COFINITENESS AND COASSOCIATED PRIMES OF LOCAL COHOMOLOGY MODULES
MOHARRAM AGHAPOURNAHR and LEIF MELKERSSON
Abstract
LetRbe a noetherian ring,ᑾan ideal ofRsuch that dimR/ᑾ=1 andMa finiteR-module. We will study cofiniteness and some other properties of the local cohomology modules Hiᑾ(M). For an arbitrary idealᑾand anR-moduleM(not necessarily finite), we will characterizeᑾ-cofinite artinian local cohomology modules. Certain sets of coassociated primes of top local cohomology modules over local rings are characterized.
1. Introduction
ThroughoutRis a commutative noetherian ring. By a finite module we mean a finitely generated module. For basic facts about commutative algebra see [3]
and [9] and for local cohomology we refer to [2].
Grothendieck [7] made the following conjecture:
Conjecture. For every idealᑾand every finiteR-moduleM, the module HomR(R/ᑾ,Hᑾn(M))is finite for alln.
Hartshorne [8] showed that this is false in general. However, he defined an R-moduleMto beᑾ-cofiniteif SuppR(M)⊂V(ᑾ)and ExtiR(R/ᑾ, M)is finite (finitely generated) for eachiand he asked the following question:
Question. Ifᑾ is an ideal of R and M is a finite R-module. When is ExtiR(R/ᑾ,Hjᑾ(M))finite for everyiandj?
Hartshorne [8] showed that if(R,ᒊ)is a complete regular local ring and M a finiteR-module, then Hiᑾ(M)isᑾ-cofinite in two cases:
(a) Ifᑾis a nonzero principal ideal, and (b) Ifᑾis a prime ideal with dimR/ᑾ=1.
Yoshida [14] and Delfino and Marley [4] extended (b) to all dimension one idealsᑾof an arbitrary local ringR.
In Corollary 2.3, we give a characterization of theᑾ-cofiniteness of these local cohomology modules whenᑾis a one-dimensional ideal in a non-local
Received September 19, 2007, in final form October 14, 2008.
ring. In this situation we also prove in Theorem 2.7, that these local cohohomo- logy modules always belong to a class introduced by Zöschinger in [16].
Our main result in this paper is Theorem 2.10, where we for an arbitrary idealᑾand anR-moduleM (not necessarily finite), characterize the artinian ᑾ-cofinite local cohomology modules (in the rangei < n). With the additional assumption thatM is finitely generated, the characterization is also given by the existence of certain filter-regular sequences.
The second author has in [10, Theorem 5.5] previously characterized arti- nian local cohomology modules (in the same range). In case the moduleMis not supposed to be finite, the two notions differ. For example letᑾbe an ideal of a local ringR, such that dim(R/ᑾ) >0 and letM be the injective hull of the residue field ofR. The module H0ᑾ(M), which is equal toM, is artinian.
However it is notᑾ-cofinite, since 0 :
Mᑾdoes not have finite length.
AnR-moduleMhasfinite Goldie dimensionifMcontains no infinite direct sum of submodules. For a commutative noetherian ring this can be expressed in two other ways, namely that the injective hull E(M)ofMdecomposes as a finite direct sum of indecomposable injective modules or thatMis an essential extension of a finite submodule.
A prime idealᒍis said to becoassociatedto M ifᒍ = AnnR(M/N)for someN ⊂ M such that M/N is artinian and is said to beattached to M if ᒍ = AnnR(M/N) for some arbitrary submodule N of M, equivalently ᒍ=AnnR(M/ᒍM). The set of these prime ideals are denoted by CoassR(M) and AttR(M)respectively. Thus CoassR(M)⊂AttR(M)and the two sets are equal whenM is an artinian module. The two sets behave well with respect to exact sequences. If 0→M→M →M →0 is an exact sequence, then
CoassR(M)⊂CoassR(M)⊂CoassR(M)∪CoassR(M) and
AttR(M)⊂AttR(M)⊂AttR(M)∪AttR(M).
There are equalities CoassR(M ⊗R N) = CoassR(M) ∩ SuppR(N) and AttR(M ⊗R N) = AttR(M)∩SuppR(N), whenever the module N is re- quired to be finite. We prove the second equality in Lemma 2.11. In particular CoassR(M/ᑾM)=CoassR(M)∩V(ᑾ)and AttR(M/ᑾM)=AttR(M)∩V(ᑾ) for every idealᑾ. Coassociated and attached prime ideals have been studied in particular by Zöschinger, [17] and [18].
