Modules with minimax Cousin cohomologies
Let R be a commutative Noetherian ring with non-zero identity and let X be an arbitrary R-module. In this paper, we show that if all the cohomology modules of the Cousin complex for X are minimax, then the following hold for any prime ideal p of R and for every integer n less than X—the height of p:...
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2020
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| Цитувати: | Modules with minimax Cousin cohomologies / A. Vahidi // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 1. — С. 143–149. — Бібліогр.: 16 назв. — англ. |
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irk-123456789-1885582023-03-06T01:27:05Z Modules with minimax Cousin cohomologies Vahidi, A. Let R be a commutative Noetherian ring with non-zero identity and let X be an arbitrary R-module. In this paper, we show that if all the cohomology modules of the Cousin complex for X are minimax, then the following hold for any prime ideal p of R and for every integer n less than X—the height of p: (i) the nth Bass number of X with respect to p is finite; (ii) the nth local cohomology module of Xp with respect to pRp is Artinian. 2020 Article Modules with minimax Cousin cohomologies / A. Vahidi // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 1. — С. 143–149. — Бібліогр.: 16 назв. — англ. 1726-3255 DOI:10.12958/adm528 2010 MSC: 13D02, 13D03, 13D45, 13E10. http://dspace.nbuv.gov.ua/handle/123456789/188558 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let R be a commutative Noetherian ring with non-zero identity and let X be an arbitrary R-module. In this paper, we show that if all the cohomology modules of the Cousin complex for X are minimax, then the following hold for any prime ideal p of R and for every integer n less than X—the height of p: (i) the nth Bass number of X with respect to p is finite; (ii) the nth local cohomology module of Xp with respect to pRp is Artinian. |
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Article |
| author |
Vahidi, A. |
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Vahidi, A. Modules with minimax Cousin cohomologies Algebra and Discrete Mathematics |
| author_facet |
Vahidi, A. |
| author_sort |
Vahidi, A. |
| title |
Modules with minimax Cousin cohomologies |
| title_short |
Modules with minimax Cousin cohomologies |
| title_full |
Modules with minimax Cousin cohomologies |
| title_fullStr |
Modules with minimax Cousin cohomologies |
| title_full_unstemmed |
Modules with minimax Cousin cohomologies |
| title_sort |
modules with minimax cousin cohomologies |
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Інститут прикладної математики і механіки НАН України |
| publishDate |
2020 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/188558 |
| citation_txt |
Modules with minimax Cousin cohomologies / A. Vahidi // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 1. — С. 143–149. — Бібліогр.: 16 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT vahidia moduleswithminimaxcousincohomologies |
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2025-07-16T10:39:52Z |
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2025-07-16T10:39:52Z |
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1837799721902538752 |
| fulltext |
“adm-n3” — 2021/1/3 — 11:37 — page 143 — #149
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 30 (2020). Number 1, pp. 143–149
DOI:10.12958/adm528
Modules with minimax Cousin cohomologies∗
A. Vahidi
Communicated by V. Lyubashenko
Abstract. Let R be a commutative Noetherian ring with
non-zero identity and let X be an arbitrary R-module. In this paper,
we show that if all the cohomology modules of the Cousin complex
for X are minimax, then the following hold for any prime ideal p
of R and for every integer n less than X—the height of p:
(i) the nth Bass number of X with respect to p is finite;
(ii) the nth local cohomology module of Xp with respect to pRp
is Artinian.
Introduction
Throughout R will denote a commutative Noetherian ring with non-
zero identity, X an arbitrary R-module which is not necessarily finite (i.e.,
finitely generated), and M a non-zero finite R-module. For basic results,
notations and terminology not given in this paper, the reader is referred
to [2], [3], and [12].
The notion of the Cousin complex for an R-module X was introduced
by Sharp [13] as an analogue of Hartshorne [8]. The Cousin cohomologies
(i.e., the cohomology modules of the Cousin complex) have been studied
by several authors. Sharp used the vanishing of Cousin cohomologies for
investigating the Cohen-Macaulay property, Serre’s Sn-condition, and the
vanishing of Bass numbers of X in [13], [14], and [15]. Dibaei, Tousi, Jafari,
and Kawasaki, in [4], [5], [6], [7], and [10], worked on the finiteness of
∗This research was in part supported by a grant from Payame Noor University.
2010 MSC: 13D02, 13D03, 13D45, 13E10.
Key words and phrases: Artinian modules, Bass numbers, Cousin complexes,
local cohomology modules, minimax modules.
https://doi.org/10.12958/adm528
“adm-n3” — 2021/1/3 — 11:37 — page 144 — #150
144 Modules with minimax Cousin cohomologies
Cousin cohomologies and, in [11, Proposition 9.3.5], Lipman, Nayak, and
Sastry generalized their results to complexes on formal schemes.
