ON ARTINIANNESS OF FORMAL LOCAL COHOMOLOGY, COLOCALIZATION
AND COASSOCIATED PRIMES
MAJID EGHBALI
Abstract
This paper at first concerns some criteria onArtinianness and vanishing of formal local cohomology modules. Then we consider the cosupport and the set of coassociated primes of these modules more precisely.
1. Introduction
Throughout,ᑾis an ideal of a commutative Noetherian ringR andM anR- module. LetV (ᑾ)be the set of prime ideals inRcontainingᑾ. For an integer i, letHᑾi(M)denote thei-th local cohomology module of M. We have the isomorphism of Hᑾi(M) to lim
−→n ExtiR(R/ᑾn, M)for every i ∈ Z, see [2] for more details.
Consider the family of local cohomology modules{Hᒊi(M/ᑾnM)}n∈N. For everynthere is a natural homomorphismHᒊi(M/ᑾn+1M)→ Hᒊi(M/ᑾnM) such that the family forms a projective system. The projective limitᑠiᑾ(M):= lim←−n Hᒊi(M/ᑾnM)is called thei-th formal local cohomology ofMwith respect toᑾ. Formal local cohomology modules were used by Peskine and Szpiro in [12] whenR is a regular ring in order to solve a conjecture of Hartshorne in prime characteristic. It is noteworthy to mention that ifU = Spec(R)\ {ᒊ} and(U , Ou)denote the formal completion ofU alongV (ᑾ)\ {ᒊ}and alsoF denotes theOu-sheaf associated to lim←−
n
M/ᑾnM, they have described the formal cohomology modulesHi(U , Ou)via the isomorphismsHi(U , Ou)∼=ᑠiᑾ(M), i≥1. See also [11, proposition (2.2)] whenRis a Gorenstein ring.
Letx= {x1, . . . , xr}denote a system of elements such thatᒊ= Rad(x). In [15], Schenzel has studied formal local cohomology module via following
Received 10 January 2011, in final form 5 December 2011.
isomorphism
lim←−n Hᒊi(M/ᑾnM)∼=Hi
lim←−n (Cˇx⊗M/ᑾnM) whereCˇxdenotes the ˇCech complex ofRwith respect tox.
When the local ring(R,ᒊ)is a quotient of a local Gorenstein ring(S,ᒋ), we have
(1.1) ᑠiᑾ(M)∼=HomR
Hᑾdim S−i(M, S), E
, i∈Z
whereEdenotes the injective hull ofR/ᒊandᑾis the preimage ofᑾinS(cf.
[15, Remark 3.6]).
Important problems concerning local cohomology modules are vanishing, finiteness and Artinianness results (see, e.g., [6]). In Section 2 we examine the vanishing and Artinianness of formal local cohomology modules. In the next theorem we give some criteria for vanishing and Artinianness of formal local cohomology modules:
Theorem1.1. Let (R,ᒊ)be a local ring and M be a finitely generated R-module. For given integersiandt >0, the following statements are equi- valent:
(1) SuppR(ᑠiᑾ(M))⊆V (ᒊR) for alli < t; (2) ᑠiᑾ(M)is Artinian for alli < t;
(3) SuppR(ᑠiᑾ(M))⊆V (ᑾR) for alli < t; (4) ᑾ⊆Rad(AnnR(ᑠiᑾ(M)))for alli < t;
Suppose thatt ≤depthM, then the above conditions are equivalent to (5) ᑠiᑾ(M)=0for alli < t;
whereRdenotes theᒊ-adic completion ofR.
It should be noted that it has been shown independently in [8] that statements (2) and (4) are equivalent.
Note that as we see in Theorem 1.1, we have the equivalence between SuppR(ᑠiᑾ(M))⊆ V (ᑾR) for alli < t andᑾ ⊆ Rad(AnnR(ᑠiᑾ(M)))for all i < t, which is not true in general for an arbitrary module.
In Section 3, we study the cosupport of formal local cohomology via Richardson’s definition of colocalization (cf. Definition 3.1). We show that when(R,ᒊ)is a local ring,M is a finiteR-module andᑠiᑾ(M)is Artinian (i∈Z), then CoSupp(ᑠiᑾ(M))⊆V (ᑾ)(cf. 3.5). As a further result we reduce to the caseM = Rwhen considering the cosupport of top formal local co- homology modules which is the analogue for formal local cohomology of the result due to Huneke-Katz-Marley in [7, Proposition 2.1]:
Theorem1.2. Let (R,ᒊ)be a local ring. Let M be a finitely generated R-module. Then
(1) CoSupp(ᑠcᑾ(M))=CoSupp(ᑠcᑾ(R/J )), (2) Supp(ᑠcᑾ(M))=Supp(ᑠcᑾ(R/J )), whereJ isAnnR(M)andc:=dimR/ᑾ.
