ADIC-COMPLETION AND SOME DUAL HOMOLOGICAL RESULTS

ADIC-COMPLETION AND SOME DUAL 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 Ha-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

In this paper A is a commutative ring, a an ideal of A and .theAmodules 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 both local cohomology and local homology.So local cohomology and local homology are not only duals of each other, but aleo intimately connected .
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.

.. 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) .

. 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.

.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 = fAl -1-aN .
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] .
1 .4.The class Ca .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 Ca, and it is wortllwile to note that flat rnodules belong tO Ca .
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 Ca ( [12, corollary 4.5]) .

.5 . Local cohomology and Matlis duality .
Recall the functor Ha' : 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 : when N has a projective resolution composed of finitely generated modules ([6, VI, 5.1 ; 5.3]) : When the ring is noetherian, Hó(M) v -(Mv ) ^and Há(M)v -U°'(Mv) 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 .

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 Ga = 0 for n < 0 and if each exact sequence 0 -+ M' ~11N1 -> M" -0 in A gives rise to a long exact sequence Dually, f-(M) for of .These tively.
Proposition .Let, a. C b be ideals in the ring A,  ) .We want an analogous result for the U-codepth u"_ using Tor instead of Ext .To achieve this, we need some preparation .
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 nI biei = 0, bi E A, we put b = (bl , . . . .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 et = 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 maps L/eL --> F/cL for each ideal c of A .As the image ei of the elements el 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 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 .

.3.
In the preceding proposition, the condition M = aM means that the Tor-codepth and the Ua-codepth of M are positive : torA (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 Ca , 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, hn = 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 --> Ml --> P --> M -> 0, where P is flat and P = aP (2.2) .The long exact sequences associated with it shows tor' (A/a, N11 ) = tor' (A/a, M) -l, .ua(Ml) = ua (M) -.l .So 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 Tort (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 .

. 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.
As Á is A-flat and as X = A ®A X when X is a finitely generated module, by tensorizing with Á we are reduced to the case where A and our finitely generated modules are complete in the a-adic topology .In that case; the sequence above is a restriction of the exact sequence 0 -> M' I + MI -> 1V1"I -> 0, so ú is injective .We already know that v is surjective (1.2) .Let w E kerb = (M( I>) ^n AI'I .By the Artin-Reos lemma, there is a natural number c such that, for all n > 0; an+°AI n M' = a7'(a°M n M') .As w E for each natural number n there is a finite subset Jn of I such that, Vi 1 J,n , wi E an+IM .So; Vi 1 Jn, wi E an+o1l n M' C anM'; this means w E (M'('» A a.nd finishes the proof.a Remark . By exercising a little more, one can prove that (M(I))^i s in fact a pure A-submodule of MI when M is finitely generated (see [18, 2.1 .9],for the case M = A) .As a consequence, for all ideal b of A ; we have an isomorphism (11VI(I>)^/b .(A4I(4)^-((A1/bNl)~I>)^. Indeed ; as b = bÁ ([5, Section 3, 4; corollary l]), we are again reduced to the case where A is complete .In that case, we apply the lemma to the sequence 0 --> bM -> M --> AI/bAl --4 0, we obtain ((bAT)') ^= (M(I))^n (bM)I = (M(I))^nb .M' = b.(M('»^and the desired isomorphism .In particular, for a free A-module L, me have (L/bL)^-L/bL .

.2 .
Proposition .Let M be a finitely generated module over the noethe- rian ring A, and let I be a set.Then the module N = MU) belongs to the classe Co : Uó (N) = Ñ, Ui (N) = 0 for i > 0.
Proof.. Let . . .L1 > Lo -.> M -> 0 be a resolution of M, where the Li are finitely generated free .Put Mi = im di .The short exact sequences 0 -> Mi+i -> Li -> NIi --> 0 give rise to exact sequences by 3.1 .But L( I) is a free resolution of M(I), so we have the result.
Theorem .Let f : R -> A be a morphism of noetherian rings such that A is finitely generated as an R-module .Let, r be an ideal of R and a = f (r)A be its extension in A .Then U? and U2`form naturally isornorphic descending connected exact sequences of functors from Amodules to R-modules .Proof. .An A-module 111 can be viewed as a R-module; on A1, the a-adic and the r-adic topology are the serme .Take a free resolution . . .L 1 -> Lo --> 111 -> 0 of M as an A-module .By (3 .2), the modules Li belong to the classes C, and C,.introduced in (1 .4) .Completing, we get the natural isomorphisms U? (111) -Hj(L .)-U"(M) .

