Local cohomology modules and their properties
UDC 512.5 Let $(R, \mathfrak{m})$ be a complete Noetherian local ring and let $M$ be a generalized Cohen-Macaulay $R$-module of dimension $d \geq 2.$We show that$$D \left(H_{\mathfrak{m}}^d\Big(D \big(H_{\mathfrak{m}}^d(D_{\mathfrak{m}}(M))\big)\Big)\right) \approx D_{\mathfrak{m}} (M),$$where $D =...
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| Дата: | 2021 |
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Institute of Mathematics, NAS of Ukraine
2021
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860506980718739456 |
|---|---|
| author | Azami, J. Hasanzad, M. Azami, J. Hasanzad, M. |
| author_facet | Azami, J. Hasanzad, M. Azami, J. Hasanzad, M. |
| author_sort | Azami, J. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:48:28Z |
| description | UDC 512.5
Let $(R, \mathfrak{m})$ be a complete Noetherian local ring and let $M$ be a generalized Cohen-Macaulay $R$-module of dimension $d \geq 2.$We show that$$D \left(H_{\mathfrak{m}}^d\Big(D \big(H_{\mathfrak{m}}^d(D_{\mathfrak{m}}(M))\big)\Big)\right) \approx D_{\mathfrak{m}} (M),$$where $D = {\rm Hom} (-, E)$ and $D_{\mathfrak{m}} (-)$ is the ideal transform functor. Also, assuming that $I$ is a proper ideal of a local ring $R$, we obtain some results on the finiteness of Bass numbers, cofinitness, and cominimaxness of local cohomology modules with respect to $I.$ |
| doi_str_mv | 10.37863/umzh.v73i2.127 |
| first_indexed | 2026-03-24T02:02:02Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
DOI: 10.37863/umzh.v73i2.127
UDC 512.5
J. Azami, M. Hasanzad (Univ. Mohaghegh Ardabili, Ardabil, Iran)
LOCAL COHOMOLOGY MODULES AND THEIR PROPERTIES
ЛОКАЛЬНI КОГОМОЛОГIЧНI МОДУЛI ТА ЇХНI ВЛАСТИВОСТI
Let (R,m) be a complete Noetherian local ring and let M be a generalized Cohen-Macaulay R-module of dimension
d \geq 2. We show that
D
\Bigl(
Hd
m
\Bigl(
D
\bigl(
Hd
m(Dm(M))
\bigr) \Bigr) \Bigr)
\approx Dm(M),
where D = \mathrm{H}\mathrm{o}\mathrm{m}( - , E) and Dm( - ) is the ideal transform functor. Also, assuming that I is a proper ideal of a local ring
R, we obtain some results on the finiteness of Bass numbers, cofinitness, and cominimaxness of local cohomology modules
with respect to I.
Нехай (R,m) — повне локальне нетерове кiльце, а M — узагальнений R-модуль Коена – Маколея, що має розмiрнiсть
d \geq 2. Доведено, що
D
\Bigl(
Hd
m
\Bigl(
D
\bigl(
Hd
m(Dm(M))
\bigr) \Bigr) \Bigr)
\approx Dm(M),
де D = \mathrm{H}\mathrm{o}\mathrm{m}( - , E) i Dm( - ) — функтор перетворення iдеалу. Також якщо I є нетривiальним iдеалом локального
кiльця R, отримано деякi результати щодо фiнiтностi чисел Басса, кофiнiтностi та комiнiмаксностi локальних
модулiв когомологiї вiдносно I.
1. Introduction. Throughout this paper, let R denote a commutative Noetherian local ring (with
identity) and I an ideal of R. For an R-module M, the ith local cohomology module of M with
respect to I is defined as
H i
I(M) = \mathrm{l}\mathrm{i}\mathrm{m} - \rightarrow
n\geq 1
\mathrm{E}\mathrm{x}\mathrm{t}iR(R/In,M).
For more details about local cohomology modules see [2, 4]. We shall refer to DI as the I -
transform functor. Note that this functor is left exact. For an R-module M, we call DI(M) =
= \mathrm{l}\mathrm{i}\mathrm{m} - \rightarrow
n\geq 1
\mathrm{H}\mathrm{o}\mathrm{m}R(I
n,M) the ideal transform of M with respect to I, or, alternatively, the I -transform
of M.
Let (R,m) be a Noetherian local ring and let M be a non-zero finitely generated R-module
of dimension n > 0. We say that M is a generalized Cohen – Macaulay R-module precisely when
H i
I(M) is finitely generated for all i \not = n. (Such modules were called ”quasi-Cohen – Macaulay
modules” by P. Schenzel in [11].)
