LINH HÓA TỬ CỦA MÔĐUN ĐỐI ĐỒNG ĐIỀU ĐỊA PHƯƠNG VÀ CẤU TRÚC VÀNH

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<b>ANNIHILATOR OF LOCAL COHOMOLOGY MODULES AND </b>

<b>STRUCTURE OF RINGS </b>

<b>Tran Nguyen An </b>

<i>TNU - University of Education </i>

ABSTRACT

<i>Let (R, m) be a Noetherian local ring, A an Artinian module, and M a finitely generated </i>
<i>R-module. It is clear that Ann R(M/ p M) = p, for all p </i>∈<i> Var(Ann R M). Therefore, it is natural to </i>
consider the following dual property for annihilator of Artinian modules:

<i>Ann R(0 : A p) = p, for all p </i>∈<i> Var(Ann R A). (</i>∗)

<i>Let i ≥ 0 be an integer. Alexander Grothendieck showed that the local cohomology module Hmi </i>
<i>(M) of M is Artinian. The property (</i>∗) of local cohomology modules is closed related to the
structure of the base ring. In this paper, we prove that for each p ∈<i> Spec(R) such that Hmi (R/ p) </i>
<i>satisfies the property (*) for all i, then R/ p is universally catenary and the formal fibre of R over p </i>
is Cohen-Macaulay.

<i><b>Keywords: Local cohomology; universally catenary; formal fibre; Artinian module; </b></i>
<i>CohenMacaulay ring </i>

<i><b>Received: 26/5/2020; Revised: 29/8/2020; Published: 04/9/2020 </b></i>

<b>LINH HĨA TỬ CỦA MƠĐUN ĐỐI ĐỒNG ĐIỀU ĐỊA PHƯƠNG VÀ </b>

<b>CẤU TRÚC VÀNH </b>

<b>Trần Nguyên An </b>

<i>Trường Đại học Sư phạm - ĐH Thái Nguyên</i>

TÓM TẮT

<i>Cho (R, m) là vành Noether địa phương, A là R-môđun Artin, và M là R-môđun hữu hạn sinh. Ta </i>
<i>có Ann R(M/ p M) = p với mọi p </i>∈<i> Var(Ann R M). Do đó rất tự nhiên ta xét tính chất sau về linh </i>
hóa tử của môđun Artin

<i>Ann R(0 : A p) = p for all p </i>∈<i> Var(Ann R A). (</i>∗)

<i>Cho i ≥ 0 là số nguyên. Alexander Grothendieck đã chỉ ra rằng môđun đối đồng điều địa phương </i>
<i>Hi m(M) là Artin. Tính chất (</i>∗) của các mơđun đối đồng điều địa phương liên hệ mật thiết với cấu
trúc vành cơ sở. Trong bài báo này, chúng tôi chỉ ra với mỗi p ∈<i> Spec(R) mà Hmi (R/ p) thỏa mãn </i>
<i>tính chất (*) với mọi i thì R/ p là catenary phổ dụng và các thớ hình thức của R trên p là </i>
Cohen-Macaulay.

<i><b>Từ khóa: Đối đồng điều địa phương; catenary phổ dụng; thớ hình thức; môđun Artin; vành </b></i>
<i>Cohen-Macaulay </i>

<i><b>Ngày nhận bài: 26/5/2020; Ngày hoàn thiện: 29/8/2020; Ngày đăng: 04/9/2020 </b></i>

<i>Email: </i>

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1. Introduction

Throughout this paper, let (R,m) be a
Noetherian local ring, A an Artinian
R-module, and M a finitely generated
R-moduleofdimensiond.ForeachidealIofR,
wedenoteby Var(I)thesetofallprime

ide-alscontaining I.Fora subsetT of Spec(R),
we denote by min(T) theset of all minimal
elements of T undertheinclusion.

It is clear that AnnR(M/pM) = p, for all
p ∈ Var(AnnRM). Therefore, it is natural
to consider the following dual property for
annihilator ofArtinian modules:

AnnR(0:Ap)=p,∀p∈Var(AnnRA).(∗)

