Adic-completion and some dual homological results

Publicacions Matemátiques, Vol 36 (1992), 965-979 . A bstract ADIC-COMPLETION AND SOMEDUAL HOMOLOGICAL RESULTS ANNE-MARIE SIMON To the memory of Pere Menal Let a be an ideal of a commutative ring A . There is a kind of duality between the left derived functors Ui of the a-adic completion functor, called local homology functors, and the local cohomology functors Há . Some dual results are obtained for these Ua, and also inequalities involving both local homology and local cohomology when the ring A is noetherian or more generally when the U° and H a -global dimensions of A are finite . In this paper A is a commutative ring, a an ideal of A and . the Amodules are given the a-adic topology . There is a certain duality between the left derived functors Ui of the a-adic completion functor and the local cohomology functors H,,, first observed by Matlis when the ideal a is generated by a (finite) regular sequence, true also for any noetherian ring . More recently, that duality has also been observed by Creenlees and May in a more general context . The purpose of this note is to pursue the analogy between the local cohomology functors and these functors Uá, called local homology functors by Creenlees and May . First we have dual results about codepth, a notion dual to the notion of homological depth or grade . To go further, we need come noetherian hypothesis in order to have a chango of rings theorem for the Ui ; arlalogous to the corresponding one in local cohomology . This brings us back to the first case studied by Matlis, namely the case of an ideal generated by a regular sequence, and allows generalizations of some Matlis results . As a consequence, we obtain vanishing results for the U°, and also inequalities involving 96 6  A .-M . SIMON both local cohomology and local homology . So local cohomology and local homology are not only duals of each other, but aleo intimately connected . As general references for commutative algebra and homological questions, we quote [1], [18] . The work below has been communicated at the International workshop on local cohomology, geometric applications and related topics . We take this opportunity to thank the organizers and the participante for useful discussions . In this first section we fix notations and collect the material we need . Though part of it appeared in different places, we think it is more convenient to have it at hand . 1 .1 . Completion . Let a be an ideal of the commutative ring A . The A-module are given the a-adic topology . The completion of an A-module M is denoted by 1Vl : thus M = lim M/a'M . Let Tn,1 : M --> M be the natural morphism . 3J) . 1 . Preliminaries and notations Here and in the next section, the ideal a is not necessarily finitely generated ; and it might happen that the A-module M, complete in its natural topology, is not complete in its a-adic topology . An example of this can be found in ([5, III, Section 2, exercise 12]) or in ([3, I ; Section Recall however the following result ([3, theorem 1 .3 .1], or [13, theorem 15], or [18, 2 .2 .5]) . Theorem . Suppose the ideal a finitely generated . Let M be an Amodule and b an open ideal in the a-adic topology of A . Then the morphism-rmoA/b : M/bM -> M/blVl is an isomorphism . So M is complete in its a-adic topology . 1 .2 . When f is onto . The a-adic completion functor, though not right exact, preserves surjection . However, we want to know precisely when the completion of a morphism is onto . The following lemma was proved in ([16, 1 .2]) for a noetherian ring A, using 1 .1 . It is true in general . Lemma . Let f : M ---> N be a morphism of A-modules .  Then f su7-jective if and only if N = f Al -1aN . ADIC-COMPLGTION, DUAL RESULTS  967 Proof .. If N = f Alf + aN, then N = f M + anN for all n >_ 1, and the projective system of sequences 0 ---> f -1 ( a 'ti N) la"M ---> Ml a't M --> N/a' N ---> 0 is exact . It is easily seen to be surjective (see [16, 1 .2]) . So, after taking limits, we get an exact sequence and f is surjective . Conversely, if f is surjective, we tensor the commutative natural diagram M N M/aM  > N/aN, where the vertical composite maps are the natural projections, with Ala . The composite vertical maps become the identity, so M/aM -> N/aN is surjective and N = f M +aN . 