For a representable moduleM, CoSupp(M) = V (AnnM)(cf. [13, The- orem 2.7]). It motivates us to see when the cosupport of formal local co- homology module is a closed subset of SpecR in Zariski topology. For this reason in Section 4 we study the set of coassociated primes of formal local cohomology more precisely. In this direction when(R,ᒊ)is a local ring and M is anR-module, the set of minimal primes in CoSupp(M)is finite if and only if CoSupp(M) is a closed subset of Spec(R)(Lemma 4.2). Hence, it is enough to ask when the CoassM is finite. We give affirmative answers to this question in some cases, see Proposition 4.4 and Theorem 1.4 below.
It is noteworthy that for a finitely generated module M over a local ring (R,ᒊ), CoassR(ᑠ0ᑾ(M))is finite sinceᑠ0ᑾ(M)is a finiteR-module (cf. [15, Lemma 4.1]) and Coass(ᑠdimᑾ (M))is finite as ᑠdimᑾ M(M)is an Artinian R- module (cf. [1, Lemma 2.2] or Proposition 2.1).
As final results in Section 4, we give the following results for top formal local cohomology modules:
Theorem1.3.Let(R,ᒊ)be a local ring of dimensiond >1. Letᑠdᑾ(R)= 0. Then:
(1) Ifᒍ∈Coass(ᑠd−ᑾ 1(R)), then it implies thatdim(R/(ᑾ,ᒍ))=d−1.
(2) Assh(R)∩Coass(ᑠd−ᑾ 1(R))⊆ {ᒍ∈Spec(R): dim(R/ᒍ)=d,Rad(ᑾ+ ᒍ)=ᒊ}.
(3) IfCoass(ᑠd−ᑾ 1(R))⊆Assh(R), then{ᒍ∈ Spec(R): dim(R/(ᑾ,ᒍ)) = d−1} ⊆Coass(ᑠd−ᑾ 1(R)).
Next result shows that for a one dimensional idealᑾof a complete local ringRof dimensiond, Cosupp(ᑠd−ᑾ 1(R))is closed.
Theorem1.4.Let(R,ᒊ)be a local complete ring of dimensiond. Letᑾ be an ideal of dimension one. Then
ᑠd−ᑾ 1(R)=0, when d >2, in particularCoassR(ᑠd−ᑾ 1(R))= ∅.
CoassR(ᑠd−ᑾ 1(R))⊆ {ᒊ}, when d =1,
and in the cased =2we have CoassR(ᑠd−ᑾ 1(R))=r
i=1
CoassR(Rᒍi)
= {ᒍ1, . . . ,ᒍr} ∪ s
j=1
{ᒎj :Rᒍi/ᒎjRᒍi is not complete}
,
whereᒍ1, . . . ,ᒍr are minimal prime ideals ofᑾ andᒎ1, . . . ,ᒎs are minimal prime ideals ofRwithᒎj ⊆ᒍi fori∈ {1, . . . , r}.
In particularCosupp(ᑠd−ᑾ 1(R))is closed for alld >0.
My thanks are due to my phd. adviser, Professor Peter Schenzel, for his guidance to prepare this paper and useful hints and to the reviewer for sug- gesting several improvements. Some parts of this paper was written while the author was at Oberwolfach: Representations of Finite Groups, Local Cohomo- logy and Support. Many thanks to the organisers.
2. On Artinianness of ᑠiᑾ(M)
Important problems concerning local cohomology modules are vanishing, fi- niteness and Artinianness results. In the present section we study the vanishing and Artinianness conditions of formal local cohomology modules as our main result. Not so much is known about the mentioned properties. In [1] Asghar- zadeh and Divani-Aazar have investigated some properties of formal local cohomology modules. For instance they showed that ᑠdᑾ(M)is Artinian for d :=dim(M). Here we give an alternative proof of it with more information on the attached primes ofᑠdᑾ(M):
Proposition2.1.Letᑾbe an ideal of a local ring(R,ᒊ)andM a finitely generatedR-module of dimensiond. Thenᑠdᑾ(M)is Artinian. Furthermore
AttR(ᑠdᑾ(M))= {ᒍ∈Ass(M): dim(R/ᒍ)=d} ∩V (ᑾ).
Proof. By Independence Theorem we may assume that Ann(M)=0 and sod =dim(R). AsHᒊd(M/ᑾnM)is right exact (n∈N), we have
Hᒊd(M/ᑾnM)∼=Hᒊd(R)⊗RM/ᑾnM
∼=Hᒊd(M)⊗RR/ᑾn
∼=Hᒊd(M)/ᑾnHᒊd(M).
SinceHᒊd(M)is an Artinian module so there exists an integern0such that for all integert ≥n0we haveᑾtHᒊd(M)=ᑾn0Hᒊd(M). Then one can see that
ᑠdᑾ(M)∼=Hᒊd(M)/ᑾn0Hᒊd(M),
which is an Artinian module. By virtue of above equations and [2, The- orem 7.3.2], the second claim is clear.