IfRiscompletewithrespecttom-adic
topol-ogy, it follows by Matlis duality that the
property (*) is satisfied for all Artinian
R-modules. However, there are Artinian
mod-ules which do not satisfy this property.For
example, by [1, Example 4.4], the Artinian
R-module H_m1(R) does notsatisfythe
prop-erty(*),whereRistheNoetherianlocal
do-main ofdimension2 constructedbyM.
Fer-rand and D. Raynaud [2] (see also [3, App.
Ex. 2] Ex. 2]) such that its m-adic

comple-b

tion R has an associated prime q of
dimen-sion 1.In [4],N.T. Cuong, L.T. Nhanand
N.T. Dungshowedthat thetoplocal
coho-mologymoduleH_md(M)satisfiesproperty(∗)
if and only if the ring R/AnnR(M/UM(0))

iscatenary,where UM(0) isthe largest
sub-module of M of dimensionless thand. The
property (∗)oflocalcohomologymodulesis
closedrelatedtothestructureofthering.In
[5], L.T. Nhanand theauthor provedthat
if Hi

m(M ) satisfies the property (*) for all i ,
then R/ p is unmixed for all p ∈ Ass M and
the ring R/ AnnRM is universally catenary.
The following conjecture was given by N. T.
Cuong in his seminar.

Conjecture 1.1. The following statements
are equivalent:

(i) H_mi(R) satisfiestheproperty (*)for alli;

(ii) R isuniversallycatenary andallits
for-mal flbers are Cohen-Macaulay.

L.T.Nhanand T.D.M.Chauprovedin[6]
thatH_mi(M)satisfiestheproperty(*)forall
i, for all finitely generated R-module M if
and only ifR isuniversallycatenary and all
its formal flbers are Cohen-Macaulay. The
followingresultisthemainresultofthis
pa-per. Wehope thatwecanuse this togive a
positive answerfortheabove conjecture.

Theorem 1.2. Assume p ∈ Spec(R) such
that H_mi(R/p) satisfies the property (*) for
all i. ThenR/p is universally catenary and
the formal fibre of R over p is
Cohen-Macaulay.

2. Proof

of

the

main

results

The theory of secondaryrepresentationwas
introduced by I. G. Macdonald (see [7])
which is in some sense dual to that of
pri-mary decomposition for Noetherian
mod-ules. Note that every Artinian R-module
A has a minimal secondary representation
A=A1+...+An,whereAi ispi-secondary.
The set {p₁,...,p_n} is independent of the
choice of theminimalsecondary
representa-tionofA.Thissetiscalledthesetofattached
prime ideals of A, and denoted by AttRA.
Note also thatA hasa natural structureas

b

an R-module. With this structure, a subset
of A is an R-submodule if and only if it is
an bR-submodule of A. Therefore, A is an
Ar-tinian bR-module.

Lemma 2.1. (i) The set of all minimal
ele-ments of AttRA is exactly the set of all

min-imal elements of Var(Ann_RA).

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R. N. Roberts introduced the concept of
Krull dimension for Artinian modules (see
[8]). D. Kirby changed the terminology of
Roberts and referred to Noetherian
dimen-sion to avoid confudimen-sion with Krull dimendimen-sion
defined for finitely generated modules (see
[9]). The Noetherian dimension of A is
de-noted by N-dimR(A). In this paper, we use
the terminology of Kirby (see [9]).

Lemma 2.2 ([1]). (i) N-dimR(A) 6
dim(R/ AnnRA), and the equality holds if A
satisfies the property (*).

(ii) N-dim_R(H_mi(M )) ≤ i, for all i.

The following property of attached primes
of the local cohomology under localization is
known as Weak general Shifted Localization
Principle (see [10]).

Lemma 2.3. We have AttRp(H

i−dim R/ p

pRp (Mp))

is the subset of {q Rp | q ∈

min AttR(Hmi(M )), q ⊆ p}, for all p ∈
Spec(R).

For an integer i ≥ 0, following M. Brodmann
and R. Y. Sharp (see [11]), the i-th pseudo
support of M , denoted by Psuppi_R(M ), is
defined by the set

{p ∈ Spec R | H_pi−dim R/ p_R

p (Mp) 6= 0}.

Note that the role of Psuppi_R(M ) for the
Artinian R-module A = Hi

m(M ) is in
some sense similar to that of Supp L for
a finitely generated R-module L, cf. [11],
[5]. Although, we always have Supp L =
Var(AnnRL), but the analogous equality
Psuppi_R(M ) = Var(AnnRHmi(M )) is not
valid in general. The following lemma gives
a necessary and sufficient conditions for the
above equality.

Lemma 2.4 ([5]). Let i ≥ 0 be an
inte-ger. Then the following statements are
equiv-alent:

(i) H_mi(M ) satisfies the property (*).

(ii) Var Ann_R(H_mi(M ))
= Psuppi

RM .

In particular, if H_mi(M ) satisfies the
prop-erty (*) then

min AttR(Hmi(M )) = min PsuppiRM.