1 .3 . The left derived functors of the completion functor . The left derived functors of the a-adic completion functor are denoted by Ui . These were first studied by Matlis when the ideal is generated by a finite regular sequence [12], [13] . We used them in [16], where the ring is noetherian . More recently, they have been computed by Greenlees and May in a more general situation [7] . Let L 1 -f > Lo -> A1  + 0 be arr exact sequence, with Lo, L 1 free . By definition Uo (NI) = coker .Í, so we have natural morphisms A1 Uó (A1) -> M whose composite is -rAr . The following lernma ([16, 5 .1]), consequence of 1 .2, is still available . Lemma . (i) The natural morj)hism UÓ (M) -> NI is onto . (ii) M = 0 if and only if M = aA1 if and only if Uó (M) = 0 . (iii) If the ideal a is finitely generated, then (Uó (M))^ = A1 . 1 .4 . The class C a . Let C, be the class of modules M such that Uó (M) = A7 and Ui (M) _ 0 for i > 0 . A standard homological argument shows that Uá(A1) can be computed using a left resolution of A1 with modules in C a , and it is wortllwile to note that flat rnodules belong tO C a . 968  A .-M . SIMON More generally, let a¡,, n > 0, be a decreasing sequence of ideals which form a basis of the a-adic topology . A module M such that TorA(A/a  ,, M)= 0 for all i > 0 and all n > 0 belongs to C a ([12, corollary 4 .5]) . When the ring is noetherian, the completion of a free module is flat ([14, p . 77], or [3, 4, 7], or [18, 2 .2 .4]) . This can be used to show that complete modules belong to C a when A is noetherian ([16, 5 .2]) . 1 .5 . Local cohomology and Matlis duality . Recall the functor H a' : H°(M) = {x E Mjanx = 0 for some natural number n}, whose right derived functors Há are the local cohomology functors . Recall also the Matlis duality . Let E be the injective hull of the direct sum of all the A/m with m, a inaximal ideal of A . The Matlis duality functor, defined by Mv = HOMA(M, E), is faithfully exact [12] ; [13], and we have the Ext-Tor duality : TorA(N, M) v - ExtÁ(N, M v ) ; when N has a projective resolution composed of finitely generated modules ([6, VI, 5 .1 ; 5 .3]) : TorA(N, M v ) - ExtÁ(N, M) v . When the ring is noetherian, Hó(M) v - (M v ) ^ and Há(M)v - U°'(M v ) for all i ([16, 4 .2 ; 5 .6]) . This is based on the fact that, over a noetherian ring, flat modules and injective modules are interchanged by Matlis duality . This was first proved by Matlis when the ideal a is generated by a regular sequence . But modules are not necessarily duals, so informations about the local cohomology functors Hi, do not always provide informations about the local homology functors Ui . 1 .6 . Formal depth, codepth and dimension . A sequence of covariant additive functors G,,, : A  + B, n E Z, between abelian categories is a descending connected exact sequence of functors if G a = 0 for n < 0 and if each exact sequence 0 -+ M' ~ 11N1 -> M" - 0 in A gives rise to a long exact sequence . . . -> Gi+i(01) -> GZ(M') ) Gi(M) -, Gi(M") -> . . . in a functorial way . ADIC-COMYLETION, DUAL RESULTS  96 9 When we have such a sequence, as in ([18, 1 .1] or [17]), we put g_(M) = inf{¡IG i( M) =,,~ o} for each object M in ,A (so that o <_ g_ (M) < oo) . Dually, we define f- (M) for an ascending connected exact sequence of functors F" in the saíne way . These numbers can be viewed as a kind of codepth or depth respectively . When a is an ideal of the ring A, we are merely concerned with the sequence Ext ;~(A/a, .), TorA(A/a, .), H,'~( .), Ui (-) and with the corresponding numbers . Here are some first remarks about them, which will be completed later (1 .7, 2 .4) . Proposition . Let, a . C b be ideals in the ring A, let M an A-module (i) ext A (A/a, AI) = ha (Ah) (ii) tor'(A/a, M) = ext-(A/a, M") (iii) extA (A/a, M) < ext A (A/b, Al1) (iv) torA (A/a, M) < torA (A/b, M) (v) the numbers ext A (A/a, M), torA(A/a, M) only depend on the topology defined by the ideal a . For (i) and (iii), see ([18, 5 .3 .1 .5, 5 .3 .11]) ; (ii) is a direct consequence of the Ext-Tor duality 1 .5 ; (iv) follows from (ii) . and (iii) . Since local cohomology only depends on the topology defined by the ideal a, so does extA(A/a, .) in view of (i), and so does also torA(A/a, .) in view of (ii) . We also define g+ (M) = sup{ilC i (M) :y~ 0} (so that g+(M) = -oo or 0 _< g + (A11) < oo) and f+(AI) in the saíne way . These last numbers are relative homological dimensions . 