Lemma2.2.Let(R,ᒊ)be a complete local ring andMa finitely generated R-module. Then Supp(ᑠ0ᑾ(M)) = ᒍ∈AssR(ᑠ0ᑾ(M))V (ᒍ). Moreover Supp(ᑠ0ᑾ(M))∩V (ᑾ)⊆V (ᒊ).
Proof. To prove the claim, it is enough to consider that AssR(ᑠ0ᑾ(M))= {ᒍ∈AssR(M): dim(R/(ᑾ+ᒍ))=0}(cf. [15, Lemma 4.1]).
Using Lemma 2.2 we are now able to prove Theorem 1.1:
Proof of Theorem1.1. (1)⇒(3) and (2)⇒(1) are obvious.
(3)⇒(2): By passing to the completion, we may assume thatRis complete (cf. [15, Proposition 3.3]).
We argue by induction ont. Whent = 1, there is nothing to prove, since Lemma 2.2 and the assumptions imply that
Supp(ᑠ0ᑾ(M))=Supp(ᑠ0ᑾ(M))∩V (ᑾ)⊆V (ᒊ).
Henceᑠ0ᑾ(M)is Artinian. To this end note thatᑠ0ᑾ(M)is a finitely generated submodule ofM. So suppose thatt > 1 and the result has been proved for smaller values oft. PutM =M/Hᑾ0(M). From the short exact sequences
0−→Hᑾ0(M)−→M −→M −→0
and by [15, Proposition 3.11]), we get the following long exact sequence
· · · −→ᑠiᑾ(Hᑾ0(M))−→ᑠiᑾ(M)−→ᑠiᑾ(M)−→ᑠi+ᑾ 1(Hᑾ0(M))−→ · · ·. Asᑠjᑾ(Hᑾ0(M))=Hᒊj(Hᑾ0(M))is an ArtinianR-module for everyj ∈Z([2, Theorem 7.1.3]) then, one can see that Supp(ᑠiᑾ(M)) ⊆ V (ᑾ)for alli < t. Hence, it is enough to show thatᑠiᑾ(M)is Artinian, so we may assume that Hᑾ0(M)= 0. Thus, there exists anM-regular elementx inᑾsuch that from the short exact sequence
0−→M −→x M −→M/xM =M−→0 we deduce the next long exact sequence
(∗) · · · −→ᑠiᑾ(M)−→x ᑠiᑾ(M)−→ᑠiᑾ(M) −→ᑠi+ᑾ 1(M)−→ · · ·. Since Supp(ᑠiᑾ(M))⊆V (ᑾ)for alli < t, it follows from the above long exact sequence that Supp(ᑠiᑾ(M)) ⊆ V (ᑾ)for alli < t−1. Hence, by induction
hypothesis we haveᑠiᑾ(M) is Artinian for alli < t−1. Therefore in the view of(∗),(0 :ᑠiᑾ(M)x)is Artinian for alli < t.
On the other hand since Supp(ᑠiᑾ(M)) ⊆ V (ᑾ)for all i < t, one can see that ᑠiᑾ(M)=
(0 :ᑠiᑾ(M) ᑾα)⊆
(0 :ᑠiᑾ(M)xα)⊆ᑠiᑾ(M)
soᑠiᑾ(M)= (0 :ᑠiᑾ(M) xα). Therefore by [9, Theorem 1.3],ᑠiᑾ(M)will be Artinian for alli < t.
(2)⇒(4): Since ᑠiᑾ(M)is ᑾ-adically complete for every i ∈ Z(cf. [15, Theorem 3.9] or [5, Remark 3.1]), we get
nᑾnᑠiᑾ(M)=0. Moreover for all i < t,ᑠiᑾ(M)is Artinian. Hence, there is an integerusuch thatᑾuᑠiᑾ(M)=0.
(4)⇒(3) is obvious.
(1)⇒(5): By passing to the completion we may assume thatRis complete.
We use induction ont. Lett =1. As Supp(ᑠ0ᑾ(M))⊆V (ᒊ)soᑠ0ᑾ(M)must be zero. Otherwise since
∅ =Ass(ᑠ0ᑾ(M))⊆Supp(ᑠ0ᑾ(M))⊆V (ᒊ) then,
ᒊ∈Ass(ᑠ0ᑾ(M))= {ᒍ∈Ass(M);dim(R/ᑾ+ᒍ)=0}, this is contradiction to depth(M) >0.
Now suppose that depth(M)≥ t >1 and that the result has been proved for smaller values of t. By this inductive assumption, ᑠiᑾ(M) = 0 for i = 0,1, . . . , t−2 and it only remains for us to prove thatᑠt−ᑾ 1(M)=0.
Since depth(M) >1 then, there existsx ∈ᒊthat is anM-regular element.
Consider the short exact sequence 0−→M x
−→l M −→M/xlM = ¯M −→0 for everyl. Thus, we have the following long exact sequence
· · · −→ᑠiᑾ−1(M)¯ −→ᑠiᑾ(M)−→xl ᑠiᑾ(M)−→ᑠiᑾ(M)¯ −→ · · · for everyl.