.
The preceding theorem is useful for the computation of the Ui (M), it brings us back to the first case studied by Matlis, namely the case where the ideal a is generated by a regular sequence.Indeed, if a = (x1) . . .; xn), we use a chango of rings B = A[X1, . . ., X,ti] f> A, where B is a polynomial ring in the indeterminates Xi, where f is defined by f (Xi ) = xi .The regular sequence XI , . . ., X on the ring B generates an ideal b and, for each A-module M, for all i, UQ(M) -Ub(M) .
This allows generalizations of sorne Matlis results.Here is a first one (see [13]) .
Corollary .If the ideal a is generated by a sequence x l , . ..,x,, regular on the module 111, then A1 belongs to Ca .
Proof. .Using a chango of rings as above, we are reduced to the case where the sequence x = x1, . . . ,x, is regular on both A and M .The sequence xt = xi, . . ., x;l generate an ideal at , the Koszul complex K. (xt) is a finite free resolution of A/at , and we have Hi (xt,A1) = Tor.i (A/at,A1) = 0 for all i > 0 since the sequence xt is regular on M .As noted in (1 .4)this implies M E Ca .

.5 .
We say that an ideal a of the noetherian ring A can be generated by n elements up to radical equivalente if there exist elements x1, . . .A.-M .SIMON Assume a = (xl, . . ., xn) .Matlis proved that U'(M) = 0 for i > n if the sequence x1, . .; xn is regular on the ring A ([12, theorem 4 .12]) .Greenless and May obtained the same result in a more general situation .The change of rings theorem allows a slight refinement of this.However ; Matlis method dualized gives also a very elementary proof of the corresponding result in local cohomology; in a somewhat unusual formulation which will be useful later.That is why we insert it here.
Lemma.Let M be a 7nodule over the noetherian ring A. If the image of the ideal a in A/ AnnA M can be generated by n elements vp to radical equivalente, then u+(M) < n and há (M) < n.

Proof::
We first use a change of rings A -> A/ AnnA 111, then a change of rings as described in 3.4, and we are reduced to the case where the ideal a is generated by a regular sequence xl, . . ., xn.Write again at = (xi ; . . ., xñ) .As A/at is of finite projective dimension n and as Ha(M) = limExtÁ(A/at, M) ( [15], or [18, 4.1.3]),we have Ha(M) = 0 for i > n.

.6 .
Here is another generalization of a Matlis result.
Note also that these inequalities might be strict .Example .
In ([16, 9.4]), we showed a complete module M over a regular local ring of dimension d > 1, such that AnnA M = 0, Supp M = Spec A (so that dimM --dim A), and such that h-(M) <_ hm (M) < d.As that module is complete, we have also u-(M) = 0 = u+ (M) (see 1 .4,M E C-,.) .For that module both inequalities are strict .
1. How can we refine the above inequalities?2. If M is artinian, then u+ (M) = dim A/ AnnA M. Is this still true for a module M with M = H°,n(lll)?
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, Mv) = 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,Mv) = 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) -Hn_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 Ca, (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

2. 5 .
, xn generating an ideal b with rad a = rad b.Then the a-adic and the b-adic topology are the saine, and U°(M) = Ub(M) .

3. 8 .
The last results are more suggestive in the local case.Theorem .Let A be a noetherian local ring of maximal ideal ?n, and N an A-module .Then (i) u+ (M) <_ dim A/ AnnA A11 l-hZ (M) (ii) há (M) < dim A/ AnnA M -u°_' (M) Note that we cannot replace dirn A/ AnnA M by dim M .Example .
the numbers ext A M) only depend on topology defined by the ideal a.