Gabriel filters in Grothendieck categories

Publicacions Ma emá iques, Vol 36 (1992), 673-683 . A bs ac GABRIEL FILTERS IN GROTHENDIECK CATEGORIES A . JEREMÍAS LÓPEZ, M . P . LÓPEZLÓPEZ AND E . VILLANUEVA NÓVOA Dedica ed o he memo y o Pe e Menal In [1] i is p o ed ha one mus ake ca e ying o copy esul s om he case o modules o an a bi a y G o hendieck ca ego y in o de o desc ibe a he edi a y o sion heo y in e ms o il e s o a gene a o . By he o he side, we usually ha e o a G o hendieck ca ego y an in ini e amily o gene a o s {Gz ; i E I} and, al hough each Gz has good p ope ies he gene a o G= ® Gi is no easy o ¡E7 handle ( o ins ance in ca ego ies like g aded modules, p eshea es o shea es o modules) . In his pape he au ho s ob ain a bi- jec i e co espondence be ween he edi a y o sion heo ies in a G o hendieck ca ego y C and a app op ia ely de ined amily o Gab iel il e s o subobje s o he gene a o s o C . This has been possible by using he na u al condi ions o local p ojec i eness and local smallness o amilies o gene a o s in a G o hendieck ca e- go y, ha he embedding heo em o Gab iel-Popescu p o ided us . In oduc ion As i is well known, G o hendiek ca ego ies p o ide a good se ing o s udy o sion heo ies, o hey can be applied in se e al di e en con ex s [4], [5] . Ne e heless, up o now he e was no a sui able cha - ac e iza ion o he edi a y o sion heo ies in e ms o Gab iel il e s in he gene a o s, like he usual one in he ca ego y R-mod . This pape is The au ho s hank he Xun a de Galicia o i s pa ial suppo unde G an XUGA 8050289 . 674  A . JEREMÍAS, M . P . LÓPEZ, E . VILLANUEVA de o ed o ob ain he bijec i e co espondence be ween he edi a y o - sion heo ies in a G o hendieck ca ego y C and he app op ia ely de ined Gab iel il e s o subobjec s o a amily o gene a o s o C . This has been possible by in oducing he na u al no ions o local p ojec i eness and local smallness, bo h de ined he e, o amilies o objec s in C . Wi h espec o his, i is con enien o emphasize he ac ha p ope ies 1) o 4) o (3 .2) ca ac e izing Gab iel il e s ha e been abs ac ed om he pa allel ones in R-mod, wi h he idempo en condi ion 4) es ic ed o co e ings o he elemen s o he il e , ins ead o he o ali y o mo - phisms be ween gene a o s and hese objec s as in (3 .5) 4') . We se he compa a ion be ween he wo posibili ies in (3 .5) p o ing ha hey a e equi alen (in p esen e o he o he condi ions 1) o 3), o cou se) in he case ha he gene a o s a e p ojec i e and small . We ake om [1] an illus a i e coun e example (3 .6) showing ha i he gene a o s a e no small he condi ions 1) o 4') a e unable o cha ac e ize he Gab iel il e s co esponding o idempo en ke nel unc o s . Le us inally say ha we ha e lea n abou he comple e gene al- i y o ou esul (3 .4), looking a he example (2 .5) in he ligh o he Gab iel-Popescu Imbedding heo em, in discussions wi h P o esso A . Ve scho en . 