Also we shall use D to denote the exact, contravariant, R-linear functor \mathrm{H}\mathrm{o}\mathrm{m}R( - , E), where
E := E(R/m) is the injective envelope of the simple R-module R/m. For each R-module L, we
denote set \{ p \in \mathrm{A}\mathrm{s}\mathrm{s}R L : \mathrm{d}\mathrm{i}\mathrm{m}R/p = \mathrm{d}\mathrm{i}\mathrm{m}L\} by \mathrm{A}\mathrm{s}\mathrm{s}\mathrm{h}RL. Also, for any ideal b of R, the radical
of b, denoted by \mathrm{R}\mathrm{a}\mathrm{d}(b), is defined to be the set \{ x \in R : xn \in b for some n \in \BbbN \} and we denote
\{ p \in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(R) : p \supseteq b\} by V (b). For any unexplained notation and terminology we refer the reader
to [2, 3, 6].
c\bigcirc J. AZAMI, M. HASANZAD, 2021
268 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
LOCAL COHOMOLOGY MODULES AND THEIR PROPERTIES 269
2. Ideal transform and local cohomology.
Lemma 2.1. Let (R,m) be a Noetherian local ring and let M be an R-module of dimension
d \geq 2. Then D
\bigl(
Hd
m
\bigl(
D
\bigl(
Hd
m(M)
\bigr) \bigr) \bigr)
\approx D
\bigl(
Hd
m
\bigl(
D
\bigl(
Hd
m (Dm(M))
\bigr) \bigr) \bigr)
, where D = \mathrm{H}\mathrm{o}\mathrm{m}( - , E) and
Dm( - ) is the ideal transform.
Proof. Consider the exact sequence
0 - \rightarrow M
\Gamma m(M)
- \rightarrow Dm
\biggl(
M
\Gamma m(M)
\biggr)
- \rightarrow H1
m
\biggl(
M
\Gamma m(M)
\biggr)
- \rightarrow 0
or
0 - \rightarrow M
\Gamma m(M)
- \rightarrow Dm(M) - \rightarrow H1
m(M) - \rightarrow 0.
The above exact sequences induce an exact sequence
H1
m
\bigl(
H1
m(M)
\bigr)
- \rightarrow H2
m
\biggl(
M
\Gamma m(M)
\biggr)
- \rightarrow H2
m(Dm(M)) - \rightarrow H2
m
\bigl(
H1
m(M)
\bigr)
- \rightarrow . . . .
Since H1
m
\bigl(
H1
m(M)
\bigr)
= H2
m
\bigl(
H1
m(M)
\bigr)
= 0, hence H2
m
\biggl(
M
\Gamma m(M)
\biggr)
\approx H2
m(Dm(M)). Now, for all
i \geq 2 and, in particular, for i = d, we have
H i
m(M) = H i
m
\biggl(
M
\Gamma m(M)
\biggr)
\approx H i
m(Dm(M)), Hd
m(M) \approx Hd
m(Dm(M)).
Consequently, D
\bigl(
Hd
m
\bigl(
D
\bigl(
Hd
m(M)
\bigr) \bigr) \bigr)
\approx D
\bigl(
Hd
m
\bigl(
D
\bigl(
Hd
m (Dm(M))
\bigr) \bigr) \bigr)
.
Lemma 2.2. Let (R,m) be a complete Noetherian local ring and let M be a generalized Co-
hen – Macaulay R-module of dimension d \geq 2. Let x1, . . . , xd be an m-filter regular sequence
for Dm(M) and for 0 \leq i \leq d, we set Li = H i
m(Dm(M)), Ki = H i
(x1,...,xi)
(Dm(M)) and
Ti =
\biggl(
Ki
Li
\biggr)
xi+1
. Then
Hd
m(D(Ld)) \approx Hd - 1
m
\biggl(
D
\biggl(
Kd - 1
Ld - 1
\biggr) \biggr)
,
Hd - 1
m (D(Kd - 1)) \approx Hd - 2
m
\biggl(
D
\biggl(
Kd - 2
Ld - 2
\biggr) \biggr)
.