As depth(M)¯ =depth(M)−1>0 and for alli < t−1, Supp(ᑠiᑾ(M))¯ ⊆ V (ᒊ)then, by inductive hypothesisᑠiᑾ(M)¯ = 0 for alli < t −1. Thus, for everyl,(0 :ᑠt−1ᑾ (M) xl)is a homomorphic image ofᑠt−ᑾ 2(M)¯ . Hence,(0 :ᑠt−1ᑾ (M) xl)=0 for everyl.
Take into account that by assumption Supp(ᑠiᑾ(M)) ⊆ V (ᒊ) for every i < t. Then,ᑠt−ᑾ 1(M)= ∪(0 :ᑠt−1ᑾ (M) xl)=0. This completes the proof.
3. Cosupport
In this section we examine the cosupport of formal local cohomology. The notion of cosupport was introduced by S. Yassemi in [17]. He defined the CosuppR(M) as the set of prime ideals ᒎ such that there exists a cocyclic homomorphic imageL of M with ᒍ ⊇ Ann(L). His definition is equival- ent to Melkersson-Schenzel’s definition for ArtinianR-modules. Melkersson- Schenzel’s definition of colocalization does not map Artinian R-module to ArtinianS−1R-module through colocalization at a multiplicative closed sub- set of R (cf. [10]). In this note we use the concept of cosupport has been introduced by A. Richardson [13]. It maps Artinian R-modules to Artinian S−1R-modules (whenRis complete). Also it is suitable to investigate formal local cohomology modules.
Definition3.1 (cf. [13]). LetRbe a ring andM anR-module.
(1) LetSbe a multiplicative closed subset ofRandDR(−):=HomR(−,ER), whereERis the injective hull of⊕R/ᒊ, the sum running over all max- imal idealsᒊofR. The colocalization ofM relative toSis theS−1R- module S−1M = DS−1R(S−1DR(M)). If S = R\ᒍ for some prime idealᒍ∈Spec(R), we writeᒍM forS−1M.
(2) The cosupport ofM is defined as follows
CosuppR(M):= {ᒍ∈Spec(R):ᒍM =0}.
For brevity we often write Cosupp(M)for CosuppR(M)when there is no ambiguity about the ringR.
Below we recall some properties of cosupport:
Lemma3.2 (cf. [13, Theorem 2.7]).LetRbe a ring andM anR-module.
(1) Cosupp(M)=Supp(DR(M)).
(2) IfM is finitely generated, thenCoSupp(M)=V (Ann(M))∩max(R). (3) Cosupp(M)= ∅if and only ifM =0.
(4) Cosupp(M)⊆V (Ann(M)).
(5) If 0 −→ M −→ M −→ M −→ 0 is exact, then Cosupp(M) = Cosupp(M)∪CoSupp(M).
(6) IfM is representable, thenCosupp(M)=V (Ann(M)).
Proposition3.3.LetR be a ring andM andN beR-modules. Then the following statements are true:
(1) Cosupp(M)is stable under specialization, i.e.
ᒍ∈Cosupp(M), ᒍ⊆ᒎ⇒ᒎ∈Cosupp(M).
(2) Let M be a finite module, then Cosupp(M ⊗R N) ⊆ Supp(M)∩ Cosupp(N).
Proof.
(1) Letᒍ ∈ Cosupp(M), then by definitionDRᒍ(DR(M)ᒍ)is nonzero and so isDR(M)ᒍ. As 0=DR(M)ᒍ=(DR(M)ᒎ)ᒍRᒎ, thenDR(M)ᒎ=0. It implies thatᒎM =0.
(2) Use [13, 2.5] to prove.
Lemma3.4.Letᑾbe an ideal of a ringR. LetN be an ArtinianR-module withAttR(N)⊆V (ᑾ). Then,Cosupp(N)⊆V (ᑾ).
Proof. SinceNis an Artinian module then, the following descending chain ᑾN ⊇ᑾ2N ⊇ · · · ⊇ᑾnN ⊇ · · ·
of submodules ofN is stable, i.e. there exists an integerkthatᑾkN =ᑾk+1N. As AttR(N/ᑾkN) = AttR(N) ∩ V (ᑾ) (cf. [10, Proposition 5.2]) and Cosupp(N/ᑾkN) ⊆ V (ᑾ) by virtue of Proposition 3.3, hence, by passing toN/ᑾkNwe may assume thatᑾkN =0.
Letᒍ∈Cosupp(N)then,ᒍN =0. Thus, for everys ∈S=R\ᒍ,sN =0 (cf. [13, 2.1]). On the other hand
nᑾnN =ᑾkN =0, hence, for everys ∈S, sN ⊆ᑾtN. It follows that for alls ∈S,s /∈ᑾt and clearlyᒍ∈V (ᑾ).
Corollary 3.5. Let i ∈ Z. Let (R,ᒊ) be a local ring and M be a fi- nitely generatedR-module. Assume thatᑠiᑾ(M)is an ArtinianR-module, then Cosupp(ᑠiᑾ(M))⊆V (ᑾ).
