Artinianness of local cohomology modules

Abstract

Let $A$ be a noetherian ring, $\fa$ an ideal of $A$, and $M$ an $A$--module. Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about the artinianness of some special local cohomology modules in the graded case.

Full text

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arXiv:0809.3814v1 [math.AC] 22 Sep 2008
ARTINIANNESS OF LOCAL COHOMOLOGY
MODULES
MOHARRAM AGHAPOURNAHR AND LEIF MELKERSSON
Abstract. Let A be a noetherian ring, a an ideal of A, and M an
A–module. Some uniform theorems on the artinianness of certain
local cohomology modules are proven in a general situation. They
generalize and imply previous results about the artinianness of
some special local cohomology modules in the graded case.
1. Introduction
Throughout A is a commutative noetherian ring. As a general refer-
ence to homological and commutative algebra we use [5] and [10]. One
of the main problems in the study of local cohomology modules is to
determine when they are artinian. Recently some results have been
proved about the artinianness of graded local cohomology modules in
[2], [3], [13] and [14].
We prove some uniform results about artinianness of local cohomol-
ogy modules in the context of an arbitrary noetherian ring A. Their
proofs are simple and they have those special cases in the above refer-
ences as immediate consequences.
A Serre subcategory of the category of A–modules is a full subcat-
egory closed under taking submodules, quotient modules and exten-
sions. An example is given by the class of artinian A–modules. A
useful method to prove that a certain module belongs to such a Serre
subcategory is to apply [12, Lemma 3.1].
An A–module M is called a–cofinite if SuppA(M) ⊂ V(a) and
Exti
troduced by Hartshorne in [8]. For more information about cofiniteness
with respect to an ideal, see [9], [6] and [12].
A(A/a,M) is finite (finitely generated) for all i. This notion was in-
2000 Mathematics Subject Classification. 13D45, 13D07.
Key words and phrases. Local cohomology, artinian modules, cofinite modules.
1

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2 MOHARRAM AGHAPOURNAHR AND LEIF MELKERSSON
2. Main results
Theorem 2.1. Let S be a Serre subcategory of the category of A–
modules. Let M be a finite A module and let a be an ideal of A such
that Hi
Hn
S.
a(M) belongs to S for all i > n. If b is an ideal of A such that
a(M/bM) belongs to S, then the module Hn
a(M)/bHn
a(M) belongs to
Proof. Suppose Hn
submodule of M such that Hn
such that Γb(M/N) = L/N. Since SuppR(L/N) ⊂ V(b) ∩ SuppR(M),
Hi
From the exact sequence 0 → L/N → M/N → M/L → 0, we get
the exact sequence
a(M)/bHn
a(M) is not in S. Let N be a maximal
a(M/N)⊗AA/b is not in S. Let L ⊃ N be
a(L/N) belongs to S for all i ≥ n by [1, Theorem 3.1].
Hn
a(L/N) −→ Hn
a(M/N)
f
−→ Hn
a(M/L) −→ Hn+1
a
(L/N).
TorA
Kerf and Cokerf are in S. It follows from [12, Lemma 3.1], that
Ker(f ⊗ A/b) and Coker(f ⊗ A/b) are in S. Since Hn
is not in S, the module Hn
maximality of N, we get L = N. We have shown that Γb(M/N) = 0
and therefore we can take x ∈ b such that the sequence 0 → M/N
M/N → M/(N + xM) → 0 is exact. Thus we get the exact sequence
i(A/b,Kerf) and TorA
i(A/b,Cokerf) are in S for all i, because
a(M/N) ⊗AA/b
a(M/L) ⊗AA/b can not be in S. By the
x→
Hn
a(M/N)
x
−→ Hn
a(M/N) −→ Hn
a(M/N + xM) −→ Hn+1
a
(M/N).
This yields the exact sequence
0 −→ Hn
a(M/N)/xHn
a(M/N) −→ Hn
a(M/N + xM) −→ C −→ 0,
where C ⊂ Hn+1
Note that x ∈ b. Hence we get the exact sequence
a
(M/N) and thus C is in S.
TorA
1(A/b,C) −→ Hn
However N ? (N + xM) and therefore Hn
belongs to S by the maximality of N. Consequently Hn
is in S which is a contradiction.
a(M/N) ⊗AA/b −→ Hn
a(M/(N + xM)) ⊗AA/b
a(M/(N + xM)) ⊗AA/b
a(M/N)⊗AA/b
?
Corollary 2.2. Let a and b be two ideals of A. Let M be a finite
A–module. If Hi
then Hn
a(M) is artinian for i > n and Hn
a(M)/bHn
a(M/bM) is artinian,
a(M) is artinian.
Proof. In 2.1 take S as the category of artinian A–modules.
?
Corollary 2.3. Let a and b be two ideals of A such that A/a + b is
artinian. If Hi
artinian.
a(M) is artinian for i > n, then Hn
a(M)/bHn
a(M) is

