Adic-completion and some dual homological results

Publicacions Ma emá iques, Vol 36 (1992), 965-979 . A bs ac ADIC-COMPLETION AND SOMEDUAL HOMOLOGICAL RESULTS ANNE-MARIE SIMON To he memo y o Pe e Menal Le a be an ideal o a commu a i e ing A . The e is a kind o du- ali y be ween he le de i ed unc o s Ui o he a-adic comple ion unc o , called local homology unc o s, and he local cohomology unc o s Há . Some dual esul s a e ob ained o hese Ua, and also inequali- ies in ol ing bo h local homology and local cohomology when he ing A is noe he ian o mo e gene ally when he U° and H a -global dimensions o A a e ini e . In his pape A is a commu a i e ing, a an ideal o A and . he A- modules a e gi en he a-adic opology . The e is a ce ain duali y be ween he le de i ed unc o s Ui o he a-adic comple ion unc o and he local cohomology unc o s H,,, i s obse ed by Ma lis when he ideal a is gene a ed by a ( ini e) egula sequence, ue also o any noe he ian ing . Mo e ecen ly, ha duali y has also been obse ed by C eenlees and May in a mo e gene al con ex . The pu pose o his no e is o pu sue he analogy be ween he local co- homology unc o s and hese unc o s Uá, called local homology unc o s by C eenlees and May . Fi s we ha e dual esul s abou codep h, a no ion dual o he no ion o homological dep h o g ade . To go u he , we need come noe he ian hypo hesis in o de o ha e a chango o ings heo em o he Ui ; a lalogous o he co esponding one in local cohomology . This b ings us back o he i s case s udied by Ma lis, namely he case o an ideal gene a ed by a egula sequence, and allows gene aliza ions o some Ma lis esul s . As a consequence, we ob ain anishing esul s o he U°, and also inequali ies in ol ing 96 6  A .-M . SIMON bo h local cohomology and local homology . So local cohomology and local homology a e no only duals o each o he , bu aleo in ima ely connec ed . As gene al e e ences o commu a i e algeb a and homological ques- ions, we quo e [1], [18] . The wo k below has been communica ed a he In e na ional wo kshop on local cohomology, geome ic applica ions and ela ed opics . We ake his oppo uni y o hank he o ganize s and he pa icipan e o use ul discussions . In his i s sec ion we ix no a ions and collec he ma e ial we need . Though pa o i appea ed in di e en places, we hink i is mo e con- enien o ha e i a hand . 1 .1 . Comple ion . Le a be an ideal o he commu a i e ing A . The A-module a e gi en he a-adic opology . The comple ion o an A-module M is deno ed by 1Vl : hus M = lim M/a'M . Le Tn,1 : M --> M be he na u al mo phism . 3J) . 1 . P elimina ies and no a ions He e and in he nex sec ion, he ideal a is no necessa ily ini ely gene a ed ; and i migh happen ha he A-module M, comple e in i s na u al opology, is no comple e in i s a-adic opology . An example o his can be ound in ([5, III, Sec ion 2, exe cise 12]) o in ([3, I ; Sec ion Recall howe e he ollowing esul ([3, heo em 1 .3 .1], o [13, heo em 15], o [18, 2 .2 .5]) . Theo em . Suppose he ideal a ini ely gene a ed . Le M be an A- module and b an open ideal in he a-adic opology o A . Then he mo - phism- moA/b : M/bM -> M/blVl is an isomo phism . So M is comple e in i s a-adic opology . 