Sharp, in [14, Theorem 2.4], showed that M is Cohen-Macaulay if and
only if the Cousin complex for M is exact. Thus we get the following
theorem.
Theorem 1. Let M be a non-zero finite R-module such that all the
cohomology modules of the Cousin complex for M are zero. Then the
followings hold for any prime ideal p of R and for every integer n less
than X—the height of p.
(i) The nth Bass number of M with respect to p is zero;
(ii) The nth local cohomology module of Mp with respect to pRp is zero.
Now, it is natural to ask whether a similar statement is valid if ‘zero’
is replaced by ‘finite’.
Question 1. Let X be an arbitrary R-module such that all the cohomology
modules of the Cousin complex for X are finite. Do the followings hold for
any prime ideal p of R and for every integer n less than X–height of p?
(i) The nth Bass number of X with respect to p is finite;
(ii) The nth local cohomology module of Xp with respect to pRp is finite.
In this paper, we answer the above question. We show that the first
part of Question 1 is true. In fact, in Theorem 2, we prove that the nth Bass
number of X with respect to p is finite for any prime ideal p of R and for
every integer n less than X–height of p, when all the cohomology modules
of the Cousin complex for X are minimax. Even though the second part
of Question 1 is false in general, we show in Theorem 3 that if all the
cohomology modules of the Cousin complex for X are minimax, then the
nth local cohomology module of Xp with respect to pRp is Artinian for
any prime ideal p of R and for every integer n less than X–height of p.
1. Main results
Suppose that X is an arbitrary R-module. Recall that, for a prime ideal
p of SuppR(X), the X–height of p is defined to be htX(p) = dimRp
(Xp). Let
i be a non-negative integer and set Ui(X) = {p ∈ SuppR(X) : htX(p) > i}.
Then SuppR(X) = U0(X), Ui(X) ⊇ Ui+1(X), and Ui(X)−Ui+1(X) (=
{p ∈ SuppR(X) : htX(p) = i}) is low with respect to Ui(X) (i.e., each
member of Ui(X)−Ui+1(X) is a minimal member of Ui(X) with respect
to inclusion). The Cousin complex CR(X) for X is of the form
CR(X) : 0
d−2
−→ X
d−1
−→ X0 d0
−→ X1 d1
−→ · · ·
di−2
−→ Xi−1 di−1
−→ Xi di
−→ · · ·
“adm-n3” — 2021/1/3 — 11:37 — page 145 — #151
A. Vahidi 145
where, for all i > 0,
• Xi =
⊕
p∈Ui(X)−Ui+1(X)(Coker d
i−2)p and
• di−1(x) = {x+Im di−2
1 }
p∈Ui(X)−Ui+1(X) for every element x of Xi−1;
and satisfies
• SuppR(X
i) ⊆ Ui(X),
• SuppR(Coker d
i−2) ⊆ Ui(X), and
• SuppR(H
i−1(CR(X))) ⊆ Ui+1(X)
(see [13] for details). Here, we use the notations Ci−2 := Coker di−2 and
Hi−1 := Hi−1(CR(X)) for all i > 0.
Recall that an R-module X is said to be minimax, if there is a finite
submodule X ′ of X such that X
X′ is Artinian [3]. Thus the class of minimax
modules includes all finite and all Artinian modules. Note that, for any
short exact sequence
0 −→ X ′ −→ X −→ X ′′ −→ 0
of R-modules, X is minimax if and only if X ′ and X ′′ are both minimax
[1, Lemma 2.1].
In the following, we state our first main result. Note that, for an
R-module X and a prime ideal p of R, the number
µn(p, X) = dim Rp
pRp
(ExtnRp
(
Rp
pRp
, Xp))
is the nth Bass number of X with respect to p.
Theorem 2. Let X be an arbitrary R-module such that Hi is minimax for
all i. Then µn(p, X) is finite for all prime ideals p of R and all n < htX(p).