In Corollary 2.13 we give a characterization of certain sets of coassociated primes of the highest nonvanishing local cohomology module Hᑾt(M), where Mis a finitely generated module over a complete local ring. In case it happens thatt =dimM, the characterization is given in [4, Lemma 3]. In that case the
top local cohomology module is always artinian, but in general the top local cohomology module is not artinian ift <dimM.
2. Main results
First we extend a result by Zöschinger [15, Lemma 1.3] with a much weaker condition. Our method of proof is also quite different.
Proposition2.1. Let M be a module over the noetherian ring R. The following statements are equivalent:
(i) M is a finiteR-module.
(ii) Mᒊis a finiteRᒊ-module for allᒊ∈MaxRandMinR(M/N)is a finite set for all finite submodulesN ⊂M.
Proof. The only nontrivial part is (ii)⇒(i).
Let F be the set of finite submodules of M. For each N ∈ F the set SuppR(M/N)is closed in Spec(R), since MinR(M/N)is a finite set. Also it follows from the hypothesis that, for eachᒍ∈Spec(R)there isN ∈F such that Mᒍ = Nᒍ, that is ᒍ ∈/ SuppR(M/N). This means that
N∈F SuppR(M/N) = ∅. Now Spec(R) is a quasi-compact topological space. Consequentlyr
i=1SuppR(M/Ni)=∅for someN1, . . . , Nr ∈F. We claim thatM = N, whereN =r
i=1Ni. Just observe that SuppR(M/N) ⊂ SuppR(M/Ni)for eachi, and therefore SuppR(M/N)=∅.
Corollary2.2. LetM be anR-module such thatSuppM ⊂ V(ᑾ)and Mᒊ isᑾRᒊ-cofinite for each maximal idealᒊ. The following statements are equivalent:
(i) M isᑾ-cofinite.
(ii) For allj,MinR(ExtjR(R/ᑾ, M)/T )is a finite set for each finite submod- uleT ofExtjR(R/ᑾ, M).
Proof. The only nontrivial part is (ii)⇒(i).
Supposeᒊ is a maximal ideal of R. By hypothesisMᒊ isᑾRᒊ-cofinite.
Therefore ExtjR(R/ᑾ, M)ᒊis a finiteRᒊ-module for allj. Hence by Proposi- tion 2.1 ExtjR(R/ᑾ, M)is finite for allj. ThusM isᑾ-cofinite.
Corollary2.3. Letᑾan ideal of Rsuch thatdimR/ᑾ = 1,M a finite R-module andi≥0. The following statements are equivalent:
(i) Hiᑾ(M)isᑾ-cofinite.
(ii) For all j, MinR(ExtjR(R/ᑾ,Hᑾi(M))/T ) is a finite set for each finite submoduleT ofExtjR(R/ᑾ,Hᑾi(M)).
Proof. For all maximal ideals ᒊ, Hiᑾ(M)ᒊ ∼= HiᑾRᒊ(Mᒊ). By [4, The- orem 1] HiᑾRᒊ(Mᒊ)isᑾRᒊ-cofinite.
A moduleM isweakly Laskerian, when for each submoduleN ofM the quotient M/N has just finitely many associated primes, see [6]. A module M isᑾ-weakly cofiniteif SuppR(M) ⊂ V(ᑾ)and ExtiR(R/ᑾ, M) is weakly Laskerian for alli. Clearly eachᑾ-cofinite module isᑾ-weakly cofinite but the converse is not true in general see [5, Example 3.5(i) and (ii)].
Corollary 2.4. If Hiᑾ(M) (withdimR/ᑾ = 1)is an ᑾ-weakly cofinite module, then it is alsoᑾ-cofinite.
Next we will introduce a subcategory of the category ofR-modules that has been studied by Zöschinger in [16, Satz 1.6].
Theorem2.5 (Zöschinger). For anyR-moduleM the following are equi- valent:
(i) M satisfies the minimal condition for submodulesN such thatM/Nis soclefree.
(ii) For any descending chainN1 ⊃ N2 ⊃ N3 ⊃ · · ·of submodules ofM, there isnsuch that the quotientsNi/Ni+1have support inMaxRfor all i ≥n.
(iii) WithL(M)=
ᒊ∈MaxRᒊ(M), the moduleM/L(M)has finite Goldie dimension, anddimR/ᒍ≤1for allᒍ∈AssR(M).
If they are fulfilled, then for each monomorphismf :M −→M, SuppR(Cokerf )⊂MaxR.
We will say thatMis in the classZifMsatisfies the equivalent conditions in Theorem 2.5.