1 . P eli nina ies Le us i s ake a quick look a some o he basic opics in o sion heo ies [2], [4] . (1 .1) I C is a ca ego y wi h ze o, he ela ion (A, B) E 4  Homc (A, B) = 0 de ines a Galois connec ion in he class ¡Cl o he objec s o C PICI z_± PICI  by means o B E (X)  Homc (A, B) = 0  o e e y A E X A E (Y) ~ Homc(A, B) = 0  o e e y B E Y, X and Y being classes o objec s in C . (1 .2) A class X E PICI ( esp . Y E PICI) is a o sion class ( esp . a o sion- ee class) i , and only i , X = (Y) ( esp . Y = (X)) o any Y E PIC) ( esp . X E PICI) . As in e e y Galois connec ion he e is an inclusion- e e sing bijec ion be ween osion classes and ee o sion classes de ined by , wi h in e se . GABRIEL FILTERS IN GROTHENDIECK CATEGORIES  67 5 A pola pai is a pai o classes (T, F), such ha T = (F) and F = (T) . I cons i u es a o sion heo y . (1 .3) I (T, F) is a o sion heo y in an abelian ca ego y C we ha e : a) F n T = {0} . b) T is closed unde quo ien objec s, ex ensions and cop oduc s . c) F is closed unde p oduc s, subobjec s and ex ensions . I T is also closed unde subobjec s, we say ha T is a he edi a y o sion class and (T, F) is a he edi a y o sion heo y . In his case, F is closed unde aking essen ial ex ensions . (1 .4) I C is an abelian AB5 ca ego y (i .e . C has exac induc i e limi s and cop oduc s) he e is a bijec ion be ween he edi a y o sion heo ies and idempo en ke nel unc o s in C [4] . I o, : C --> C is an idempo en ke nel unc o in C, he co esponding o sion heo y (T Q , F Q ) is gi en by aking o T Q he class o objec s M in C, such ha o,(M) = M, and o F Q hose M wi h u(M) = 0 . Con e sely, i (T, F) is a he edi a y o sion heo y, he ke nel unc- o which co esponds o (T, F) is de ined, o M E ICI, by o,(M) _ N ; i .e . u(M) is he bigge o sion subobjec o M . MDNET (1 .5) In an abelian AB5 ca ego y C i T C_ ¡Cl is closed unde quo- ien s, ex ensions, cop oduc s and subobjec s, we ha e ha (T, (T)) is a he edi a y o sion heo y . 2 . Local smallness and local p ojec i enesso sys ems o gene a o s Local smallness and local p ojec i eness a e in oduced in his sec- ion, and we see ha hey become na u al concep s in a G o hendieck ca ego y . A is always a ca ego y wi h a bi a y cop oduc s . (2 .1) De ini ion . Le 13 = {A ¡ / i E V} be a amily o objec s in A . We shall say ha 13 is a locally p ojec i e amily in A, i o each : A i --> M' wi h A i E 13 and o e e y epimo phism cp : M --~ M' in A he e exis s an epimo phism  : Ij Aj  A ¡ , whe e A j is also in 13 (hence o h a jeUi 13-co e ing  _ (~ j ) : ú A j -~ A ¡ ), such ha each o~ j ac o s h ough jEO i M, i .e . o each j E 0? he e is a j : A j--> M such ha cp o .7 = o ~ . (2 .2) Lemma . I A is an abelian ca ego y wi h a sys em o gene a o s ~, hen 9 is a locally p ojec i e amily . 676  A . JEREMÍAS, M . P . LÓPEZ, E . VILLANUEVA P oo . Since A has uni e sal epimo phisms [3], i cp : A -> A' is an epimo phism and EHomA(G, A') we ha e a pullback diag am , o~ j a e he equi ed ac o iza ions . A ~ A' w wi h ~o' : B -> G epimo phism . We can co e B by (~ j ) : jjG ; -> B and 7 (2 .3) Rema k . An objec A o A is said o be small i o any cop oduc jj Ma and o any E Hom A (A, j1 Ma) he e is a ini e ,NEA  aEA subse F o A such ha ac o s h ough jj MA [3] . ,NEF This is a s ong and e y es ic i e concep . In ac , i A is a G o hendieck ca ego y wi h a small p ojec i e gene a o U, A is equi - alen o R-mod whe e R= HomA(U, U) [3] . Le us now se a weake no ion . (2 .4) De ini ion . We say ha a amily o objec s 13 ={A 2 /i E 0} o A is locally small i o any i E 1 and o e e y E HomA(A2, jj Ma) aEA he e exis s a 13-co e ing (Sk) :  JJ Ak --> A Z such ha each o Sk kEJi Ak ` jj Ma ac o s h ough a ini e subcop oduc , Le . o each k E .IZ >,EA he e is a ini e subse Fk o A and a k E HomA(Ak, jI Ma) such ),EFk ha jF,, o k = o ~k, whe e jF,, :  jj Ma ->  jj MA is he canonical , EFk >,EFk inclusion . The ollowing example will be used in he main esul o his sec ion . (2 .5) Example . I u : R-mod --> R-mod is an idempo en ke nel unc o , we deno e by (R, o,)-mod he ull subca ego y o he o-closed R- modules, Le . he R-modules ha a e bo h o , - o sion ee and Q-injec i e . (R, u)-mod is a G o hendieck ca ego y wi h gene a o Q,(R)  = lim HOMR(I, R/u(R)), whe e C Q deno es he Gab iel il e o le ide- IEG o als I o R, such ha o,(R/I) = R/I . In gene al Q Q (R) is no a small no p ojec i e objec in (R, u)-mod, bu i yields a locally small sys- em o gene a o s, i we ake enough copies o QQ(R) . Explici ely, i {Ma/a E A} is a amily o objec s in (R, o,)-mod we deno e by jI Ma >,EA GABRIEL FILTERS IN GROTHENDIECK CATEGORIES  677 his cop oduc in (R, o,)-mod, Le .  Ll Ma= Qo( ® Ma) and, since >,EA  >,EA he class o u- o sion ee is closed unde p oduc s and subobjec s we ha e ha he localiza ion mo phism jA :  ® M>, --> Ll Ma is injec- aEA >,EA i e .  Le E HOMR(Q,(R), jj Ma) and y = (1) .  Since coke (jA) aEA is a - o sion module, he e is an I E G Q such ha o E e y aE I he mo phism o a : Q,(R) --> jj Ma ac o s h ough ® M, , whe e aEA  >,EA a : Q,(R) - Q Q (R) is he igh -mul iplica ion by a . Thus we ha e, o each aE I a R-linea map a : Q, (R) -> ® MA such ha he diag am , EA commu es . Now, a (1) E ® Ma~ and so we ha e a (Q,(R)) C n a n a 1] M>,, since ® M, i is also Q-closed .  The mo phism ~ = (a)aEI i=1  i=1 j1QQ(R) -> QQ(R) is an epimo phism in (R,u)-mod, Le . coke (1) is I a a- o sion R-module, because I E ,C, . This p e es ha a amily o enough copies o Q Q (R) is a locally small sys em o gene a o s o (R, o' )- mod . The o me example p o ides a e y su p ising consequence ia he Gab iel-Popescu heo em [4] . (2 .6) Theo em . E e y G o hendieck ca ego y C wi h a gene a o U has a locally small sys em o gene a o s . P oo . By he Gab iel-Popescu heo em, C is equi alen , by means o he unc o Homc (U, -) : C -~ R-mod, e he ca ego y (R, u)-mod, whe e R = Homc (U, U) and a is an idempo en ke nel unc o o which R= Q, (R) . U co esponding e R= Homc (U, U) in his equi alen e, and R being a Q-closed module, he esul ollows om (2 .5) because he no ion o local smallness is p ese ed by equi alen es . I hen su ices o ake enough copies o U o ob ain he equi ed locally small sys em o gene a o s . (2 .7) Co olla y . I C is a G o hendieck ca ego y wi h a sys em o gene a o s G= {Gil¡ E 1} hen G is a locally small amily . Qo (R) a Q (R) a 1 1 ® Ma jj Ma aEA .7A aEA 678  A . JEREMÍAS, M . P . LÓPEZ, E . VILLANUEVA P oo .. Le U= jI G i be he big gene a o o C . We deno e by pi ¡E0 U = Gi x (jjGj) -4 Gi he canonical p ojec ion . Le : Gi -> jI M>, js4i  EA be an a bi a y mo phism and ' = o p i .  