1.7 . The depth-codepth sensitivity of the Koszul complex . The depth sensitivity of the Koszul complex was proved by Barger and Hochster for a coherent ring [2], [8], by Kirby and Mehran for any commutative ring [10] . An approach involving both depth and codepth can be found in ([18, 6 .1] or [17]) . Let x = x I ,. . . , x, 1 be a sequence of eleinents of the ring A generating an ideal a, and let K . (x) be the associated Koszul complex . For an A-module M, ;ve consider the descending and ascending Koszul complexes K . (x, All) = K . (x) ®A A7, K" (x,1Vl) = HornA (K . (x), M) ; we note their homologies by I-Ii (x, M) and IIi (x, All) respectively . These functors Hj(x, .) and H'(x, .) are descending ; ascending connected exact sequences of functors . In the notations of 1 .6, we have the following result ([18 ., 6 .1 .6 ; 6 .1 .7]) . 97 0  A .-M . SIMON Theorem . Let x = xl, . . . , xn . generate an ideal a in the ring A and ¡el M be an A-module . Then h_ (x, M) = tor' (A/a, M), h- (x, M) _ extí (A/a, M) . Corollary .  Let a = (XI, . . . , xn) be a finitely generated ideal of the ring A, and let M be an A-module . (i) torA (A/a, M v ) = ext~ (A/a, M) (ii) The numbers h~ (M), ext_ (A/a, M), torA (A/a, M) are finite simultaneously . In that case, extA (A/a, M) + torA (A/a, M) <_ n . (iii) If the numbers h~l (M), ext~(A/a, M), torA(A/a, M) are infinite, then, for any ideal a', open in the a-adic topology of A, the numbers ext - (A/a', M), torA (A/a',111), uá' (M) are also infinite . (iv) If f : A ---> B is a morphism of rings, if b = f (a)B, then, for each B-module N, we have hb (N) = ext, (B/b, N) = ext A (A/a, N) _ h a - (N), torB(B/b, N) = torA(A/a, N) . Proof . (i) We have an isomorphism K'(x,M)v-K(x,M'), so torA(A/a,M v ) = h_ (x,117") = h - (x, M)= ext A (Ala, Al) . (ii) This is a consequence of the self-duality of the Kosmil complex : H'(x, M) - H n_ 2 (x, M) (see [18 ; 6 .1 .8]) . (iii) When an ideal a' is open in the a-adic topology, we have a' D ar for a certain natural number r . Using 1 .6, we obtain oo = torA(A/a, M) = torA(A/ar, M) _< torA (A/a', M) = oo .  The open ideal a' being fixed now, we have also torA(A/a", M) = oc for all ideals a", open in the a'-adic topology . So the module M belongs to the class C a , (1 .4) and, as M = a'M, we have also Uó' (M) = 0 (1 .3) and uá' (M) = oo . (iv) Take the imago y in B of the sequence x generating a : yi . = f (xi) . There are obvious isomorphisms K . (y) - K . (x) ®A B, K . (y) ®B N ,_ ., K . (x) ®A N, HOMB (K . (y), N) - HomA(K . (x), N) . So this is another consequence of the theorem above . Note that we have obtained here a change of rings result for the depth h a (-) (a finitely generated) in a situation where we don't have a change of rings theorem for the local cohomology functors 2 . U-codepth In local cohomology, we have the equality h a (M) = ext A (A/a, M) already mentionned (1 .6) . We want an analogous result for the U-codepth ADIC-COMPLETION, DUAL RESULTS  97 1 u"_ using Tor instead of Ext . To achieve this, we need some preparation . 2 .1 . The following lifting proposition was proved in ([4, 3 .5]) in the local case . Proposition . Let a be an ideal contained in the Jacobson radical of the ring A, and let F be a fíat A-'module such that F/aF is free as an A/a-module . If {ei¡i E I} is a set, of elements of F such that its image {~ili E I} in F/aF is a basis of F/aF, then the set {ei¡i E I} generates a pure free submodule L of F, , and F =L + aF . Proof ... We first prove the freeness of the el in F .  If r n I biei = 0, bi E A, we put b = (b l , . . . . b, y ), e = (e], . . . . en) ; in matrlclal language, we have b .e' = 0 . By a flatness criterium ([5, I ; Section 2, proposition 13, corollary 1]), there is a matrix X E A'n" and a vector f E FI "1 such that e t = X . f c, b .X = 0 .  Denoting the images modulo the ideal a by -), we have é= Ñ j' . But the éi forrn a basis of F/aF, the matrix X is thus right-invertible, and so is the matrix X since a is contained in the Jacobson radical of A . From b .X = 0 we deduce b = 0 and the freeness of the el in F . We now prove the purity of L in F . As F is flat, it is enough to check the injectivity of the mapsL/eL --> F/cL for each ideal c of A . As the image ei of the elements e l of L form a basis of F/aF, the natural morphism L/aL -> F/aF is an isomorphism, and so is L/(a + c)L -> F/(a + c)F . But the ideal (a + c)/c of A/c is contained in the Jacobson radical of A/c . We apply the first part of the proof to the fíat A/c-module F/cF and to the images of the el in F/cF : there images generate a free submodule of F/cF, so the morphism L/cL -> F/cL is injective and L is pure in F . Now F = L+ aF is clear . 