1 .2 . When is on o . The a-adic comple ion unc o , hough no igh exac , p ese es su - jec ion . Howe e , we wan o know p ecisely when he comple ion o a mo phism is on o . The ollowing lemma was p o ed in ([16, 1 .2]) o a noe he ian ing A, using 1 .1 . I is ue in gene al . Lemma . Le : M ---> N be a mo phism o A-modules .  Then su7-jec i e i and only i N = Al -1- aN . ADIC-COMPLGTION, DUAL RESULTS  967 P oo .. I N = Al + aN, hen N = M + anN o all n >_ 1, and he p ojec i e sys em o sequences 0 ---> -1 ( a ' i N) la"M ---> Ml a' M --> N/a' N ---> 0 is exac . I is easily seen o be su jec i e (see [16, 1 .2]) . So, a e aking limi s, we ge an exac sequence and is su jec i e . Con e sely, i is su jec i e, we enso he commu a i e na u al dia- g am M N M/aM  > N/aN, whe e he e ical composi e maps a e he na u al p ojec ions, wi h Ala . The composi e e ical maps become he iden i y, so M/aM -> N/aN is su jec i e and N = M +aN . 1 .3 . The le de i ed unc o s o he comple ion unc o . The le de i ed unc o s o he a-adic comple ion unc o a e deno ed by Ui . These we e i s s udied by Ma lis when he ideal is gene a ed by a ini e egula sequence [12], [13] . We used hem in [16], whe e he ing is noe he ian . Mo e ecen ly, hey ha e been compu ed by G eenlees and May in a mo e gene al si ua ion [7] . Le L 1 - > Lo -> A1  + 0 be a exac sequence, wi h Lo, L 1 ee . By de ini ion Uo (NI) = coke .Í, so we ha e na u al mo phisms A1 Uó (A1) -> M whose composi e is - A . The ollowing le nma ([16, 5 .1]), consequence o 1 .2, is s ill a ailable . Lemma . (i) The na u al mo j)hism UÓ (M) -> NI is on o . (ii) M = 0 i and only i M = aA1 i and only i Uó (M) = 0 . (iii) I he ideal a is ini ely gene a ed, hen (Uó (M))^ = A1 . 1 .4 . The class C a . Le C, be he class o modules M such ha Uó (M) = A7 and Ui (M) _ 0 o i > 0 . A s anda d homological a gumen shows ha Uá(A1) can be compu ed using a le esolu ion o A1 wi h modules in C a , and i is wo llwile o no e ha la nodules belong O C a . 968  A .-M . SIMON Mo e gene ally, le a¡,, n > 0, be a dec easing sequence o ideals which o m a basis o he a-adic opology . A module M such ha To A(A/a  ,, M)= 0 o all i > 0 and all n > 0 belongs o C a ([12, co olla y 4 .5]) . When he ing is noe he ian, he comple ion o a ee module is la ([14, p . 77], o [3, 4, 7], o [18, 2 .2 .4]) . This can be used o show ha comple e modules belong o C a when A is noe he ian ([16, 5 .2]) . 1 .5 . Local cohomology and Ma lis duali y . Recall he unc o H a' : H°(M) = {x E Mjanx = 0 o some na u al numbe n}, whose igh de i ed unc o s Há a e he local cohomology unc o s . Recall also he Ma lis duali y . Le E be he injec i e hull o he di ec sum o all he A/m wi h m, a inaximal ideal o A . The Ma lis duali y unc o , de ined by M = HOMA(M, E), is ai h ully exac [12] ; [13], and we ha e he Ex -To duali y : To A(N, M) - Ex Á(N, M ) ; when N has a p ojec i e esolu ion composed o ini ely gene a ed mod- ules ([6, VI, 5 .1 ; 5 .3]) : To A(N, M ) - Ex Á(N, M) . When he ing is noe he ian, Hó(M) - (M ) ^ and Há(M) - U°'(M ) o all i ([16, 4 .2 ; 5 .6]) . This is based on he ac ha , o e a noe he ian ing, la modules and injec i e modules a e in e changed by Ma lis duali y . This was i s p o ed by Ma lis when he ideal a is gene a ed by a egula sequence . Bu modules a e no necessa ily duals, so in o ma ions abou he local cohomology unc o s Hi, do no always p o ide in o ma ions abou he local homology unc o s Ui . 