Then by (2 .6) we ha e a co e ing cW : j1Uk --> U induced by cp k : Uk = U --> U such ha o k e e y k E IK he e exis s a k : Uk -->  LI Ma (Fk is a ini e subse o aEFk A) such ha jF k o k = o Wk . Fo e e y kE IK le G~ = Gj and le h y ~ : G 3 ~ -> Uk be he canonical monomo phism . I we pu 3 ~ = k o h 3 ~, and  pi o <Pk o h 7 ~ : G 3 ~ -+ Gi we ob ain a commu a i e diag am : 7 U k . k (Pk U pi Gi , V 1 M a - > JJ M a aEF k . jF, AEA Then o ~. ac o s h ough he ini e cop oduc . jI M>,, and u he - AEFk mo e, he mo phism  :  jI jI G~ = j1Uk --> Gi induced by (1~ )j,k kEKjEJ k G 7 ~ ---> Gi is he epimo phism pi o W . 3 . Gab iel il e s and o sion heo ies This pa is de o ed o es ablish he bijec ion be ween idempo en ke nel unc o s (Le . he edi a y o sion heo ies) and some amilies o il e s o subobjec s o he gene a o s in a G o hendieck ca ego y ha we will deno e by C . (3 .1) Le C be a ca ego y and Q : C ---> C an idempo en ke nel unc o . We de ine, o each M E ¡C¡, he Gab iel il e o subobjec s o M, GM, ela i e o u, by N E GM <=> M/N E T Q . (3 .2) P oposi ion . I C is a G o hendieck ca ego y wi h a sys em o gene a o s 9 = {Gi/i E V } and i o : C -> C is an idempo en ke nel unc o hen 1) GG :~ 0 o e e y G E 9 . 2) ICJCG,IE£G=~>JEGG . 3) I G, G' E 9, E Hom c (G', G) and I E GG hen -1 (I) E £ 7 G) GABRIEL FILTERS IN GROTHENDIECK CATEGORIES  679 4) Le G E 9 and I C_ J E GG . I he e is a G-co e ing (Sj) jj Gj - " J s ch ha ~~1 (I) E GG j o each j E ,D, hen I E GG . jEJ P oo . Le (T Q , F a ) be he he edi a y o sion heo y co esponding o o, . Then 1) is clea since 0 E T Q . Fo 2) we conside he canonical epimo phism G/I -> G/J and, since G/I E T Q we ha e G/J E T Q because T Q is closed unde quo ien s . 3) is a consequence o he ac s ha T Q is closed unde aking subobjec s and ha he e is a monomo phism G o l -1 (I) - GIL We can p o e 4) by conside ing he commu a i e diag am : 0  V 0 S3- .I ( I )  , . V G j jEi 0 V G j V GjlSj 1(I) V / L  , V ' LjGjl, 1 (I)  V  ~/  >  J/I jEi  0 wi h exac columns because he cop oduc s a e exac in C . ~~ : Gj / ~ 1(I) -~ J/I is de ined by Sj, and j' is an epimo phism by commu a i i y . Then, as T Q is closed unde cop oduc s and quo ien s, J/I E T Q since Gj l ~-1 (I) E T Q by hypo hesis . As well, he canonical exac sequence 0 --> J/I - G/I , G/J - 0 and he ac ha T Q is closed unde ex ensions, shows ha G/I E T Q i G/J is so . No e ha i I, J E GG hen I 1 J E GG . In ac , i (~ 2 ) : jjG Z ----> I is ¡Ea a g-co e ing o I hen , - . 1 (I 1 J) = 0,-1(J), being 4'j = ho ~j wi h h 680  A . JEREMÍAS, M . P . LÓPEZ, E . VILLANUEVA I -j G he inclusion . Hence, condi ion 4) wo ks because I (J) E GG j o e e y j E V, by condi ion 3) . Con e sely : (3 .3) P oposi ion . I C is a G o hendieck ca ego y wi h a sys em o gene a o s 9, and i {,CG/G E 9} is a amily o il e s o subobjec s o he objec s G such ha e i ies join ly he p ope ies 1) o 4) o (3 .2) hen he e exis s an idempo en ke nel unc o o, : C --> C such ha Gc = GG o e e y G E ~ . P oo . We de ine a class T o objec s in C saying ha A is a membe o T i ke ( ) E Cc o e e y EHomc (G, A) and G E ~ . Wi h his de ini ion T is a he edi a y o sion class . In ac , T is closed by subobjec s because i A' C_ A E T and i ' E Homc(G, A'), ke ( ) = ke ( ') being = i o ', i  A' C A . To show ha T is closed unde epimo phic images we shall use ha 9 is a locally p ojec i e sys em o gene a o s in C (2 .2) . I cp : A ---> A' is an epimo phism wi h A E T and i E Homc (G, A') wi h G E 9 le %) :  1] G 9 --> G be a G-co e ing such ha o~9 ac o s h ough 9EG 9 : G 9 --> A .  