In 2010, N. T. Cuong, L. T. Nhan and N.
T. K. Nga (see [12]) used pseudo support
to describe the non-Cohen-Macaulay locus
of M . Recall that M is equidimensional if
dim(R/ p) = d, for all p ∈ min(Ass M ).

Lemma 2.5 ([12]). Suppose that M is
equidimensional and the ring R/ Ann_RM
is catenary. Then Psuppi_R(M ) is closed for

i = 0, 1, d and nCM(M ) =
d−1

[

i=0

Psuppi_R(M ),

where nCM(M ) is the Non Cohen-Macaulay
locus of M .

Following M. Nagata ([3]), we say that M
is unmixed if dim( bR /bp) = d for all prime
idealsbp∈ Ass cM , and M is quasi unmixed if

c

M is equidimensional. The next lemma show
that the property (*) for the local
cohomol-ogy modules H_mi(M ) of levels i < d is closed
related to the universal catenaricity and
un-mixedness of certain local rings.

Lemma 2.6 ([5]). Assume that H_mi(M )
sat-isfies the property (*) for all i < d. Then
R/ p is unmixed for all p ∈ Ass M and the
ring R/ Ann_RM is universally catenary.

Proof of Theorem 1.2. It follows from the
Lemma 2.6 that R/ p = R/ AnnR(R/ p) is
universally catenary.

Set S to be the image of R \ p in bR. We have

Rp/ p Rp⊗RR ∼b= S−1( bR/ p bR).

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that (bq∩ R) ∩ S = ∅. Assume that the
state-ment is not true. Since

(S−1( bR/ p bR))_S−1
b

q∼= ( bR/ p bR)bq

as bR

bq-module, there existsbq∈ Spec( bR),bq∩
S = ∅ such that ( bR/ p bR)

bq is not

Cohen-Macaulay. Then there exists bp ∈
Spec(R),bq ⊇ bp, (bp∩ R) ∩ S = ∅ and bp ∈
Min nCM( bR/bp bR). Hence,

nCM(( bR/bp bR)
b
p) =

n

b
p bR

bp
o

.

We have R/ p is unmixed by Lemma
2.6. So bR/bp bR is equidimensional. Hence
( bR/bp bR)

b

p is equidimensional. On the other
hand, since ( bR/bp bR)

bp is the image of a
Cohen-Macaulay ring, ( bR/bp bR)

bp is
general-ized Cohen-Macaulay.

Set s = dim bR/bp bR = ht(bp/ p bR). By Lemma
2.5, we have

nCM( bR/bp bR)
b
p =
s−1
[
i=0
Psuppi
b

R(( bR/bp bR)bp).

Therefore, there exists i < s such that
Hi

bp bRbp

( bR/ p bR)
b

p6= 0. On the other hand,

`(Hi
bp bR

bp

( bR/ p bR)
b
p) < ∞.

Then

Att
b

R(H

i

bp bR
b

p

( bR/ p bR)
b
p) =

n
p bR

bp
o

.

It is followed by Weak general Shifted
Lo-calization Principle (Lemma 2.3) that bp ∈

Att
b

R(H

i+dim bR/bp

m ( bR/ p bR)). Set j = i +
dim bR/bp. We have

j < htbp/ p bR + dim bR/bp≤ dim bR/ p bR

= dim R/ p .

Hence, p ∈ AttR(Hmj(R/ p)) by Lemma 2.1.
By Lemma 2.2

N-dim H_mj(R/ p) ≤ j < dim R/ p

≤ R/ Ann_RH_mj(R/ p).

This impliesthat Hmj(R/p) does notsatisfy
theproperty(*).Itisincontradictiontothe
hypothesis. Therefore, all its formal fibers
over pareCohen-Macaulay.

3. Conclusion

The paper gives a relation between the
property (*) of local cohomology module
and structure of base ring. In detail, we
prove that for each p ∈ Spec(R) such that
H_mi(R/ p) satisfies the property (*) for all
i, then R/ p is universally catenary and the
formal fibre of R over p is Cohen-Macaulay.

References

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the Noetherian dimension of Artinian
modules," Vietnam Journal of
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2002.

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formelles d’un anneau local
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[3]. M. Nagata, Local rings, Interscience,
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[4]. C. T. Nguyen, D. T. Nguyen and N.
T. Le, "Toplocal cohomology and the
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afinitely generatedmodule,"
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[6]. N.T. Le andC. D.M.Tran,
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[7]. I.G.Macdonald,"Secondary
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[8]. R. N. Roberts, "Krull dimension for
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