Proof. Let N = Dm(M). Since \Gamma m(N) = H1
m(N) = 0, so N is a generalized Cohen – Macualay
module of dimension d \geq 2. Now, let x1, . . . , xd be an m-filter regular sequence for N. Consider
the following exact sequences:
0 - \rightarrow N - \rightarrow T0 - \rightarrow K1 - \rightarrow 0,
0 - \rightarrow K1 - \rightarrow T1 - \rightarrow K2 - \rightarrow 0,
0 - \rightarrow K2
L2
- \rightarrow T2 - \rightarrow K3 - \rightarrow 0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
270 J. AZAMI, M. HASANZAD
0 - \rightarrow Kd - 2
Ld - 2
- \rightarrow Td - 2 - \rightarrow Kd - 1 - \rightarrow 0,
0 - \rightarrow Kd - 1
Ld - 1
- \rightarrow Td - 1 - \rightarrow Ld - \rightarrow 0,
where, by [10] (Corollary 2.6), K2 \approx H1
Rx2
(K1) and also we have the following:
L2 = H2
m(N) \approx \Gamma Rx3
\bigl(
H2
Rx1+Rx2
(N)
\bigr)
\approx \Gamma m
\Bigl(
H2
(x1,x2)
(N)
\Bigr)
\approx \Gamma m(K2).
The above exact sequences induces the following exact sequences:
0 - \rightarrow D(Ld) - \rightarrow D(Td - 1) - \rightarrow D
\biggl(
Kd - 1
Ld - 1
\biggr)
- \rightarrow 0,
0 - \rightarrow D(Kd - 1) - \rightarrow D(Td - 2) - \rightarrow D
\biggl(
Kd - 2
Ld - 2
\biggr)
- \rightarrow 0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0 - \rightarrow D(K3) - \rightarrow D(T2) - \rightarrow D
\biggl(
K2
L2
\biggr)
- \rightarrow 0,
0 - \rightarrow D(K2) - \rightarrow D(T1) - \rightarrow D(K1) - \rightarrow 0,
0 - \rightarrow D(K1) - \rightarrow D(T0) - \rightarrow D(N) - \rightarrow 0.
Since for xi+1 \in m, the map Ti
xi+1 - - - \rightarrow Ti is an isomorphism, so, for all j \geq 0, the map
Hj
m(D(Ti))
xi+1 - - - \rightarrow Hj
m(D(Ti))
is an isomorphism. On the other hand, Hj
m(D(Ti)) is m-torsion and so is an Rxi+1-torsion. It
follows that Hj
m(D(Ti)) = 0 for all j \geq 0. Therefore, we have
Hd
m(D(Ld)) \approx Hd - 1
m
\biggl(
D
\biggl(
Kd - 1
Ld - 1
\biggr) \biggr)
,
Hd - 1
m (D(Kd - 1)) \approx Hd - 2
m
\biggl(
D
\biggl(
Kd - 2
Ld - 2
\biggr) \biggr)
, (2.1)
H2
m(D(K2)) \approx H1
m(D(K1)),
H1
m(D(K1)) \approx D(N).
Theorem 2.1. Let the situation and notation be as in Lemma 2.2. Then
D
\Bigl(
Hd
m
\Bigl(
D
\Bigl(
Hd
m(Dm(M))
\Bigr) \Bigr) \Bigr)
\approx Dm(M).
Proof. Note that, for all 2 \leq i \leq d - 1, the R-module Li is of finite length and so is D(Li).
Hence, for all j \geq 1 and 2 \leq i \leq d - 1, Hj
m(D(Li)) = 0. By the notation of previous lemma, the
exact sequence
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
LOCAL COHOMOLOGY MODULES AND THEIR PROPERTIES 271
0 - \rightarrow Li - \rightarrow Ki - \rightarrow
Ki
Li
- \rightarrow 0
induces the exact sequence
0 - \rightarrow D
\biggl(
Ki
Li
\biggr)
- \rightarrow D(Ki) - \rightarrow D(Li) - \rightarrow 0.
Now we can write
Hj
m
\biggl(
D
\biggl(
Ki
Li
\biggr) \biggr)
\approx Hj
m(D(Ki)) (2.2)
for j \geq 2. Finally, from (2.1) and (2.2), we have the following:
Hd
m
\Bigl(
D
\Bigl(
Hd
m(N)
\Bigr) \Bigr)
\approx Hd - 1
m
\biggl(
D
\biggl(
Kd - 1
Ld - 1
\biggr) \biggr)
\approx Hd - 1
m (Kd - 1) \approx Hd - 2
m
\biggl(
Kd - 2
Ld - 2
\biggr)
\approx
\approx Hd - 2
m (Kd - 2) \approx \cdot \cdot \cdot \approx H2
m(D(K2)) \approx H1
m(D(K1)) \approx D(N).