A moduleM issoclefreeif it has no simple submodules, or in other terms AssM∩MaxR=∅. For example ifMis a module over the local ring(R,ᒊ) then the moduleM/ᒊ(M), whereᒊ(M)is the submodule ofMconsisting of all elements ofM annihilated by some high powerᒊnof the maximal ideal ᒊ, is always soclefree.
Proposition 2.6. The class Z is a Serre subcategory of the category of R-modules, that is Z is closed under taking submodules, quotients and extensions.
Proof. The only difficult part is to show thatZ is closed under taking extensions. To this end let 0 −→ M −→f M −→g M −→ 0 be an ex- act sequence with M, M ∈ Z and let N1 ⊃ N2 ⊃ · · · be a descend- ing chain of submodules ofM. Consider the descending chainsf−1(N1) ⊃
f−1(N2)⊃ · · ·andg(N1) ⊃ g(N2)⊃ · · ·of submodules ofMandMre- spectively. By (ii) there isnsuch that SuppR(f−1(Ni)/f−1(Ni+1))⊂MaxR and SuppR(g(Ni)/g(Ni+1))⊂MaxRfor alli ≥n. We use the exact sequence
0−→f−1(Ni)/f−1(Ni+1)−→Ni/Ni+1−→g(Ni)/g(Ni+1)−→0.
to conclude that SuppR(Ni/Ni+1)⊂MaxRfor alli ≥n.
Theorem2.7. LetN be a module over a noetherian ringRandᑾan ideal ofRsuch thatdimR/ᑾ= 1. IfNᒊ isᑾRᒊ-cofinite for allᒊ∈ MaxR, then Nis in the classZ. In particular, ifM is a finiteR-module, thenHiᑾ(M)is in the classZ for alli.
Proof. LetX =N/L(N). Note that AssR(X)⊂Minᑾand therefore is a finite set. Since
E(X)=
ᒍ∈AssR(X)
E(R/ᒍ)μi(ᒍ,X),
it is enough to prove thatμi(ᒍ, X)is finite for allᒍ∈AssR(X). This is clear, since eachᒍ∈ AssR(X)is minimal overᑾand thereforeXᒍ ∼= Nᒍwhich is, ᑾRᒍ-cofinite, i.e. artinian overRᒍ.
Given elementsx1, . . . , xr inR, we denote by Hi(x1, . . . , xr;M)thei’th Koszul cohomology module of theR-moduleM. The following lemma is used in the proof of Theorem 2.10.
Lemma2.8. LetEbe an injective module. If H0(x1, . . . , xr;E)=0, then Hi(x1, . . . , xr;E)=0for alli.
Proof. We may assume that E = E(R/ᒍ)for some prime idealᒍ, since Eis a direct sum of modules of this form, and Koszul cohomology preserves (arbitrary) direct sums.
Put ᑾ = (x1, . . . , xr). By hypothesis 0 :E ᑾ = 0, which means that ᑾ ⊂ ᒍ. Take an elements ∈ ᑾ\ᒍ. It acts bijectively on E, hence also on Hi(x1, . . . , xr;E)for eachi. Butᑾ⊂AnnR(Hi(x1, . . . , xr;E))for alli, so the elementstherefore acts as the zero homomorphism on each Hi(x1, . . . , xr;E).
The conclusion follows.
First we state the definition, given in [10], of the notion of filter regularity on modules (not necessarily finite) over any noetherian ring. When(R,ᒊ)is local andMis finite, it yields the ordinary notion of filter-regularity, see [12].
Definition2.9. LetMbe a module over the noetherian ringR. An element xofRis called filter-regular onM if the module 0 :M xhas finite length.
A sequencex1, . . . , xs is said to be filter regular onMifxjis filter-regular onM/(x1, . . . , xj−1)M forj =1, . . . , s.
The following theorem yields a characterization of artinian cofinite local cohomology modules.
Theorem2.10. Let ᑾ = (x1, . . . , xr)be an ideal of a noetherian ringR and letnbe a positive integer. For eachR-moduleMthe following conditions are equivalent:
(i) Hiᑾ(M)is artinian andᑾ-cofinite for alli < n.
(ii) ExtiR(R/ᑾ, M)has finite length for alli < n.
(iii) The Koszul cohomology moduleHi(x1, . . . , xr;M)has finite length for alli < n.
WhenMis finite these conditions are also equivalent to:
(iv) Hiᑾ(M)is artinian for alli < n.
(v) There is a sequence of lengthninᑾthat is filter-regular onM.
Proof. We use induction on n. Whenn = 1 the conditions (ii) and (iii) both say that 0 :M ᑾhas finite length, and they are therefore equivalent to (i) [10, Proposition 4.1].