1 .6 . Fo mal dep h, codep h and dimension . A sequence o co a ian addi i e unc o s G,,, : A  + B, n E Z, be ween abelian ca ego ies is a descending connec ed exac sequence o unc o s i G a = 0 o n < 0 and i each exac sequence 0 -+ M' ~ 11N1 -> M" - 0 in A gi es ise o a long exac sequence . . . -> Gi+i(01) -> GZ(M') ) Gi(M) -, Gi(M") -> . . . in a unc o ial way . ADIC-COMYLETION, DUAL RESULTS  96 9 When we ha e such a sequence, as in ([18, 1 .1] o [17]), we pu g_(M) = in {¡IG i( M) =,,~ o} o each objec M in ,A (so ha o <_ g_ (M) < oo) . Dually, we de ine - (M) o an ascending connec ed exac sequence o unc o s F" in he saíne way . These numbe s can be iewed as a kind o codep h o dep h espec- i ely . When a is an ideal o he ing A, we a e me ely conce ned wi h he sequence Ex ;~(A/a, .), To A(A/a, .), H,'~( .), Ui (-) and wi h he co e- sponding numbe s . He e a e some i s ema ks abou hem, which will be comple ed la e (1 .7, 2 .4) . P oposi ion . Le , a . C b be ideals in he ing A, le M an A-module (i) ex A (A/a, AI) = ha (Ah) (ii) o '(A/a, M) = ex -(A/a, M") (iii) ex A (A/a, M) < ex A (A/b, Al1) (i ) o A (A/a, M) < o A (A/b, M) ( ) he numbe s ex A (A/a, M), o A(A/a, M) only depend on he opology de ined by he ideal a . Fo (i) and (iii), see ([18, 5 .3 .1 .5, 5 .3 .11]) ; (ii) is a di ec consequence o he Ex -To duali y 1 .5 ; (i ) ollows om (ii) . and (iii) . Since local cohomology only depends on he opology de ined by he ideal a, so does ex A(A/a, .) in iew o (i), and so does also o A(A/a, .) in iew o (ii) . We also de ine g+ (M) = sup{ilC i (M) :y~ 0} (so ha g+(M) = -oo o 0 _< g + (A11) < oo) and +(AI) in he saíne way . These las numbe s a e ela i e homological dimensions . 1.7 . The dep h-codep h sensi i i y o he Koszul complex . The dep h sensi i i y o he Koszul complex was p o ed by Ba ge and Hochs e o a cohe en ing [2], [8], by Ki by and Meh an o any commu a i e ing [10] . An app oach in ol ing bo h dep h and codep h can be ound in ([18, 6 .1] o [17]) . Le x = x I ,. . . , x, 1 be a sequence o eleinen s o he ing A gene a - ing an ideal a, and le K . (x) be he associa ed Koszul complex . Fo an A-module M, ; e conside he descending and ascending Koszul com- plexes K . (x, All) = K . (x) ®A A7, K" (x,1Vl) = Ho nA (K . (x), M) ; we no e hei homologies by I-Ii (x, M) and IIi (x, All) espec i ely . These unc- o s Hj(x, .) and H'(x, .) a e descending ; ascending connec ed exac se- quences o unc o s . In he no a ions o 1 .6, we ha e he ollowing esul ([18 ., 6 .1 .6 ; 6 .1 .7]) . 97 0  A .-M . SIMON Theo em . Le x = xl, . . . , xn . gene a e an ideal a in he ing A and ¡el M be an A-module . Then h_ (x, M) = o ' (A/a, M), h- (x, M) _ ex í (A/a, M) . Co olla y .  Le a = (XI, . . . , xn) be a ini ely gene a ed ideal o he ing A, and le M be an A-module . (i) o A (A/a, M ) = ex ~ (A/a, M) (ii) The numbe s h~ (M), ex _ (A/a, M), o A (A/a, M) a e ini e si- mul aneously . In ha case, ex A (A/a, M) + o A (A/a, M) <_ n . (iii) I he numbe s h~l (M), ex ~(A/a, M), o A(A/a, M) a e in ini e, hen, o any ideal a', open in he a-adic opology o A, he num- be s ex - (A/a', M), o A (A/a',111), uá' (M) a e also in ini e . (i ) I : A ---> B is a mo phism o ings, i b = (a)B, hen, o each B-module N, we ha e hb (N) = ex , (B/b, N) = ex A (A/a, N) _ h a - (N), o B(B/b, N) = o A(A/a, N) . P oo . (i) We ha e an isomo phism K'(x,M) -K(x,M'), so o A(A/a,M ) = h_ (x,117") = h - (x, M)= ex A (Ala, Al) . (ii) This is a consequence o he sel -duali y o he Kosmil complex : H'(x, M) - H n_ 2 (x, M) (see [18 ; 6 .1 .8]) . (iii) When an ideal a' is open in he a-adic opology, we ha e a' D a o a ce ain na u al numbe . Using 1 .6, we ob ain oo = o A(A/a, M) = o A(A/a , M) _< o A (A/a', M) = oo .  