Consequently, Hd
m
\bigl(
D
\bigl(
Hd
m(N)
\bigr) \bigr)
\approx D(N). Since R is complete, it follows that
D
\Bigl(
Hd
m
\Bigl(
D
\Bigl(
Hd
m(N)
\Bigr) \Bigr) \Bigr)
\approx D(D(N)) \approx N = Dm(M).
Corollary 2.1. Let (R,m) be a complete Noetherian local ring and let M be a generalized Co-
hen – Macaulay R-module of dimension d \geq 2. Then
\bigl(
D(Hd
m(M)
\bigr)
is Cohen – Macaulay iff Dm(M)
is Cohen – Macaulay and this is equivalent to the following:\bigl\{
i \in \BbbN 0 : H i
m(M) \not = 0
\bigr\}
\subseteq \{ 0, 1, d\} .
Proof. The assertion follows immediately from above theorem.
3. Cofiniteness and cominimaxness of local cohomology modules.
Lemma 3.1. Let I be an ideal of a commutative Noetherian ring R of dimension one. Let
M(R, I)com denote the category of I -cominimax modules over R. Then M(R, I)com forms an
Abelian subcategory of the category of all R-modules. That is, if f : M - \rightarrow N is an R-homomorphism
of I -cominimax modules, then \mathrm{k}\mathrm{e}\mathrm{r} f and \mathrm{c}\mathrm{o}\mathrm{k}\mathrm{e}\mathrm{r}f are I -cominimax.
Proof. See [5] (Theorem 2.6).
Remark 3.1. For Noetherian local ring (R,m) of dimension d \geq 1 and proper ideal I of R,
we set
T1 = \{ p \in \mathrm{A}\mathrm{s}\mathrm{s}\mathrm{h}(R)| \mathrm{R}\mathrm{a}\mathrm{d}(p+ I) = m\} ,
T2 = \mathrm{A}\mathrm{s}\mathrm{s}R(R)\setminus T1.
Let 0 =
\bigcap
pi\in Ass(R)
qi be a minimal primary decomposition for the zero ideal of R such that qi is
pi-primary. If L1 =
\bigcap
qi\in T1
qi and L2 =
\bigcap
qi\in T2
qi, then \mathrm{A}\mathrm{s}\mathrm{s}
R
L1
= T1 and \mathrm{A}\mathrm{s}\mathrm{s}
R
L2
= T2. By [2]
(Theorem 8.2.1), Hd
I
\biggl(
R
L2
\biggr)
= 0 and H i
I
\biggl(
R
L1
\biggr)
\approx H i
m
\biggl(
R
L1
\biggr)
for all i \geq 0. Thus, H i
I
\biggl(
R
L1
\biggr)
is
Artinian for all i \geq 0 and \mathrm{c}\mathrm{d}
\biggl(
I,
R
L2
\biggr)
\leq d - 1.
On the other hand, \mathrm{A}\mathrm{n}\mathrm{n}(L2) \subseteq \mathrm{A}\mathrm{n}\mathrm{n}(L2M), so \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(L2M) \subseteq \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(L2). Therefore, H i
I(L2M) \approx
\approx H i
m(L2M) for all i \geq 0 and H i
I(L2M) is Artinian for i \geq 0.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
272 J. AZAMI, M. HASANZAD
Theorem 3.1. Let (R,m) be a Noetherian local ring of dimension d \geq 1 and I be a proper
ideal of R. Then the following statements are equivalent:
(1) the Bass-numbers of Hd - 1
I (R) are finite;
(2) for any finitely generated R-module M, the Bass numbers of Hd - 1
I (M) are finite;
(3) Hd - 1
I (R) is I -cominimax;
(4) for any finitely generated R-module M, the R-module Hd - 1
I (M) is I -cominimax.
Proof. 1 \updownarrow 3. Follows from [1] (Theorem 2.12).
2 \rightarrow 1. Is clear.
2 \updownarrow 4. If \mathrm{d}\mathrm{i}\mathrm{m}M = d the assertion follows from [8] (Proposition 5.1). If \mathrm{d}\mathrm{i}\mathrm{m}M \leq d - 1, then
Hd - 1
I (M) is Artinian.
1 \rightarrow 2. Let M be a finitely generated R-module. If \mathrm{d}\mathrm{i}\mathrm{m}M < d - 1, then Hd - 1
I (M) = 0 and
the result follows. If \mathrm{d}\mathrm{i}\mathrm{m}M = d - 1, then Hd - 1
I (M) is Artinian and by [8] (Proposition 5.1).
