lus
More informationChapter 4 (Part 1): NonParametric Classification (Sections ) Pattern Classification 4.3) Announcements
Aoucemets NoParametrc Desty Estmato Techques HW assged Most of ths lecture was o the blacboard. These sldes cover the same materal as preseted DHS Bometrcs CSE 90a Lecture 7 CSE90a Fall 06 CSE90a Fall
More informationLecture 02: Bounding tail distributions of a random variable
CSCIB609: A Theorst s Toolkt, Fall 206 Aug 25 Lecture 02: Boudg tal dstrbutos of a radom varable Lecturer: Yua Zhou Scrbe: Yua Xe & Yua Zhou Let us cosder the ubased co flps aga. I.e. let the outcome
More informationA Primer on Summation Notation George H Olson, Ph. D. Doctoral Program in Educational Leadership Appalachian State University Spring 2010
Summato Operator A Prmer o Summato otato George H Olso Ph D Doctoral Program Educatoal Leadershp Appalacha State Uversty Sprg 00 The summato operator ( ) {Greek letter captal sgma} s a structo to sum over
More informationComplex Numbers Primer
Complex Numbers Prmer Before I get started o ths let me frst make t clear that ths documet s ot teded to teach you everythg there s to kow about complex umbers. That s a subject that ca (ad does) take
More informationOn the construction of symmetric nonnegative matrix with prescribed Ritz values
Joural of Lear ad Topologcal Algebra Vol. 3, No., 14, 6166 O the costructo of symmetrc oegatve matrx wth prescrbed Rtz values A. M. Nazar a, E. Afshar b a Departmet of Mathematcs, Arak Uversty, P.O. Box
More informationPolyphase Filters. Section 12.4 Porat
Polyphase Flters Secto.4 Porat .4 Polyphase Flters Polyphase s a way of dog saplgrate coverso that leads to very effcet pleetatos. But ore tha that, t leads to very geeral vewpots that are useful buldg
More informationCOMPROMISE HYPERSPHERE FOR STOCHASTIC DOMINANCE MODEL
Sebasta Starz COMPROMISE HYPERSPHERE FOR STOCHASTIC DOMINANCE MODEL Abstract The am of the work s to preset a method of rakg a fte set of dscrete radom varables. The proposed method s based o two approaches:
More information( ) ( ) A number of the form x+iy, where x & y are integers and i = 1 is called a complex number.
A umber of the form y, where & y are tegers ad s called a comple umber. Dfferet Forms )Cartesa Form y )Polar Form ( cos s ) r or r cs )Epoetal Form r e Demover s Theorem If s ay teger the cos s cos s If
More informationLecture 3. Sampling, sampling distributions, and parameter estimation
Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called
More informationProbabilistic Meanings of Numerical Characteristics for Single Birth Processes
A^VÇÚO 32 ò 5 Ï 206 c 0 Chese Joural of Appled Probablty ad Statstcs Oct 206 Vol 32 No 5 pp 452462 do: 03969/jss00426820605002 Probablstc Meags of Numercal Characterstcs for Sgle Brth Processes LIAO
More informationOn generalized fuzzy mean code word lengths. Department of Mathematics, Jaypee University of Engineering and Technology, Guna, Madhya Pradesh, India
merca Joural of ppled Mathematcs 04; (4): 734 Publshed ole ugust 30, 04 (http://www.scecepublshggroup.com//aam) do: 0.648/.aam.04004.3 ISSN: 3300043 (Prt); ISSN: 330006X (Ole) O geeralzed fuzzy mea
More informationSTK3100 and STK4100 Autumn 2017
SK3 ad SK4 Autum 7 Geeralzed lear models Part III Covers the followg materal from chaters 4 ad 5: Sectos 4..5, 4.3.5, 4.3.6, 4.4., 4.4., ad 4.4.3 Sectos 5.., 5.., ad 5.5. Ørulf Borga Deartmet of Mathematcs
More information3D Geometry for Computer Graphics. Lesson 2: PCA & SVD
3D Geometry for Computer Graphcs Lesso 2: PCA & SVD Last week  egedecomposto We wat to lear how the matrx A works: A 2 Last week  egedecomposto If we look at arbtrary vectors, t does t tell us much.