The open ideal a' being ixed now, we ha e also o A(A/a", M) = oc o all ideals a", open in he a'-adic opology . So he module M belongs o he class C a , (1 .4) and, as M = a'M, we ha e also Uó' (M) = 0 (1 .3) and uá' (M) = oo . (i ) Take he imago y in B o he sequence x gene a ing a : yi . = (xi) . The e a e ob ious isomo phisms K . (y) - K . (x) ®A B, K . (y) ®B N ,_ ., K . (x) ®A N, HOMB (K . (y), N) - HomA(K . (x), N) . So his is ano he consequence o he heo em abo e . No e ha we ha e ob ained he e a change o ings esul o he dep h h a (-) (a ini ely gene a ed) in a si ua ion whe e we don' ha e a change o ings heo em o he local cohomology unc o s 2 . U-codep h In local cohomology, we ha e he equali y h a (M) = ex A (A/a, M) al- eady men ionned (1 .6) . We wan an analogous esul o he U-codep h ADIC-COMPLETION, DUAL RESULTS  97 1 u"_ using To ins ead o Ex . To achie e his, we need some p epa a ion . 2 .1 . The ollowing li ing p oposi ion was p o ed in ([4, 3 .5]) in he local case . P oposi ion . Le a be an ideal con ained in he Jacobson adical o he ing A, and le F be a ía A-'module such ha F/aF is ee as an A/a-module . I {ei¡i E I} is a se , o elemen s o F such ha i s image {~ili E I} in F/aF is a basis o F/aF, hen he se {ei¡i E I} gene a es a pu e ee submodule L o F, , and F =L + aF . P oo ... We i s p o e he eeness o he el in F .  I n I biei = 0, bi E A, we pu b = (b l , . . . . b, y ), e = (e], . . . . en) ; in ma lclal language, we ha e b .e' = 0 . By a la ness c i e ium ([5, I ; Sec ion 2, p oposi ion 13, co olla y 1]), he e is a ma ix X E A'n" and a ec o E FI "1 such ha e = X . c, b .X = 0 .  Deno ing he images modulo he ideal a by -), we ha e é= Ñ j' . Bu he éi o n a basis o F/aF, he ma ix X is hus igh -in e ible, and so is he ma ix X since a is con ained in he Jacobson adical o A . F om b .X = 0 we deduce b = 0 and he eeness o he el in F . We now p o e he pu i y o L in F . As F is la , i is enough o check he injec i i y o he mapsL/eL --> F/cL o each ideal c o A . As he image ei o he elemen s e l o L o m a basis o F/aF, he na u al mo phism L/aL -> F/aF is an isomo phism, and so is L/(a + c)L -> F/(a + c)F . Bu he ideal (a + c)/c o A/c is con ained in he Jacobson adical o A/c . We apply he i s pa o he p oo o he ía A/c-module F/cF and o he images o he el in F/cF : he e images gene a e a ee submodule o F/cF, so he mo phism L/cL -> F/cL is injec i e and L is pu e in F . Now F = L+ aF is clea . 2 .2 . P oposi ion . Le a be an ideal con ained in he Jacobson adical o he ing A, and le M be an A-module wi h M = aM . Then he e exis s an epimo phism P -> M whe e P is a ía A-module wi h P= aP . P oo . Le 0 --> K -> F -> M -> 0 be an exac sequence, whe e F is ee . As M = aM, he sequence K/aK -> F/aF --> 0 is exac . Choose in K elemen s yi whose images in he ee A/a-module F/aF o n a basis 97 2  A .-M . SIMON o F/aF . By 2 .1, hese yi gene a e a ee pu e submodule L o F, and L C K . So P = F/L is ia , and P= aP since F= L+aF . Now he epimo phism F -> M induces an epimo phism P= F/L -> M . 