2 .2 . Proposition . Let a be an ideal contained in the Jacobson radical of the ring A, and let M be an A-module with M = aM . Then there exists an epimorphism P -> M where P is a fíat A-module with P= aP . Proof . Let 0 --> K -> F -> M -> 0 be an exact sequence, where F is free . As M = aM, the sequence K/aK -> F/aF --> 0 is exact . Choose in K elements yi whose images in the free A/a-module F/aF forrn a basis 97 2  A .-M . SIMON of F/aF . By 2 .1, these yi generate a free pure submodule L of F, and L C K . So P = F/L is fiat, and P= aP since F= L+aF . Now the epimorphism F -> M induces an epimorphism P= F/L -> M . 2 .3 . In the preceding proposition, the condition M = aM means that the Tor-codepth and the Ua-codepth of M are positive : tor A (A/a, M) > 0, ua (M) > 0 (1 .3) . On the other hand, for the flat module P, we have torA(A/a, P) = oo = uQ (P) (P belongs to C a , see 1 .4) . So this shows that the functions torA(A/a, .) and ua ( .) satisfy the duals of the axioms of Itoh characterizing a homological grade [9] . It will be used to prove the equality between the U'-codepth and the Tor-codepth . When the ring is noetherian, we get rid of the assumption on the ideal a by tensorizing with Á . Indeed, in that case, Á is A-flat, h n = OÁ, á is contained in the Jacobson radical of Á [1], and we have the following easy observation, extending ([18, 2 .2 .2]) . Lemma . Let a be an ideal of the noetherian ring A . Then, for each A-module M, the module 11!1 is isomorphic to the á-adic completion of the Á-module Á ®A M, and Ua(M) - UA(Á ®A M), TorA(A/a n , M) - TorA(Á/á', Á (DA M) for all n> 0 . (If L . --> M -> 0 is a free resolution of M, then A OA L is a free resolution of the Á-module Á ®A M, and Á/án ®á (Á &A L .) - Á/án ®A L . -- A/an (DA L . . This gives the result, after taking limits for the U-part of it) . 2 .4 . Theorem . Let a be an ideal of the ring A . ff a is contained in the Jacobson radical of A or if A is noetherian, then, for each A-module M ; uá (M) = torA (A/a, M) . Proof . . We already know that uá (M) and torA (A/a, M) vanish simultaneously, exactly when M =A a .1V1 (1 .3) . With 2 .3, we are reduced to the first case, where a is contained in the Jacobson radical of A . In that case, if one of the numbers above is positive finite, we have an exact sequence 0 --> M l --> P --> M -> 0, where P is flat and P = aP (2 .2) . The long exact sequences associated with it shows tor' (A/a, N1 1 ) = tor' (A/a, M) - l, .ua (Ml) = ua (M) - .l . So 2 .5 . ADIC-COMPLETCON, DUAL RESULTS  97 3 by an induction argument we have u' (AI) = torA(A/a, Al1) . This shows also tliat these two numbers are infinito simultaneously . Proposition .  Under the hypothesis of 2 .4, if t = u°_ (M) < oo and if the ideal a is finitely generated, then U,"(M) ^ = lim Tor t (A/a", M) . Proof .. This is done by induction on t, using an exact sequence as in 2 .4 (after having tensored by A in the noetherian case), the case t = 0 is 1 .3 . 3 . U-dimension and H-dimension over a noetherian ring We now study the dimensions u+ (M) and há (M) as defined in 1 .6 . Our rings are now noetherian . In local cohornology, it is known that h a (M) <_ dirn A4I for each Amodule M [15] (moreover, if All is finitely generated and if a = m is the maximal ideal of a local ring ; then h ,- 4 , - ,(M) = dim AJ [11]) . We stablish an analogous inequality for the U-dimension . This in turn allows i_is to refine the inequality above . To achieve this, we need a chango of rings theorem for the U ¡ ', analogous to the corresponding one in local cohomology . 3 .1 . Let Ali, i E I, be a family of modules over the noetherian ring A . Then (®ilhli) ^ = {w E lliÑ i l for all n, all but finitely many components ivi of w belong to a n ú¿} ([16, 9 .4]), so (®iA4i) ^ = When 1Vl i - A1 for all i, we write as usual AI(') = ® i M i , MI= IIiMi . . The following lemina was observed in ([14, p . 77 . 2 .4 .2]) . Lemma . Let I be a set . Each short exact sequence 0 -> M' - Al1 -> AI" -> 0 of finitely generated modules oven the noetherian ring A gives rise to an exact sequence 0 -)  (M1(1))^ -- *U , (Aj(I))^ -- V --> (A4,ir(1))^  > 0