Therefore we assume that \mathrm{d}\mathrm{i}\mathrm{m}M = d. By [1] (Theorem 2.12), the Bass numbers of Hd - 1
I (M)
are finite iff \mathrm{H}\mathrm{o}\mathrm{m}R
\biggl(
R
m
, Hd - 1
I (M)
\biggr)
be a finitely generated. Therefore with out lose of generality,
we may assume that (R,m) is a complete Noetherian local ring. By notation in Remark 3.1, from
the exact sequence
0 - \rightarrow L2 - \rightarrow
R
L1
- \rightarrow R
L1 + L2
- \rightarrow 0,
we have \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(L2) \subseteq \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}
\biggl(
R
L1
\biggr)
and H i
I(L2) \approx H i
m(L2) for i \geq 0. Also, the exact sequence
0 - \rightarrow L2 - \rightarrow R - \rightarrow R
L2
- \rightarrow 0
induces the following exact sequence:
Hd - 1
I (L2) - \rightarrow Hd - 1
I (R) - \rightarrow Hd - 1
I
\biggl(
R
L2
\biggr)
- \rightarrow Hd
I (L2) - \rightarrow . . . ,
which implies that Hd - 1
I
\biggl(
R
L2
\biggr)
is I -cominimax. Since \mathrm{c}\mathrm{d}
\biggl(
I,
R
L2
\biggr)
\leq d - 1, it follows that
Hd - 1
I
\biggl(
R
L2
\biggr)
\otimes M \approx Hd - 1
I
\biggl(
R
L2
\otimes M
\biggr)
\approx Hd - 1
I
\biggl(
M
L2M
\biggr)
.
Thus, by Lemma 3.1, Hd - 1
I
\biggl(
M
L2M
\biggr)
is I -cominimax (for this we consider a free resolution of M ).
Also, from the exact sequence
0 - \rightarrow L2M - \rightarrow M - \rightarrow M
L2M
- \rightarrow 0
we have the following exact sequence:
Hd - 1
I (L2M) - \rightarrow Hd - 1
I (M) - \rightarrow Hd - 1
I
\biggl(
M
L2M
\biggr)
- \rightarrow Hd
I (L2M).
By Remark 3.1, it follows that Hd - 1
I (M) is I -cominimax.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
LOCAL COHOMOLOGY MODULES AND THEIR PROPERTIES 273
Lemma 3.2. Let R be a Noetherian ring, I an ideal of R and M an R-module such that
\mathrm{d}\mathrm{i}\mathrm{m}M \leq 1. Then for all n \geq 0 and all finitely generated R-module K, the R-module \mathrm{T}\mathrm{o}\mathrm{r}Rn (K,M)
is I -cofinite.
Proof. See [9] (Lemma 3.3).
Lemma 3.3. Let (R,m) be a Noetherian local ring and M be a non-zero finitely generated
R-module such that
\surd
I +\mathrm{A}\mathrm{n}\mathrm{n}M = m. Then R-module Hn
I (M) is Artinian and I -cofinite, for all
n \geq 0.
Proof. We have the following relations:
Hn
I (M) \simeq Hn
I+AnnM/AnnM (M) = Hn
m/AnnM (M) \simeq Hn
m(M).
So the R-module Hn
I (M) is Artinian. On the other hand, we have
\mathrm{H}\mathrm{o}\mathrm{m}R(R/I,Hn
I (M)) \simeq \mathrm{H}\mathrm{o}\mathrm{m}R (R/I,\mathrm{H}\mathrm{o}\mathrm{m}R (R/\mathrm{A}\mathrm{n}\mathrm{n}M,Hn
I (M))) \simeq
\simeq \mathrm{H}\mathrm{o}\mathrm{m}R (R/\mathrm{A}\mathrm{n}\mathrm{n}M + I,Hn
I (M)) .
Now, since the R-module \mathrm{H}\mathrm{o}\mathrm{m}R (R/I +\mathrm{A}\mathrm{n}\mathrm{n}M,Hn
I (M)) is of finite length and Hn
I (M) is Arti-
nian, so Hn
I (M) is I -cofinite by [8] (Proposition 4.1).