Proof. Let p be a prime ideal of R and let n < htX(p). Let i be an integer
such that 0 6 i 6 n. By considering the short exact sequences
0 −→
Ci−2
Hi−1
−→ Xi −→ Ci−1 −→ 0 (1)
and
0 −→ Hi−1 −→ Ci−2 −→
Ci−2
Hi−1
−→ 0, (2)
we have the long exact sequences
0 −→ HomR(
R
p
,
Ci−2
Hi−1
) −→ HomR(
R
p
, Xi) −→ HomR(
R
p
,Ci−1)
“adm-n3” — 2021/1/3 — 11:37 — page 146 — #152
146 Modules with minimax Cousin cohomologies
−→ Ext1R(
R
p
,
Ci−2
Hi−1
) −→ Ext1R(
R
p
, Xi) −→ Ext1R(
R
p
,Ci−1)
−→ · · ·
−→ Extn−i−1
R (
R
p
,
Ci−2
Hi−1
) −→ Extn−i−1
R (
R
p
, Xi) −→ Extn−i−1
R (
R
p
,Ci−1)
−→ Extn−i
R (
R
p
,
Ci−2
Hi−1
) −→ Extn−i
R (
R
p
, Xi) −→ Extn−i
R (
R
p
,Ci−1)
−→ · · ·
and
0 −→ HomR(
R
p
,Hi−1) −→ HomR(
R
p
,Ci−2) −→ HomR(
R
p
,
Ci−2
Hi−1
)
−→ Ext1
R
(
R
p
,Hi−1) −→ Ext1
R
(
R
p
,Ci−2) −→ Ext1
R
(
R
p
,
Ci−2
Hi−1
)
−→ · · ·
−→ Extn−i−1
R
(
R
p
,Hi−1) −→ Extn−i−1
R
(
R
p
,Ci−2) −→ Extn−i−1
R
(
R
p
,
Ci−2
Hi−1
)
−→ Extn−i
R
(
R
p
,Hi−1) −→ Extn−i
R
(
R
p
,Ci−2) −→ Extn−i
R
(
R
p
,
Ci−2
Hi−1
)
−→ · · · .
Since Hi is minimax for all i, Extn−i
R (R
p
,Hi−1) is minimax for all 0 6 i 6 n.
On the other hand, by [13, Lemma 4.5],Extn−i
R (R
p
, Xi) = 0 for all 0 6 i 6 n.
Thus, from the above long exact sequences, Extn−i
R (R
p
,Ci−2) is minimax
whenever Extn−i−1
R (R
p
,Ci−1) is minimax. Hence ExtnR(
R
p
,C−2) is minimax.
Therefore ExtnR(
R
p
, X) is minimax. Thus there is a finite submodule E′ of
ExtnR(
R
p
, X) such that
ExtnR(R
p
,X)
E′ is Artinian. Since pRp(
ExtnRp
(
Rp
pRp
,Xp)
E′
p
) = 0,
ExtnRp
(
Rp
pRp
,Xp)
E′
p
is finite. Thus ExtnRp
(
Rp
pRp
, Xp) is finite. Hence µn(p, X) is
finite as we desired.
For an R-module X and an ideal a of R, we write Hn
a (X) as the nth
local cohomology module of X with respect to a. An important problem
in commutative algebra is to determine when Hn
a (X) is Artinian. In the
second main result of this paper, we show that for an arbitrary R-module
X (not necessarily finite) with minimax Cousin cohomologies, Hn
pRp
(Xp)
is Artinian for all prime ideals p of R and all n < htX(p), which is related
to the third of Huneke’s four problems in local cohomology modules [9].
“adm-n3” — 2021/1/3 — 11:37 — page 147 — #153
A. Vahidi 147
Theorem 3. Let X be an arbitrary R-module such that Hi is minimax
for all i. Then Hn
pRp
(Xp) is Artinian for all prime ideals p of R and all
n < htX(p).
Proof. The proof is similar to that of Theorem 2. We bring it here for
the sake of completeness. Let p be a prime ideal of R and let n < htX(p).
Let i be an integer such that 0 6 i 6 n. By considering the short exact
sequences (1) and (2), we have the long exact sequences
0 −→ ΓpRp
(
Ci−2
p
Hi−1
p
) −→ ΓpRp
(Xi
p) −→ ΓpRp
(Ci−1
p )
−→ H1
pRp
(
Ci−2
p
Hi−1
p
) −→ H1
pRp
(Xi
p) −→ H1
pRp
(Ci−1
p )
−→ · · ·
−→ Hn−i−1
pRp
(
Ci−2
p
Hi−1
p
) −→ Hn−i−1
pRp
(Xi
p) −→ Hn−i−1
pRp
(Ci−1
p )
−→ Hn−i
pRp
(
Ci−2
p
Hi−1
p
) −→ Hn−i
pRp
(Xi
p) −→ Hn−i
pRp
(Ci−1
p )
−→ · · ·
and
0 −→ ΓpRp
(Hi−1
p ) −→ ΓpRp
(Ci−2
p ) −→ ΓpRp
(
Ci−2
p
Hi−1
p
)
−→ H1
pRp
(Hi−1
p ) −→ H1
pRp
(Ci−2
p ) −→ H1
pRp
(
Ci−2
p
Hi−1
p
)
−→ · · ·
−→ Hn−i−1
pRp
(Hi−1
p ) −→ Hn−i−1
pRp
(Ci−2
p ) −→ Hn−i−1
pRp
(
Ci−2
p
Hi−1
p
)
−→ Hn−i
pRp
(Hi−1
p ) −→ Hn−i
pRp
(Ci−2
p ) −→ Hn−i
pRp
(
Ci−2
p
Hi−1
p
)
−→ · · · .