Proof. Asᑠiᑾ(M)is Artinian andᑾ-adically complete so, there exists an integer k such that
n≥1ᑾnᑠiᑾ(M) = ᑾkᑠiᑾ(M) = 0. Hence, [2, Proposi- tion 7.2.11] implies that Att(ᑠiᑾ(M))⊆ V (ᑾ)and in the light of Lemma 3.4 Cosupp(ᑠiᑾ(M))⊆V (ᑾ).
Remark 3.6. Converse of Corollary 3.5 is not true in general. LetR = k[|x|] denote the formal power series ring over a fieldk. Putᑾ=(x)R. Then
Cosupp(ᑠ0ᑾ(R))=Supp(DR(DR(Hᑾ1(R))))=Supp(Hᑾ1(R))⊆V (ᑾ) butᑠ0ᑾ(R)is not Artinian.
We now turn our attention to prove Theorem 1.2. For this reason we give a preliminary Lemma:
Lemma3.7.Let(R,ᒊ)be ad-dimensional local ring. LetM be a finitely generatedR-module. Then
ᑠcᑾ(M)∼=ᑠcᑾ(R)⊗RM, where c:=dim(R/ᑾ).
Proof. At first note that by definition of inverse limit, ᑠjᑾ(−) preserves finite direct sum, for everyj ∈Z. Furthermoreᑠcᑾ(−)is a right exact functor (cf. [15, Theorem 4.5]). Hence, by Watts’ Theorem ([14, Theorem 3.33]) the claim is proved.
Lemma 3.7 declares thatᑠcᑾ(R)=0 if and only ifᑠcᑾ(M)=0 for all finitely generatedR-moduleM.
In order to prove Theorem 1.2 we utilize the useful consequence of Gruson’s Theorem (see, e.g., [16, Corollary 4.3]) allows us to reduce to the caseM =R when considering the cosupport of top formal local cohomology modules:
Proof of Theorem1.2. (1) Sinceᑠcᑾ(M)∼= ᑠcᑾ(R/J )(M), by Independ- ence Theorem [2, 4.2.1], we may replace R by R/J to assume that M is faithful. Note that for dim(R/(ᑾ, J )) < c, there is nothing to prove because, ᑠcᑾ(M)=0.
In the view of Lemma 3.7 and [13, Proposition 2.5], for everyᒍ∈Spec(R)
ᒍᑠcᑾ(M)∼=Mᒍ⊗Rᒍᒍᑠcᑾ(R).
As Mᒍ is a faithful Rᒍ-module, [16, Corollary 4.3] implies thatMᒍ⊗R ᒍᑠcᑾ(R)=0 if and only ifᒍᑠcᑾ(R)=0, which completes the proof.
(2) To prove, we use the localization instead of colocalization in the proof of (1).
4. Coassociated primes
LetMbe anR-module. A prime idealᒍofRis called a coassociated prime of Mif there exists a cocyclic homomorphic imageLofMsuch thatᒍ=Ann(L). The set of coassociated prime ideals ofMis denoted by CoassR(M)(cf. [17]).
When the ambient R is understood, we will often write Coass(M)instead of CoassR(M).
Note that for an Artinian module the set of coassociated primes is fi- nite. In this section (R,ᒊ) is a local ring and we denote by DR(M) = HomR(M, E(R/ᒊ))the Matlis dual ofR-moduleM, whereE(R/ᒊ)is the injective hull of residue field, so in this case Coass(M)=Ass(DR(M)).
Among other results, we will see that under certain assumptions CosuppR(ᑠiᑾ(M))as a subset of Spec(R)is closed in the Zariski topology for somei ∈Z.
Lemma 4.1. Let (R,ᒊ) be a local ring and M an R-module. Then the following statements are true:
(1) Coass(M)⊆Cosupp(M).
(2) Every minimal element ofCosupp(M)belongs toCoass(M).
(3) For any NoetherianR-module M, Coass(M) = Cosupp(M) ⊆ {ᒊ}, whereRdenotes theᒊ-adic completion ofR.
Proof.
(1) Letᒍ∈Coass(M), then, it implies that 0=HomRᒍ(Rᒍ/ᒍRᒍ, DR(M)ᒍ). Note that it remains nonzero by taking HomRᒍ(−, ERᒍ(Rᒍ/ᒍRᒍ))and consequentlyᒍ∈Cosupp(M).
(2) Letᒍ∈min Cosupp(M)=min Supp(DR(M)), so, ᒍ∈min Ass(DR(M)). It follows thatᒍ∈min Coass(M). (3) It is clear by (1) and (2).
It should be noted that Supp(ᑠiᑾ(M))is closed when Ass(ᑠiᑾ(M))is finite. In fact for a local Gorenstein ring(R,ᒊ), Ass(ᑠiᑾ(R))=Ass(DR(HᑾdimR−i(R))) see [4] for details. Take into account that it is not finite in general (see [4] or [1, Remark 2.8(vi)]).