Letn >1 and assume that the conditions are equivalent whennis replaced by n−1. Put L = ᑾ(M) and M = M/L and form the exact sequence 0−→L−→M −→ M −→0. We haveᑾ(M)= 0 and Hiᑾ(M)∼=Hiᑾ(M) for alli >0. There are exact sequences
ExtiR(R/ᑾ, L)→ExtiR(R/ᑾ, M)→ExtiR(R/ᑾ, M)→ExtiR+1(R/ᑾ, L) and
Hi(x1, . . . , xr;L)→Hi(x1, . . . , xr;M)
→Hi(x1, . . . , xr;M)→Hi+1(x1, . . . , xr;L) BecauseLis artinian andᑾ-cofinite the outer terms of both exact sequences have finite length. HenceM satisfies one of the conditions if and only ifM satisfies the same condition. We may therefore assume thatᑾ(M)=0.
LetE be the injective hull ofM and putN = E/M. Consider the exact sequence 0 −→ M −→ E −→ N −→ 0. We know that 0 :M ᑾ = 0.
Therefore 0 :E ᑾ =0 andᑾ(E) =0. Consequently there are isomorphisms for alli ≥0:
Hᑾi+1(M)∼=Hᑾi(N), ExtiR+1(R/ᑾ, M)∼=ExtiR(R/ᑾ, N) and
Hi+1(x1, . . . , xr;M)∼=Hi(x1, . . . , xr;N).
In order to get the third isomorphism, we used that Hi(x1, . . . , xr;E)=0 for alli ≥ 0 (Lemma 2.8). HenceM satisfies one of the three conditions if and only ifNsatisfies the same condition, withnreplaced byn−1. By induction, we may therefore conclude that the moduleM satisfies all three conditions if it satisfies one of them.
Let nowM be a finite module.
(ii)⇔(iv) Use [10, Theorem 5.5(i)⇔(ii)].
(v)⇒(i) Use [10, Theorem 6.4].
(i)⇒(v) We give a proof by induction onn. PutL=ᑾ(M)andM =M/L.
Then AssRL=AssRM∩V(ᑾ)and AssRM = AssRM \V(ᑾ). The module Lhas finite length and therefore AssRL⊂MaxR. By prime avoidance take an elementy1∈ᑾ\
ᒍ∈AssR(M)ᒍ. Then AssR(0 :M y1)=AssR(M)∩V(y1)= (AssRL∩V(y1))∪(AssRM ∩V(y1)) ⊂ MaxR. Hence 0 :M y1 has finite length, so the elementy1∈ᑾis filter regular onM.
Supposen >1 and takey1as above.
Note that Hiᑾ(M)∼= Hᑾi(M)for alli ≥1. Thus we may replaceM byM, [10, Proposition 6.3(b)], and we may assume thaty1is a non-zerodivisor on M.
The exact sequence 0→M →y1 M →M/y1M →0 yields the long exact sequence
· · · −→Hiᑾ−1(M)−→Hiᑾ−1(M/y1M)−→Hiᑾ(M)−→ · · ·.
Hence Hiᑾ(M/y1M)isᑾ-cofinite and artinian for alli < n−1, by [11, Corol- lary 1.7]. Therefore by the induction hypothesis there existsy2, . . . , yn inᑾ, which is filter-regular onM/y1M. Thusy1, . . . , ynis filter-regular onM.
Remark. In [1] we studied the kernel and cokernel of the natural homo- morphism f : ExtnR(R/ᑾ, M) → HomR(R/ᑾ,Hnᑾ(M)). Applying the cri- terion of Theorem 2.10 we get that if ExttR−j(R/ᑾ,Hjᑾ(M))has finite length fort =n, n+1 and for allj < n, then ExtnR(R/ᑾ, M)has finite length if and only if Hnᑾ(M)isᑾ-cofinite artinian.
Next we will study attached and coassociated prime ideals for the last non- vanishing local cohomology module. First we prove a lemma used in Corol- lary 2.13
Lemma2.11. For allR-modulesMand for every finiteR-moduleN, AttR(M⊗RN)=AttR(M)∩SuppR(N).
Proof. Let ᒍ ∈ AttR(M ⊗R N), so ᒍ = AnnR((M ⊗R N)⊗R R/ᒍ).
However this ideal contains both AnnR(M/ᒍM)and AnnR(N)and therefore ᒍ=AnnR(M/ᒍM)andᒍ∈SuppR(N).