Since Hi is minimax for all i, there is a finite submodule Hi′ of Hi such
that Hi
Hi′
is Artinian. Therefore, from the exact sequence
Hn−i
pRp
(Hi−1′
p ) −→ Hn−i
pRp
(Hi−1
p ) −→ Hn−i
pRp
(
Hi−1
p
Hi−1′
p
),
“adm-n3” — 2021/1/3 — 11:37 — page 148 — #154
148 Modules with minimax Cousin cohomologies
Hn−i
pRp
(Hi−1
p ) is Artinian for all 0 6 i 6 n. On the other hand, by [13,
Lemma 4.5], for all 0 6 i 6 n and all j > 0, Extn−i
R ( R
pj
, Xi) = 0 and so
Hn−i
pRp
(Xi
p)
∼= (Hn−i
p (Xi))p = 0 because
Hn−i
p (Xi) ∼= lim
−→
j>0
Extn−i
R (
R
pj
, Xi).
Thus, from the above long exact sequences, Hn−i
pRp
(Ci−2
p ) is Artinian when-
ever Hn−i−1
pRp
(Ci−1
p ) is Artinian. Hence Hn
pRp
(C−2
p ) is Artinian. Therefore
Hn
pRp
(Xp) is Artinian.
The following corollaries are immediate applications of the above
theorems.
Corollary 1. Let X be an arbitrary R-module such that Hi is finite for
all i. Then
(i) µn(p, X) is finite and
(ii) Hn
pRp
(Xp) is Artinian
for all prime ideals p of R and all n < htX(p).
Corollary 2. Let X be an arbitrary R-module such that Hi is Artinian
for all i. Then
(i) µn(p, X) is finite and
(ii) Hn
pRp
(Xp) is Artinian
for all prime ideals p of R and all n < htX(p).
References
[1] K. Bahmanpour, R. Naghipour, On the cofiniteness of local cohomology modules,
Proc. Amer. Math. Soc. 136 (2008), 2359–2363.
[2] M.P. Brodmann, R.Y. Sharp, Local Cohomology: An Algebraic Introduction with
Geometric Applications, Cambridge, Cambridge University Press, 1998.
[3] W. Bruns, J. Herzog, Cohen-Macaulay Rings, Cambridge, Cambridge University
Press, 1998.
[4] M.T. Dibaei, M. Tousi, The structure of dualizing complex for a ring which is (S2),
J. Math. Kyoto Univ. 38 (1998), 503–516.
[5] M.T. Dibaei, M. Tousi, A generalization of the dualizing complex structure and its
applications, J. Pure Appl. Algebra 155 (2001), 17–28.
[6] M.T. Dibaei, A study of Cousin complexes through the dualizing complexes, Comm.
Algebra 33 (2005), 119–132.
[7] M.T. Dibaei, R. Jafari, Modules with finite Cousin cohomologies have uniform local
cohomological annihilators, J. Algebra 319 (2008), 3291–3300.
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A. Vahidi 149
[8] R. Hartshorne, Residues and Duality, Springer, 1966.
[9] C. Huneke, Problems on Local Cohomology: Free Resolutions in Commutative
Algebra and Algebraic Geometry, Jones and Bartlett, 1992.
[10] T. Kawasaki, Finiteness of Cousin cohomologies, Trans. Amer. Math. Soc. 360
(2008), 2709–2739.
[11] J. Lipman, S. Nayak, P. Sastry, Pseudofunctorial behavior of Cousin complexes
on formal schemes, Contemp. Math. 375 (2005), 3–133.
[12] J. Rotman, An Introduction to Homological Algebra, Academic Press, 1979.
[13] R.Y. Sharp, The Cousin complex for a module over a commutative Noetherian
ring, Math. Z. 112 (1969), 340–356.
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Noetherian rings, J. London Math. Soc. 3 (1971), 621–624.
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Contact information
Alireza Vahidi Department of Mathematics, Payame Noor
University (PNU), P.O. Box 19395-4697,
Tehran, Iran
E-Mail(s): vahidi.ar@pnu.ac.ir
Received by the editors: 25.08.2017
and in final form 15.08.2018.
mailto:vahidi.ar@pnu.ac.ir
A. Vahidi
|