Lemma4.2.Let(R,ᒊ)be a local ring andM be anR-module. The set of minimal primes inCosupp(M)is finite if and only ifCosupp(M)is a closed subset ofSpec(R).
Proof. Let Cosupp(M)=V (ᑿ)for some idealᑿofR. AsRis Noetherian then so isR/ᑿ. It turns out that the set of minimal elements of Cosupp(M)is finite.
For the reverse direction, let ᒍ1, . . . ,ᒍt be the minimal prime ideals of Cosupp(M). Putᒎ:= ∩iᒍi. We claim that Cosupp(M)=V (ᒎ).
It is clear that Cosupp(M)⊆V (ᒎ). For the opposite direction assume that there is a prime idealQ⊃ᒎ. Then,Q⊃ᒍj, for some 1≤j ≤t so the proof follows by 3.3(1).
We deduce from above lemma that the cosupport of formal local cohomo- logy module is closed, whenever its set of coassociated primes is finite. There- fore if one of the situations in Theorem 1.1 is true, the cosupport of formal local cohomology module is closed. Also Cosupp(ᑠdimᑾ (M)(M))is closed, as ᑠdimᑾ (M)(M)is Artinian, wheneverMis a finitely generated module over a local ring(R,ᒊ)(cf. [1, Lemma 2.2]).
Take into account that when R is a complete local Gorenstein ring and ᑠiᑾ(M)is assumed to be either Noetherian or Artinian module, then
Cosupp(ᑠiᑾ(M))=Supp(HᑾdimR−i(M, R)).
By virtue of [1, Theorem 2.7], for a Cohen-Macaulay ringRwith ht(ᑾ) >0, ᑠdimᑾ (R/ᑾ)(R)is not Artinian. Moreoverᑠdimᑾ (M/ᑾM)(M)is not finitely generated for dim(M/ᑾM) >0 (cf. [1, Theorem 2.6(ii)]). Below we give an alternative proof:
Theorem4.3. Letᑾ be an ideal of a local ring(R,ᒊ)andM a finitely generatedR-module. Assume thatdim(M/ᑾM) >0. Thenᑠdimᑾ (M/ᑾM)(M)is not a finitely generatedR-module.
Proof. Put c := dim(M/ᑾM). In the contrary assume that ᑠcᑾ(M)is a finitely generatedR-module. Letx ∈ ᒊbe a parameter element ofM/ᑾM. Hence, [15, Theorem 3.15] implies the following long exact sequence
· · · −→Hom(Rx,ᑠcᑾ(M))−→ᑠcᑾ(M)−→ᑠc(ᑾ,x)(M)−→ · · ·, wherei∈Z. As dim(M/(ᑾ, x)M) <dim(M/ᑾM)then,ᑠc(ᑾ,x)(M)=0. Now letf ∈Hom(Rx,ᑠcᑾ(M)). Fix an arbitrary integern, so
f (1/xn)=xmf (1/xm+n)∈xmᑠcᑾ(M), for every integerm. It implies thatf (1/xn)∈
mxmᑠcᑾ(M)= 0 by Krull’s Theorem and hence, f = 0. Now it follows that ᑠcᑾ(M) = 0, which is a contradiction, see [15, Theorem 4.5].
Now we examine the set of coassociated primes of top formal local co- homology to show that by some assumptions onR, it could be finite.
Proposition4.4. Let ᑾ be an ideal of a complete Gorenstein local ring (R,ᒊ)andc:=dim(R/ᑾ). LetM be a finitely generatedR-module. Then
CoassR(ᑠcᑾ(M))=SuppR(M)∩Ass(Hᑾhtᑾ(R)).
In particular,CoassR(ᑠcᑾ(M))is finite.
Proof.
CoassR(ᑠcᑾ(M))=CoassR(ᑠcᑾ(R)⊗RM)
=SuppR(M)∩CoassR(ᑠcᑾ(R))
=SuppR(M)∩AssR(Hᑾhtᑾ(R))
where the first equality is clear by Lemma 3.7, the second equality follows by [17, Theorem 1.21].
It should be noted that by hypotheses in Proposition 4.4, ht(ᑾ)=gradeR(ᑾ) and it is well-known that AssR(HᑾgradeRᑾ(R))is finite, cf. [3].
Corollary4.5.Keep the notations and hypotheses in Proposition 4.4, ᑠcᑾ(M)=0⇐⇒SuppR(M)∩Ass(Hᑾht(ᑾ)(R))= ∅.
Proposition 4.6. Let i ∈ Z. Let ᑾ ⊂ R be an ideal of a ring R. If CoassR(ᑠiᑾ(R))is finite, then so isCoassR(ᑠiᑾ(R/Hᑾ0(R))).