More informationOn Fuzzy Arithmetic, Possibility Theory and Theory of Evidence
O Fuzzy rthmetc, Possblty Theory ad Theory of Evdece suco P. Cucala, Jose Vllar Isttute of Research Techology Uversdad Potfca Comllas C/ Sata Cruz de Marceado 6 8 Madrd. Spa bstract Ths paper explores
More informationLecture Notes Types of economic variables
Lecture Notes 3 1. Types of ecoomc varables () Cotuous varable takes o a cotuum the sample space, such as all pots o a le or all real umbers Example: GDP, Polluto cocetrato, etc. () Dscrete varables fte
More informationStatistics Descriptive and Inferential Statistics. Instructor: Daisuke Nagakura
Statstcs Descrptve ad Iferetal Statstcs Istructor: Dasuke Nagakura (agakura@z7.keo.jp) 1 Today s topc Today, I talk about two categores of statstcal aalyses, descrptve statstcs ad feretal statstcs, ad
More informationComplete Convergence for Weighted Sums of Arrays of Rowwise Asymptotically Almost Negative Associated Random Variables
A^VÇÚO 1 32 ò 1 5 Ï 2016 c 10 Chese Joural of Appled Probablty ad Statstcs Oct., 2016, Vol. 32, No. 5, pp. 489498 do: 10.3969/j.ss.10014268.2016.05.005 Complete Covergece for Weghted Sums of Arrays of
More informationComputational Geometry
Problem efto omputatoal eometry hapter 6 Pot Locato Preprocess a plaar map S. ve a query pot p, report the face of S cotag p. oal: O()sze data structure that eables O(log ) query tme. pplcato: Whch state
More information12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model
1. Estmatg Model parameters Assumptos: ox ad y are related accordg to the smple lear regresso model (The lear regresso model s the model that says that x ad y are related a lear fasho, but the observed
More information(b) By independence, the probability that the string 1011 is received correctly is
Soluto to Problem 1.31. (a) Let A be the evet that a 0 s trasmtted. Usg the total probablty theorem, the desred probablty s P(A)(1 ɛ ( 0)+ 1 P(A) ) (1 ɛ 1)=p(1 ɛ 0)+(1 p)(1 ɛ 1). (b) By depedece, the probablty
More informationOn the characteristics of partial differential equations
Sur les caractérstques des équatos au dérvées artelles Bull Soc Math Frace 5 (897) 8 O the characterstcs of artal dfferetal equatos By JULES BEUDON Traslated by D H Delhech I a ote that was reseted to
More information3. Basic Concepts: Consequences and Properties
: 3. Basc Cocepts: Cosequeces ad Propertes Markku Jutt Overvew More advaced cosequeces ad propertes of the basc cocepts troduced the prevous lecture are derved. Source The materal s maly based o Sectos.6.8
More informationHomework 1: Solutions Sid Banerjee Problem 1: (Practice with Asymptotic Notation) ORIE 4520: Stochastics at Scale Fall 2015
Fall 05 Homework : Solutos Problem : (Practce wth Asymptotc Notato) A essetal requremet for uderstadg scalg behavor s comfort wth asymptotc (or bgo ) otato. I ths problem, you wll prove some basc facts
More informationSufficiency in Blackwell s theorem
Mathematcal Socal Sceces 46 (23) 21 25 www.elsever.com/locate/ecobase Suffcecy Blacwell s theorem Agesza BelsaKwapsz* Departmet of Agrcultural Ecoomcs ad Ecoomcs, Motaa State Uversty, Bozema, MT 59717,
More informationGeneralized Convex Functions on Fractal Sets and Two Related Inequalities
Geeralzed Covex Fuctos o Fractal Sets ad Two Related Iequaltes Huxa Mo, X Su ad Dogya Yu 3,,3School of Scece, Bejg Uversty of Posts ad Telecommucatos, Bejg,00876, Cha, Correspodece should be addressed
More informationDebabrata Dey and Atanu Lahiri
RESEARCH ARTICLE QUALITY COMPETITION AND MARKET SEGMENTATION IN THE SECURITY SOFTWARE MARKET Debabrata Dey ad Atau Lahr Mchael G. Foster School of Busess, Uersty of Washgto, Seattle, Seattle, WA 9895 U.S.A.