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ARTINIANNESS OF LOCAL COHOMOLOGY MODULES3
Proof. Note that Hn
a(M/bM)∼= Hn
a+b(M/bM) is artinian.
?
Corollary 2.4. If Hi
Hn
a(M) is artinian for i > n, where n ≥ 1 then
a(M) is artinian.
a(M)/aHn
Proof. Note that Hn
a(M/aM) = 0 for n ≥ 1
?
Remark 2.5. In 2.4 we must assume that n ≥ 1. Take any ideal a in a
ring A such that A/a is not artinian. Let M = A/a. Then Hi
for i ≥ 1, and Γa(M) = M. On the other hand M/aM∼= M. Thus
H0
a(M) = 0
a(M)/aH0
a(M) is not artinian.
Corollary 2.6. Let a and b be two ideals of A. Let M be a finite A–
module. If Hi
i > n, and Hn
then Hn
In particular, if n ≥ 1 then Hn
has finite support).
a(M) is a–cofinite artinian (resp. has finite support) for
a(M/bM) is a–cofinite artinian (resp. has finite support)
a(M)/bHn
a(M)/aHn
a(M) is a–cofinite artinian (resp. has finite support).
a(M) has finite length (resp.
Corollary 2.7. If c = cd(a,M) > 0, then Hc
a(M) = aHc
a(M).
As a corollary we recover Yoshida’s theorem [15, Proposition 3.1].
Corollary 2.8. c = cd(a,M) > 0, then Hc
a(M) is not finite.
Proof. We may assume that (A,m) is a local ring. Use corollary 2.7
and Nakayama’s lemma.
?
Proposition 2.9. Let a and b be ideals of A with A/a+b artinian. If
the module M is a–cofinite, then Hi
b(M) is artinian for all i.
Proof. Use [12, Theorem 5.5 (i)⇔(iv)] and the fact that an a–cofinite
module has finite Bass numbers.
?
Corollary 2.10. Let a and b be two ideals of A such that A/a + b
is artinian. If a is a principle ideal and M is a finite module, then
Hi
b(Hj
a(M)) is artinian for all i,j.
Proof. Note that Hj
a(M) is a–cofinite for all j when a is principle.
?
Theorem 2.11. Let a and b be two ideals of A such that A/a + b is
artinian. Let M be a finite module such that dimM/aM ≤ 1. Then
Hi
b(Hj
a(M)) is an artinian module for all i,j.
Proof. Since A/a+b is artinian, SuppA(Hj
consists of finitely many maximal ideals. Thus we may assume that
(A,m) is local.
Passing to the ring A/Ann(M) from [6, Theorem 1] and the change
of rings principle [12, Corollary 2.6], we get that Hj
a(M)) ⊂ V(a+b) and V(a+b)
a(M) is a–cofinite