Proof. Consider the exact sequence
0−→Hᑾ0(R)−→R−→R/Hᑾ0(R)=R−→0. It provides the following long exact sequence
(∗) · · · −→ᑠiᑾ(Hᑾ0(R))−→ψ ᑠiᑾ(R)
−→ϕ ᑠiᑾ(R)−→ᑠi+ᑾ 1(Hᑾ0(R))−→ · · ·,
for everyi.
Asᑠiᑾ(Hᑾ0(R))=Hᒊi(Hᑾ0(R))is Artinian, it follows that Coass(ᑠiᑾ(Hᑾ0(R))) is finite.
By virtue of(∗), we get the following short exact sequence 0−→U −→ᑠiᑾ(R)−→U−→0,
whereU = coker(ψ)andU = coker(ϕ). It implies that Coass(ᑠiᑾ(R))is finite. To this end note that Coass(U ) is finite by [17, Theorem 1.10] and Coass(U)is finite asᑠi+ᑾ 1(Hᑾ0(R))is Artinian.
Now we are going to give more information on the last non-vanishing formal local cohomology module.
Theorem4.7.Let(R,ᒊ)be a local ring of dimensiond >1. Letᑠdᑾ(R)= 0. Then:
(1) Ifᒍ∈Coass(ᑠd−ᑾ 1(R)), then it implies thatdim(R/(ᑾ,ᒍ))=d−1.
(2) Assh(R)∩Coass(ᑠd−ᑾ 1(R))⊆ {ᒍ∈Spec(R): dim(R/ᒍ)=d,Rad(ᑾ+ ᒍ)=ᒊ}.
(3) IfCoass(ᑠd−ᑾ 1(R))⊆Assh(R), then{ᒍ∈ Spec(R): dim(R/(ᑾ,ᒍ)) = d−1} ⊆Coass(ᑠd−ᑾ 1(R)).
Proof.
(1) Letᒍ∈Coass(ᑠd−ᑾ 1(R)). Asᑠdᑾ(R)=0 then, by [15, Theorem 4.5] we have
dim(R/(ᑾ,ᒍ))≤dim(R/ᑾ)≤d−1. On the other handᒍ∈Coass(R/ᒍ⊗Rᑠd−ᑾ 1(R)), because
Coass(R/ᒍ⊗Rᑠd−ᑾ 1(R))=Supp(R/ᒍ)∩Coass(ᑠd−ᑾ 1(R)).
It yields with the similar argument to Lemma 3.7 that 0 = R/ᒍ⊗R
ᑠd−ᑾ 1(R) = ᑠd−ᑾ 1(R/ᒍ). So, we have dim(R/(ᑾ,ᒍ)) ≥ d−1. It com- pletes the proof.
(2) Letᒍ∈Assh(R)∩Coass(ᑠd−ᑾ 1(R)). Then, similar to (1),ᑠd−ᑾ 1(R/ᒍ)= 0 and moreover Rad(ᑾ+ᒍ)=ᒊ. To this end note that if Rad(ᑾ+ᒍ)=ᒊ then,ᑠd−ᑾ 1(R/ᒍ)=0 by Grothendieck’s vanishing Theorem.
(3) Letᒍ ∈ Spec(R)and dim(R/(ᑾ,ᒍ)) = d −1. Then, it follows that
∅ = Coass(ᑠd−ᑾ 1(R/ᒍ)) = Supp(R/ᒍ)∩Coass(ᑠd−ᑾ 1(R)). Let ᒎ ∈ Coass(ᑠd−ᑾ 1(R))then,ᒎ ⊇ ᒍ, but by assumption ᒎ is minimal so we deduce thatᒎ=ᒍ.
Remark4.8. The inclusion in Theorem 4.7(2) is not an equality in general.
For example Let R = k[[x, y, z]] denote the formal power series ring in three variables over a field k. Letᑾ = (x, y)be an ideal of R which is of dimension one and putᒍ = 0. It is clear thatᑠ3ᑾ−1(R)= 0 = ᑠ3ᑾ(R), that is Coass(ᑠ3ᑾ−1(R))= ∅.
Lemma4.9.Let(R,ᒊ)be a complete local ring andᑾan ideal ofR. Let ᒍbe a minimal prime ideal ofᑾ. Thenᒎ ∈ CoassR(Rᒍ)implies thatᒎ ⊆ ᒍ, where the functor. denotes the completion functor.
Proof. The proof is straightforward. Letᒎ∈CoassR(Rᒍ), then
0=HomR(R/ᒎ,HomR(Rᒍ, ER(R/ᒊ)))=HomR(R/ᒎ⊗RRᒍ, ER(R/ᒊ)).
It yields that
0=R/ᒎ⊗RRᒍ=R/ᒎ⊗RRᒍ⊗RᒍRᒍ. It is clear thatRᒍ/ᒎRᒍ=0 and soᒎmust be contained inᒍ.
Proof of Theorem1.4. Ford >2 andd =1, the claim is clear.