More informationON THE STRUCTURE OF THE SPREADING MODELS OF A BANACH SPACE
ON THE STRUCTURE OF THE SPREADING MODELS OF A BANACH SPACE G. ANDROULAKIS, E. ODELL, TH. SCHLUMPRECHT, N. TOMCZAKJAEGERMANN Abstract We study some questos cocerg the structure of the set of spreadg models
More informationIncreasing Kolmogorov Complexity
Icreasg Kolmogorov Complexty Harry Buhrma Lace Fortow Ila Newma Nkola Vereshchag September 7, 2004 classfcato: Kolmogorov complexty, computatoal complexty 1 Itroducto How much do we have to chage a strg
More informationMultiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Regrades
STAT 101 Dr. Kar Lock Morga 11/20/12 Exam 2 Grades Multple Regresso SECTIONS 9.2, 10.1, 10.2 Multple explaatory varables (10.1) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (10.2) Trasformatos
More information8.1 Hashing Algorithms
CS787: Advaced Algorthms Scrbe: Mayak Maheshwar, Chrs Hrchs Lecturer: Shuch Chawla Topc: Hashg ad NPCompleteess Date: September 21 2007 Prevously we looked at applcatos of radomzed algorthms, ad bega
More informationSolution of General Dual Fuzzy Linear Systems. Using ABS Algorithm
Appled Mathematcal Sceces, Vol 6, 0, o 4, 637 Soluto of Geeral Dual Fuzzy Lear Systems Usg ABS Algorthm M A Farborz Aragh * ad M M ossezadeh Departmet of Mathematcs, Islamc Azad Uversty Cetral ehra Brach,
More informationChain Rules for Entropy
Cha Rules for Etroy The etroy of a collecto of radom varables s the sum of codtoal etroes. Theorem: Let be radom varables havg the mass robablty x x.x. The...... The roof s obtaed by reeatg the alcato
More informationLimiting Distributions of Scaled Eigensections in a GITSetting
Lmtg Dstrbutos of Scaled Egesectos a GITSettg Dssertato zur Erlagug des Doktorgrades der Naturwsseschafte a der Fakultät für Mathematk der RuhrUverstät Bochum vorgelegt vo Dael Berger m Oktober 014 Cotets
More information( ) 2 2. MultiLayer Refraction Problem Rafael Espericueta, Bakersfield College, November, 2006
MultLayer Refracto Problem Rafael Espercueta, Bakersfeld College, November, 006 Lght travels at dfferet speeds through dfferet meda, but refracts at layer boudares order to traverse the leasttme path.
More informationarxiv: v1 [math.st] 24 Oct 2016
arxv:60.07554v [math.st] 24 Oct 206 Some Relatoshps ad Propertes of the Hypergeometrc Dstrbuto Peter H. Pesku, Departmet of Mathematcs ad Statstcs York Uversty, Toroto, Otaro M3J P3, Caada Emal: pesku@pascal.math.yorku.ca
More informationMARKOV CHAINS. 7. Convergence to equilibrium. Longrun proportions. Part IB Michaelmas 2009 YMS. Proof. (a) state j we have π (i) P ) = π
Part IB Mchaelmas 2009 YMS MARKOV CHAINS Emal: yms@statslabcamacuk 7 Covergece to equlbrum Logru proportos Covergece to equlbrum for rreducble, postve recurret, aperodc chas ad proof by couplg Logru
More informationChannel Models with Memory. Channel Models with Memory. Channel Models with Memory. Channel Models with Memory
Chael Models wth Memory Chael Models wth Memory Hayder radha Electrcal ad Comuter Egeerg Mchga State Uversty I may ractcal etworkg scearos (cludg the Iteret ad wreless etworks), the uderlyg chaels are
More informationLecture 1. (Part II) The number of ways of partitioning n distinct objects into k distinct groups containing n 1,
Lecture (Part II) Materals Covered Ths Lecture: Chapter 2 (2.6  2.0) The umber of ways of parttog dstct obects to dstct groups cotag, 2,, obects, respectvely, where each obect appears exactly oe group
More informationAnswer key to problem set # 2 ECON 342 J. Marcelo Ochoa Spring, 2009
Aswer key to problem set # ECON 34 J. Marcelo Ochoa Sprg, 009 Problem. For T cosder the stadard pael data model: y t x t β + α + ǫ t a Numercally compare the fxed effect ad frst dfferece estmates. b Compare
More informationUnsupervised Learning and Other Neural Networks
CSE 53 Soft Computg NOT PART OF THE FINAL Usupervsed Learg ad Other Neural Networs Itroducto Mture Destes ad Idetfablty ML Estmates Applcato to Normal Mtures Other Neural Networs Itroducto Prevously, all
More informationECE 559: Wireless Communication Project Report Diversity Multiplexing Tradeoff in MIMO Channels with partial CSIT. Hoa Pham
ECE 559: Wreless Commucato Project Report Dversty Multplexg Tradeoff MIMO Chaels wth partal CSIT Hoa Pham. Summary I ths project, I have studed the performace ga of MIMO systems. There are two types of
More information2006 Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America
SOLUTION OF SYSTEMS OF SIMULTANEOUS LINEAR EQUATIONS GaussSedel Method 006 Jame Traha, Autar Kaw, Kev Mart Uversty of South Florda Uted States of Amerca kaw@eg.usf.edu Itroducto Ths worksheet demostrates
More informationLogistic regression (continued)
STAT562 page 138 Logstc regresso (cotued) Suppose we ow cosder more complex models to descrbe the relatoshp betwee a categorcal respose varable (Y) that takes o two (2) possble outcomes ad a set of p explaatory
More informationWe have already referred to a certain reaction, which takes place at high temperature after rich combustion.