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4 MOHARRAM AGHAPOURNAHR AND LEIF MELKERSSON
for all j. We use theorem 2.9 to deduce that Hi
all i,j
b(Hj
a(M)) is artinian for
?
Theorem 2.12. Let a and b be two ideals of A such that A/a + b is
artinian. Let M be a finite module such that dimM/aM ≤ 2. Then
H1
b(Hi
a(M)) is an artinian module for all i.
Proof. We first reduce to the case Γb(M) = 0. From the exact sequence
0 −→ Γb(M) −→ M
f
−→ M −→ 0,
where M = M/Γb(M), we get the exact sequence
Hi
a(Γb(M)) −→ Hi
a(M)
Hi
−→ Hi
a(f)
a(M) −→ Hi+1
a (Γb(M)).
Note that Hi
KerHi
that KerH1
a(Γb(M))∼= Hi
a(f) and CokerHi
b(Hi
a+b(Γb(M)), and so is artinian for all i. Since
a(f) are artinian, it follows from [12, Lemma 3.1]
a(f)) and CokerH1
b(Hi
a(f)) are artinian. Hence
H1
b(Hi
a(M)) is artinian if and only if H1
b(Hi
a(M)) is artinian.
Thus we may assume that Γb(M) = 0.
Take x ∈ b outside all associated prime ideals of M and outside
all prime ideals p ⊃ a + Ann(M), such that dimA/p = 2.
dimM/(a+xA)M ≤ 1 and x is a non-zerodivisor on M. Hence we get
the long exact sequence
Then
··· → Hi−1
a (M/xM) → Hi
a(M)
x→ Hi
a(M) → Hi
a(M/xM) → ....
Consider the map f which is multiplication by x on Hi
the exact sequence
a(M). We have
0 −→ Hi−1
a (M)/xHi−1
a (M) −→ Hi−1
a (M/xM) −→ Kerf −→ 0
and hence the exact sequence
H1
b(Hi−1
a (M/xM)) −→ H1
b(Kerf) −→ H2
b(Hi−1
a (M)/xHi−1
a (M)).
However the last term is zero since
SuppA(Hi−1
a (M)/xHi−1
a (M)) ⊂ SuppA(M/(a + xA)M)
Consequently H1
and dimM/(a + xA)M ≤ 1.
since H1
actness of 0 → Cokerf → Hi
Γb(Cokerf) → Γb(Hi
Γb(Hi
Since we now have shown that H1
tinian, we are able to apply [12, Lemma 3.1] to deduce that KerH1
b(Kerf) is artinian,
From the ex-
b(Hi−1
a (M/xM)) is artinian by theorem 2.11.
a(M/xM), we get the exactness of 0 →
a(M/xM)). Again we use 2.11 to deduce that
a(M/xM) is artinan . Hence Γb(Cokerf) is artinian.
b(Kerf) and Γb(Cokerf) are ar-
b(f)

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ARTINIANNESS OF LOCAL COHOMOLOGY MODULES5
is artinian. But H1
follows from [4, Theorem 7.1.2] that H1
b(f) is just multiplication by x on H1
b(Hi
a(M)). It
b(Hi
a(M)) is artinian.
?
The following lemma is a generalization of [11, Corollary 1.8].
Lemma 2.13. Let a and b be two ideals of A such that A/a + b is
artinian. If M is a module such that SuppA(M) ⊂ V(a) and 0 :Ma is
finite, then Γb(M) is a–cofinite artinian.
Proof. The module 0 :Γb(M)a ⊂ 0 :Ma and is therefore finite. But the
support of 0 :Γb(M)a is contained in V(a + b), thus 0 :Γb(M)a has finite
length. Therefore Γb(M) is a–cofinite artinian, by [12, Proposition
4.1].
?
Corollary 2.14. Let a and b be two ideals of A such that A/a + b is
artinian. If M is a module such that
Extn
A(A/a,M) and Extn+1−j
A
(A/a,Hj
a(M))
for all j < n, are finite, then Γb(Hn
a(M)) is a–cofinite artinian.
Proof. Note that by [7, Theorem 6.3.9 (b)] HomA(A/a,Hn
hence by lemma 2.13, Γb(Hn
a(M)) is finite,
a(M)) is a–cofinite artinian.
?
Corollary 2.15. Let a and b be two ideals of A such that A/a + b
is artinian. If M is a finite module such that Hi
i < n, then Γb(Hi
a(M) is a–cofinite for
a(M)) is a–cofinite artinian for all i ≤ n.
Theorem 2.16. Let a and b be two ideals of A such that A/a + b
is artinian. For each finite module M, the modules Γb(H1
H1
a(M)) and
b(H1
a(M)) are artinian.
Proof. Corollary 2.15 with n = 1 implies that Γb(H1
We may assume that Γa(M) = 0, so there is an M–regular element x
in a. From the exact sequence 0 → M
the exact sequence
a(M)) is artinian.
x
→ M → M/xM → 0, we get
(1)0 −→ Γa(M/xM) −→ H1
a(M)
x
−→ H1
a(M) −→ H1
a(M/xM).
Consider the map f defined as multiplication with x on H1
curing in (1). We get
a(M) oc-
H1
b(Kerf)∼= H1
b(Γa(M/xM))∼= H1
a+b(Γa(M/xM)),
which is artinian and the exact sequence
0 −→ Γb(Cokerf) −→ Γb(H1
a(M/xM))