Letd =2. Suppose thatᒍ1, . . . ,ᒍr are the minimal prime ideals ofᑾ. Put S = r
i=1(R\ᒍi)and choosey ∈ ᒊ\ ri=1ᒍi. By [2, Theorem 2.2.4], for anyn∈Nwe have
0−→Hᒊ0(R/ᑾn)−→R/ᑾn−→D(y)(R/ᑾn)−→Hᒊ1(R/ᑾn)−→0, whereD(y)(R/ᑾn)is the(y)-transform functor. One can see thatD(y)(R/ᑾn)∼= RS/ᑾnRS, so we get the following exact sequence
0−→Hᒊ0(R/ᑾn)−→R/ᑾn −→RS/ᑾnRS −→Hᒊ1(R/ᑾn)−→0. FurthermoreRS/ᑾnRS ∼= ⊕ri=1Rᒍi/ᑾnRᒍi. All the modules in the above exact sequence satisfy the Mittag-Leffler condition so by applying inverse limits we get 0−→R/ᑠ0ᑾ(R)−→ ⊕ri=1Rᒍi −→ᑠ1ᑾ(R)−→0.
It yields that CoassR(ᑠ1ᑾ(R))⊆ ri=1CoassR(Rᒍi)⊆CoassR(ᑠ1ᑾ(R))∪ {ᒊ}. In the view of Lemma 4.9, CoassR(ᑠ1ᑾ(R)) = ri=1CoassR(Rᒍi). Now the claim is proved by [18, Beispiel 2.4]. To this end note that CoassR(Rᒍi) = CoassRᒍ
i(Rᒍi)∩Rfor everyi∈ {1, . . . , r}.
Remark4.10. Keep the notations and hypotheses in Theorem 1.4 and let Mbe a finitely generatedR-module. AsRis complete so by Cohen’s structure Theorem, there exists a Gorenstein local ring(S,ᒋ)whereRis a homomorphic image ofSand dim(R)=dim(S). Then by virtue of 3.7 we have
AssR(Hᑾ1S(M, S))⊆Coass(ᑠd−ᑾ 1(R)) is finite.
REFERENCES
1. Asgharzadeh, M., and Divaani-Aazar, K.,Finiteness properties of formal local cohomology modules and Cohen-Macaulayness, Comm. Alg. 39 (2011), 1082–1103.
2. Brodmann, M., and Sharp, R.Y.,Local cohomology: an algebraic introduction with geometric applications, Cambridge Univ. Press, Cambridge 1998.
3. Brodmann, M., and Faghani, A. L.,A finiteness result for associated primes of local cohomo- logy modules, Proc. Amer. Math. Soc. 128 (2000), 2851–2853.
4. Hellus, M.,Local Cohomology and Matlis Duality, Habilitationsschrift, Leipzig 2006.
5. Hellus, M., and Stückrad, J.,On endomorphism rings of local cohomology modules, Proc.
Amer. Math. Soc. 136 (2008), 2333–2341.
6. Huneke, C.,Problems on local cohomology, pp. 93–108 in:Free resolutions in commutative algebra and algebraic geometry, Sundance 1990, Jones and Bartlett 1992.
7. Huneke, C., Katz, D., and Marley, T.,On the support of local cohomology, J. Algebra 322 (2009), 3194–3211.
8. Mafi, A., Results of formal local cohomology modules, Bull. Malays. Math. Sci. Soc. 36 (2013), 173–177.
9. Melkersson, L.,on asymptotic stability for sets of prime ideals connected with the powers of an ideal, Math. Proc. Camb. Phil. Soc. 107 (1990), 267–271.
10. Melkersson, L., and Schenzel, P.,The co-localization of an Artinian module, Proc. Edinburgh Math. Soc. 38 (1995), 121–131.
11. Ogus, A.,Local cohomological dimension of Algebraic Varieties, Ann. of Math. 98 (1973), 327–365.
12. Peskine, C., and Szpiro, L.,Dimension projective finie et cohomologie locale. Applications á la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 47–119.
13. Richardson, A. S., Co-localization, co-support and local cohomology, Rocky Mountain J. Math. 36 (2006), 1679–1703.
14. Rotman, J.,An Introduction to Homological Algebra, Academic Press, New York 1979.
15. Schenzel, P.,On formal local cohomology and connectedness, J. Algebra 315 (2007), 894–
923.
16. Vasconcelos, W.,Divisor Theory in Module Categories, North-Holland Math. Stud. 14, North- Holland, Amsterdam 1974.
17. Yassemi, S.,Coassociated primes, Comm. Alg. 23 (1995), 1473–1498.
18. Zöschinger, H.,Der Krullsche Durchschnittssatz für kleine Untermoduln, Arch. Math. (Basel) 62 (1994), 292–299.
MARTIN-LUTHER-UNIVERSITÄT HALLE-WITTENBERG INSTITUT FÜR INFORMATIK
D-06099 HALLE (SAALE) GERMANY
ISLAMIC AZAD UNIVERSITY BRANCH SOFIAN
SOFIAN IRAN
E-mail:m.eghbali@yahoo.com