ME 41 Day 13 Topcs Chemcal Equlbrum  Theory Chemcal Equlbrum Example #1 Equlbrum Costats Chemcal Equlbrum Example #2 Chemcal Equlbrum of Hot Bured Gas 1. Chemcal Equlbrum We have already referred to a
More informationHUYGENS S ENVELOPINGWAVE PRINCIPLE
CONTRIBUTION TO THE THEORY OF HUYGENS S ENVELOPINGWAVE PRINCIPLE BY J. VAN MIEGHEM TRANSLATED BY D. H. DELPHENICH BRUSSELS PALAIS DES ACADEMIES RUE DUCAL, 96 TABLE OF CONTENTS INTRODUCTION FIRST CHAPTER
More informationCorrelation and Regression Analysis
Chapter V Correlato ad Regresso Aalss R. 5.. So far we have cosdered ol uvarate dstrbutos. Ma a tme, however, we come across problems whch volve two or more varables. Ths wll be the subject matter of the
More informationGeneral Method for Calculating Chemical Equilibrium Composition
AE 6766/Setzma Sprg 004 Geeral Metod for Calculatg Cemcal Equlbrum Composto For gve tal codtos (e.g., for gve reactats, coose te speces to be cluded te products. As a example, for combusto of ydroge wt
More informationChapter 11 Systematic Sampling
Chapter stematc amplg The sstematc samplg techue s operatoall more coveet tha the smple radom samplg. It also esures at the same tme that each ut has eual probablt of cluso the sample. I ths method of
More informationAn Introduction to Relevant Graph Theory and Matrix Theory
Chapter 0 A Itroducto to Relevat Graph heory ad Matrx heory hs boo s cocered wth the calculato of the umber of spag trees of a multgraph usg algebrac ad aalytc techques. We also clude several results o
More informationarxiv: v3 [math.ra] 17 May 2017
ON HSIMPLE NOT NECESSARILY ASSOCIATIVE ALGEBRAS A. S. GORDIENKO arxv:508.03764v3 [math.ra] 7 May 207 Abstract. At frst glace the oto of a algebra wth a geeralzed Hacto may appear too geeral, however
More information: At least two means differ SST
Formula Card for Eam 3 STA33 ANOVA FTest: Completely Radomzed Desg ( total umber of observatos, k = Number of treatmets,& T = total for treatmet ) Step : Epress the Clam Step : The ypotheses: :... 0 A
More informationAnalysis of System Performance IN2072 Chapter 5 Analysis of Non Markov Systems
Char for Network Archtectures ad Servces Prof. Carle Departmet of Computer Scece U Müche Aalyss of System Performace IN2072 Chapter 5 Aalyss of No Markov Systems Dr. Alexader Kle Prof. Dr.Ig. Georg Carle
More information( ) ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) = ( ) ( ) + ( ) ( ) = ( ( )) ( ) + ( ( )) ( ) Review. Second Derivatives for f : y R. Let A be an m n matrix.