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6 MOHARRAM AGHAPOURNAHR AND LEIF MELKERSSON
Since Γb(H1
can use [12, Lemma 3.1] with S = Γb(−) and T = H1
that KerH1
x ∈ a on H1
H1
a(M/xM)) artinian by 2.15, Γb(Cokerf) is artinian. We
b(−) to conclude
b(f) is artinian. But H1
b(H1
a(M)) is artinian.
b(f) is multiplication by the element
a(M)). Again using [4, Theorem 7.1.2], we conclude that
b(H1
?
Lemma 2.17. Let f : L −→ M be A–linear and c an ideal of A. Let
S be a Serre subcategory of the category of A–modules. If KerHj
and CokerHj
Hi+2
c(f)
c(f) are in S for all j, then for each i:
c (Kerf) is in S if and only if Hi
c(Cokerf) is in S.
Proof. Let K = Kerf, I = Imf and C = Cokerf. Factorize f as
f = h ◦ g, where
0 −→ K −→ L
g
−→ I −→ 0
and
0 −→ I
h
−→ M −→ C −→ 0
are exact. For each j, we have the exact sequence
0 → KerHj
c(g) → KerHj
CokerHj
c(f) → KerHj
c(f) → CokerHj
c(h) → CokerHj
c(h) → 0.
c(g) →
The hypothesis implies that KerHj
each j. For each j, KerHj
Moreover there are exact sequences
c(g) and CokerHj
c(h) are in S for
c(g) is in S.
c(h) is in S if and only if CokerHj
Hi
c(I)
Hi
−→ Hi
c(h)
c(M) −→ Hi
c(C) −→ Hi+1
c
(I)
Hi+1
c
−→ Hi+1
(h)
c
(M)
and
Hi+1
c (L)
Hi+1
c
−→ Hi+1
(g)
c (I) −→ Hi+2
c (K) −→ Hi+2
c
(L)
Hi+2
c
−→ Hi+2
(g)
c
(I).
It follows that
Hi
c(C) ∈ S⇔
⇔ CokerHi+1
⇔
Hi+2
KerHi+1
c (h) ∈ S
(g) ∈ S
(K) ∈ S
c
c
.
?
Theorem 2.18. Let a and b be two ideals of A such that A/a+b is ar-
tinian. Assume that ara(a) = 2. Then for each finite A–module M, the
module Hi
is artinian.
b(H2
a(M)) is artinian if and only if the module Hi+2
b(H1
a(M))

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ARTINIANNESS OF LOCAL COHOMOLOGY MODULES7
Proof. We may assume that Γa(M) = 0. Then a can be generated by
M–regular elements x and y on M. (See [10, Exercise 16.8]). Then
there is an exact sequence [4, Proposition 8.1.2]
0 −→ H1
a(M) −→ H1
xA(M)
f
−→ H1
xA(My) −→ H2
a(M) −→ 0.
In order to apply lemma 2.17 we prove that KerHj
are artinian for all j. By [4, Proposition 8.1.2], there is an exact se-
quence
b(f) and CokerHj
b(f)
Hj
b+yA(H1
xA(M)) −→ Hj
b(H1
b+yA(H1
xA(M))
xA(M)).
Hj
−→ Hj
b(f)
b(H1
xA(M)y) −→
Hj+1
Since H1
Hence KerHj
xA(M) is xA–cofinite, the outer modules are artinian, by 2.9.
b(f) and CokerHj
b(f) are artinian for all j.
?
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8 MOHARRAM AGHAPOURNAHR AND LEIF MELKERSSON
[15] K. I. Yoshida, Cofiniteness of local cohomology modules for ideals of dimen-
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Moharram Aghapournahr, Arak University, Beheshti St, P.O. Box:879,
Arak, Iran
E-mail address: m-aghapour@araku.ac.ir
Leif Melkersson, Department of Mathematics, Link¨ oping Univer-
sity, S-581 83 Link¨ oping, Sweden
E-mail address: lemel@mai.liu.se