Conversely letᒍ∈ AttR(M)∩SuppR(N). Thenᒍ= AnnM/ᒍM and we want to show thatᒍ=AnnR((M⊗RN)⊗RR/ᒍ). Since
(M⊗RN)⊗RR/ᒍ∼=M/ᒍM⊗R/ᒍN/ᒍN,
we may assume thatRis a domain andᒍ=(0). LetKbe the field of fractions of R. Then AnnM =0 andN⊗RK=0. Therefore the natural homomorphism f :R −→EndR(M)is injective and we have the following exact sequence
0−→HomR(N, R)−→HomR(N,EndR(M)).
But HomR(N,EndR(M))∼=HomR(M ⊗RN, M). Hence we get AnnR(M⊗RN)⊂AnnRHomR(M ⊗RN, M)
⊂AnnRHomR(N, R)⊂AnnR(HomR(N, R)⊗RK).
On the other hand HomR(N, R)⊗RK ∼= HomR(N ⊗RK, K), which is a nonzero vector space overK. Consequently AnnR(M⊗RN)=0.
Theorem2.12. Let(R,ᒊ)be a complete local ring and letᑾbe an ideal ofR. Lettbe a nonnegative integer such thatHiᑾ(R)=0for alli > t.
(a) Ifᒍ∈AttR(Htᑾ(R))thendimR/ᒍ≥t.
(b) Ifᒍis a prime ideal such thatdimR/ᒍ=t, then the following conditions are equivalent:
(i) ᒍ∈CoassR(Hᑾt(R)).
(ii) ᒍ∈AttR(Htᑾ(R)).
(iii) Htᑾ(R/ᒍ)=0.
(iv) √
ᑾ+ᒍ=ᒊ.
Proof. (a) By the right exactness of the functor Htᑾ(−)we have (1) Htᑾ(R/ᒍ)∼=Htᑾ(R)/ᒍHtᑾ(R)
Ifᒍ ∈ AttR(Htᑾ(R)), then Htᑾ(R)/ᒍHtᑾ(R) = 0. Hence Htᑾ(R/ᒍ) = 0 and dimR/ᒍ≥t.
(b) SinceR/ᒍis a complete local domain of dimensiont, the equivalence of (iii) and (iv) follows from the local Lichtenbaum Hartshorne vanishing theorem.
If Htᑾ(R/ᒍ) = 0, then by (1) Htᑾ(R)/ᒍHᑾt(R) = 0. Therefore ᒍ ⊂ ᒎ for some ᒎ ∈ CoassR(Htᑾ(R)) ⊂ AttR(Htᑾ(R)). By (a) dimR/ᒎ ≥ t = dimR/ᒍ, so we must have ᒍ = ᒎ. Thus (iii) implies (i) and since always CoassR(Htᑾ(R))⊂AttR(Htᑾ(R)), (i) implies (ii).
If (ii) holds then the module Htᑾ(R)/ᒍHtᑾ(R)= 0, since its annihilator is ᒍ. Hence, using again the isomorphism (1), (ii) implies (iii).
Corollary2.13. Let(R,ᒊ)be a complete local ring,ᑾan ideal ofRand Ma finiteR-module andt a nonnegative integer such thatHiᑾ(M)=0for all i > t.
(a) Ifᒍ∈AttR(Htᑾ(M))thendimR/ᒍ≥t.
(b) If ᒍ is a prime ideal in SuppR(M) such that dimR/ᒍ = t, then the following conditions are equivalent:
(i) ᒍ∈CoassR(Hᑾt(M)).
(ii) ᒍ∈AttR(Htᑾ(M)).
(iii) Htᑾ(R/ᒍ)=0.
(iv) √ᑾ+ᒍ=ᒊ.
Proof. Passing fromR to R/AnnM, we may assume that AnnM = 0 and therefore using Gruson’s theorem, see [13, Theorem 4.1], Hiᑾ(N)=0 for alli > t and everyR-moduleN. Hence the functor Hᑾt(−)is right exact and therefore, since it preserves direct limits, we get
Htᑾ(M)∼=M ⊗RHtᑾ(R).
The claims follow from Theorem 2.12 using the following equalities CoassR(Htᑾ(M))=CoassR(Htᑾ(R))∩SuppR(M) by [16, Folgerung 3.2] and
AttR(Htᑾ(M))=AttR(Htᑾ(R))∩SuppR(M) by Lemma 2.11.
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ARAK UNIVERSITY BEHESHTI ST P.O. BOX: 879 ARAK IRAN
E-mail:m-aghapour@araku.ac.ir m.aghapour@gmail.com
DEPARTMENT OF MATHEMATICS LINKÖPING UNIVERSITY SE-581 83 LINKÖPING SWEDEN
E-mail:lemel@mai.liu.se