2 .3 . In he p eceding p oposi ion, he condi ion M = aM means ha he To -codep h and he Ua-codep h o M a e posi i e : o A (A/a, M) > 0, ua (M) > 0 (1 .3) . On he o he hand, o he la module P, we ha e o A(A/a, P) = oo = uQ (P) (P belongs o C a , see 1 .4) . So his shows ha he unc ions o A(A/a, .) and ua ( .) sa is y he duals o he axioms o I oh cha ac e izing a homological g ade [9] . I will be used o p o e he equali y be ween he U'-codep h and he To -codep h . When he ing is noe he ian, we ge id o he assump ion on he ideal a by enso izing wi h Á . Indeed, in ha case, Á is A- la , h n = OÁ, á is con ained in he Jacobson adical o Á [1], and we ha e he ollowing easy obse a ion, ex ending ([18, 2 .2 .2]) . Lemma . Le a be an ideal o he noe he ian ing A . Then, o each A-module M, he module 11!1 is isomo phic o he á-adic comple ion o he Á-module Á ®A M, and Ua(M) - UA(Á ®A M), To A(A/a n , M) - To A(Á/á', Á (DA M) o all n> 0 . (I L . --> M -> 0 is a ee esolu ion o M, hen A OA L is a ee esolu ion o he Á-module Á ®A M, and Á/án ®á (Á &A L .) - Á/án ®A L . -- A/an (DA L . . This gi es he esul , a e aking limi s o he U-pa o i ) . 2 .4 . Theo em . Le a be an ideal o he ing A . a is con ained in he Jacobson adical o A o i A is noe he ian, hen, o each A-module M ; uá (M) = o A (A/a, M) . P oo . . We al eady know ha uá (M) and o A (A/a, M) anish simul ane- ously, exac ly when M =A a .1V1 (1 .3) . Wi h 2 .3, we a e educed o he i s case, whe e a is con ained in he Jacobson adical o A . In ha case, i one o he numbe s abo e is posi i e ini e, we ha e an exac sequence 0 --> M l --> P --> M -> 0, whe e P is la and P = aP (2 .2) . The long exac sequences associa ed wi h i shows o ' (A/a, N1 1 ) = o ' (A/a, M) - l, .ua (Ml) = ua (M) - .l . So 2 .5 . ADIC-COMPLETCON, DUAL RESULTS  97 3 by an induc ion a gumen we ha e u' (AI) = o A(A/a, Al1) . This shows also lia hese wo numbe s a e in ini o simul aneously . P oposi ion .  Unde he hypo hesis o 2 .4, i = u°_ (M) < oo and i he ideal a is ini ely gene a ed, hen U,"(M) ^ = lim To (A/a", M) . P oo .. This is done by induc ion on , using an exac sequence as in 2 .4 (a e ha ing enso ed by A in he noe he ian case), he case = 0 is 1 .3 . 3 . U-dimension and H-dimension o e a noe he ian ing We now s udy he dimensions u+ (M) and há (M) as de ined in 1 .6 . Ou ings a e now noe he ian . In local coho nology, i is known ha h a (M) <_ di n A4I o each A- module M [15] (mo eo e , i All is ini ely gene a ed and i a = m is he maximal ideal o a local ing ; hen h ,- 4 , - ,(M) = dim AJ [11]) . We s ablish an analogous inequali y o he U-dimension . This in u n allows i_is o e ine he inequali y abo e . To achie e his, we need a chango o ings heo em o he U ¡ ', analogous o he co esponding one in local cohomology . 3 .1 . Le Ali, i E I, be a amily o modules o e he noe he ian ing A . Then (®ilhli) ^ = {w E lliÑ i l o all n, all bu ini ely many compo- nen s i i o w belong o a n ú¿} ([16, 9 .4]), so (®iA4i) ^ = When 1Vl i - A1 o all i, we w i e as usual AI(') = ® i M i , MI= IIiMi . . The ollowing lemina was obse ed in ([14, p . 77 . 2 .4 .2]) . Lemma . Le I be a se . Each sho exac sequence 0 -> M' - Al1 -> AI" -> 0 o ini ely gene a ed modules o en he noe he ian ing A gi es ise o an exac sequence 0 -)  (M1(1))^ -- *U , (Aj(I))^ -- V --> (A4,i (1))^  > 0