Revew + v, + y = v, + v, + y, + y, Cato! v, + y, + v, + y geeral Let A be a atr Let f,g : Ω R ( ) ( ) R y R Ω R h( ) f ( ) g ( ) ( ) ( ) ( ( )) ( ) dh = f dg + g df A, y y A Ay = = r= c= =, : Ω R he Proof
More informationSTA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1
STA 08 Appled Lear Models: Regresso Aalyss Sprg 0 Soluto for Homework #. Let Y the dollar cost per year, X the umber of vsts per year. The the mathematcal relato betwee X ad Y s: Y 300 + X. Ths s a fuctoal
More informationROOTLOCUS ANALYSIS. Lecture 11: Root Locus Plot. Consider a general feedback control system with a variable gain K. Y ( s ) ( ) K
ROOTLOCUS ANALYSIS Coder a geeral feedback cotrol yte wth a varable ga. R( Y( G( + H( RootLocu a plot of the loc of the pole of the cloedloop trafer fucto whe oe of the yte paraeter ( vared. Root locu
More informationRisk management of hazardous material transportation
Maagemet of atural Resources, Sustaable Developmet ad Ecologcal azards 393 Rs maagemet of hazardous materal trasportato J. Auguts, E. Uspuras & V. Matuzas Lthuaa Eergy Isttute, Lthuaa Abstract I recet
More informationJohns Hopkins University Department of Biostatistics Math Review for Introductory Courses
Johs Hopks Uverst Departmet of Bostatstcs Math Revew for Itroductor Courses Ratoale Bostatstcs courses wll rel o some fudametal mathematcal relatoshps, fuctos ad otato. The purpose of ths Math Revew s
More informationChapter 3. Linear Equations and Matrices
Vector Spaces Physcs 8/6/05 hapter Lear Equatos ad Matrces wde varety of physcal problems volve solvg systems of smultaeous lear equatos These systems of lear equatos ca be ecoomcally descrbed ad effcetly
More informationRecall MLR 5 Homskedasticity error u has the same variance given any values of the explanatory variables Var(u x1,...,xk) = 2 or E(UU ) = 2 I
Chapter 8 Heterosedastcty Recall MLR 5 Homsedastcty error u has the same varace gve ay values of the eplaatory varables Varu,..., = or EUU = I Suppose other GM assumptos hold but have heterosedastcty.
More informationChapter 2 Simple Random Sampling
Chapter  Smple Radom Samplg Smple radom samplg (SRS) s a method of selecto of a sample comprsg of umber of samplg uts out of the populato havg umber of samplg uts such that every samplg ut has a equal
More informationi 2 σ ) i = 1,2,...,n , and = 3.01 = 4.01
ECO 745, Homework 6 Le Cabrera. Assume that the followg data come from the lear model: ε ε ~ N, σ,,..., 6. .5 7. 6.9 . . .9. ..6.4.. .6 .7.7 Fd the mamum lkelhood estmates of,, ad σ ε s.6. 4. ε
More informationFor combinatorial problems we might need to generate all permutations, combinations, or subsets of a set.
Addtoal Decrease ad Coquer Algorthms For combatoral problems we mght eed to geerate all permutatos, combatos, or subsets of a set. Geeratg Permutatos If we have a set f elemets: { a 1, a 2, a 3, a } the
More informationOn Signed Product Cordial Labeling
Appled Mathematcs 5553 do:.436/am..6 Publshed Ole December (http://www.scrp.or/joural/am) O Sed Product Cordal Label Abstract Jayapal Baskar Babujee Shobaa Loaatha Departmet o Mathematcs Aa Uversty Chea
More informationUnique Common Fixed Point of Sequences of Mappings in GMetric Space M. Akram *, Nosheen
Vol No : Joural of Facult of Egeerg & echolog JFE Pages 9 Uque Coo Fed Pot of Sequeces of Mags Metrc Sace M. Ara * Noshee * Deartet of Matheatcs C Uverst Lahore Pasta. Eal: ara7@ahoo.co Deartet of Matheatcs
More informationBezier curve and its application
, 4955 Receved: 20141112 Accepted: 20150206 Ole publshed: 20151116 DOI: http://dx.do.org/10.15414/meraa.2015.01.02.4955 Orgal paper Bezer curve ad ts applcato Duša Páleš, Jozef Rédl Slovak Uversty
More informationSupplementary Material for Limits on Sparse Support Recovery via Linear Sketching with Random Expander Matrices
Joata Scarlett ad Volka Cever Supplemetary Materal for Lmts o Sparse Support Recovery va Lear Sketcg wt Radom Expader Matrces (AISTATS 26, Joata Scarlett ad Volka Cever) Note tat all ctatos ere are to
More information