Le be I = ke ( ) and pu I 9 =  s 1 (I) .  So Ig E GG9 since ke ( 9 ) C_ I 9 , and he condi ion 2) wo ks . Also, by condi ion 4) applied o I C_ G E G G we ob ain I E GG . So A' E T and T is closed by quo ien s . Le us now ake 0 - A' 2G> A w> A" -> 0 an exac sequence wi h A', A" E T and le E Homc (G, A) o G E ~ . We mus ha e I = ke ( ) E Gc . To show his we obse e ha , i " = co o , ke ( ") is he pullback o and 0  , " W A'  > A  >A" Le (~ j ) : J1G j - ke ( ") be a C-co e ing and pu j = o ~ j . Then J GABRIEL FILTERS IN GROTHENDIECK CATEGORIES  6)81 ~ 3 -1 (I) = ke ( j) and since A' E T we ha e ~, -1 (I) E GG j. Since A" E T we ob ain ke ( ") E GG and so, he condi ion 4) applied o I C_ ke ( ") p o ides I E GG . Hence A E T and so T is closed unde ex ensions . The local smallness p ope y o G (2 .7) is now applied in p o ing ha T is closed unde cop oduc s . Indeed, i Aa E T o e e y A E A and E Homc(G, IJAa) he e exis s a 9-co e ing (~j) :  jj Gj --> G such a  jEJ ha o e e y j E .D he mo phism Sj o ac o s h ough a ini e sub- cop oduc . So, o e e y j E .D he e is a ini e subse A j o A and j E Home(Gj, Ll Aa) such ha o~j = hj o j , h j :  jj Aa --> jj Aa aEA j aEA j aEA being he canonical monomo phism . As T is closed by ex ensions we ha e  jj Aa E T and so ke ( j) = ~,- .1(ke ( )) E Cc . . . The condi ion 4) aEAj applied o ke ( ) C G yields ke ( ) E GG . Hence jj Aa E T and so T is jeA closed unde a bi a y cop oduc s . Finally, le us see ha o e e y G E 9, i I is a subobjec o G, I E Cc i G/I E T . F om he de ini ion o T, we ha e GII E T implies I E GG . Con e sely, suppose I E G G and le : G' --~ G/I be an a bi a y mo phism wi h G' E 9 also . We mus p o e ha ke ( ) E GG, . Le (Sj) : jj Gj - G' be a G-co e ing such ha o each j E J he jEj mo phism o ~j : Gj -~ GII ac o s h ough he canonical p ojec i n cp : G -~ GIL Le j : Gj -> G be his ac o iza ion . Then we ha e 3 -1 (I) = ke (cp o j ) = ke ( o .Sj) = ~,-1(ke ( )), and so, as I E GG, condi ion 3) yields . - .' (I) E Cc, o e e y j E J . As a consequence o condi ion 4) applied o ke ( ) C G' we ha e ke ( ) E .C G , . The asse ions (1 .5) and (1 .4) can now be applied, and he p oo is inished . So we can s a e : (3 .4) Co olla y . I C is a G o hendieck ca ego y, he e exis s a bi- jec ion be ween he edi a y o sion heo ies in C and amilies o Gab iel il e s GG i o subobjec s o he gene a o s p o ided ha hey sa is y he condi ions 1) o 4) o (3 .2) . No e ha i he gene a o s a e p ojec i e and small, condi ion 4) o (3 .2) can be e o mula ed . (3 .5) P oposi ion . I C is a G o hendieck ca ego y wi h a sys em 9 o small and p ojec i e gene a o s, hen a amily o il e s { .CG/G E 9} e i ies he condi ions 1) o 4) o (3 .2) i , and only i , sa is ies he condi ions 1), 2), 3) and he ollowing 4') :