Remark 3.2. Let (R,m) be a Noetherian complete local ring of dimension d \geq 1 and let I be
an ideal of R. If \mathrm{s}\mathrm{u}\mathrm{p}\{ n \in \BbbN 0 : Hn
I (M) \not = 0\} = d, since R is complete, then from Lichtenbaum –
Hartshorn vanishing theorem, the set A =
\bigl\{
p \in \mathrm{A}\mathrm{s}\mathrm{s}R|
\surd
p+ I = m
\bigr\}
is non empty. Set J =
\bigcap
p\in A
p.
Also we have m\mathrm{A}\mathrm{s}\mathrm{s}M/\Gamma J(M) \subseteq \mathrm{A}\mathrm{s}\mathrm{s}M\setminus V (J) \subseteq \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(R)\setminus A. Then by Lichtenbaum – Hartshorn
vanishing theorem, Hd
I (R/\Gamma J(R)) = 0. Since M/\Gamma J(M) is an R/\Gamma J(R)-module, it follows that
Hd
I (M/\Gamma J(M)) = 0.
On the other hand, \mathrm{A}\mathrm{s}\mathrm{s} \Gamma J(R) = \mathrm{A}\mathrm{s}\mathrm{s}R \cap V (J) = A. So,
\sqrt{}
\mathrm{A}\mathrm{n}\mathrm{n}\Gamma J(R) + I = m. In particular,
by Lemma 3.3, the R-module H i
I(\Gamma J(R)) is Artinian and I -cofinite for each i.
Theorem 3.2. Let (R,m) be a Noetherian complete local ring of dimension d \geq 1 and let I be
an ideal of R. Then the following statements are equivalent:
(i) Hd - 1
I (R) is I -cofinite;
(ii) for every finitely generated R-module M, the R-module Hd - 1
I (M) is I -cofinite.
Proof.
(ii)\rightarrow (i) is clear.
(i)\rightarrow (ii) If \mathrm{d}\mathrm{i}\mathrm{m}M < d - 1, then Hd - 1
I (M) = 0. If \mathrm{d}\mathrm{i}\mathrm{m}M = d - 1, then by [8] (Proposition 5.1),
Hd - 1
I (M) is I -cofinite.
Now, let \mathrm{d}\mathrm{i}\mathrm{m}M = d and \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
n \in \BbbN 0 : Hn
I (M) \not = 0
\bigr\}
= d - 1. Then by [2] (Excercise 6.1.8),
Hd - 1
I (M) \simeq Hd - 1
I (R)\otimes R M and by Lemma 3.1, Hd - 1
I (M) is I -cofinite. Note that, in view of [7]
(Corollary 2.5), \mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}Hd - 1
I (R) is finite and so its dimension is at most one.
Therefore, we assume that \mathrm{s}\mathrm{u}\mathrm{p} \{ n \in \BbbN 0 : Hn
I (M) \not = 0\} = d. By notation in Remark 3.2, we
have the following exact sequence:
0 - \rightarrow \Gamma J(R) - \rightarrow R - \rightarrow R/\Gamma J(R) - \rightarrow 0,
that induces the long exact sequence
. . . - \rightarrow Hd - 1
I (\Gamma J(R))
f - \rightarrow Hd - 1
I (R)
g - \rightarrow Hd - 1
I (R/\Gamma J(R))
h - \rightarrow Hd
I (\Gamma J(R)) - \rightarrow . . . .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
274 J. AZAMI, M. HASANZAD
Note that the category of Artinian I -cofinite modules is a Serre category and so \mathrm{I}\mathrm{m}f is I -cofinite.
Now from the exact sequence
0 - \rightarrow \mathrm{I}\mathrm{m} f - \rightarrow Hd - 1
I (R) - \rightarrow \mathrm{I}\mathrm{m} g - \rightarrow 0,
we deduce that \mathrm{I}\mathrm{m}g is I -cofinite.
Since \mathrm{I}\mathrm{m}h is also I -cofinite, it follows from the exact sequence
0 - \rightarrow \mathrm{I}\mathrm{m} g - \rightarrow Hd - 1
I (R/\Gamma J(R)) - \rightarrow \mathrm{I}\mathrm{m}h - \rightarrow 0
that Hd - 1
I (R/\Gamma J(R)) is I -cofinite and of dimension at most one. Now, by Lemma 3.2 and the fact
that Hd - 1
I (M/\Gamma J(M)) \simeq Hd - 1
I (R/\Gamma J(R))\otimes RM, we deduce that Hd - 1
I (M/\Gamma J(M)) is I -cofinite.
From the exact sequence
0 - \rightarrow \Gamma J(M) - \rightarrow M - \rightarrow M/\Gamma J(M) - \rightarrow 0,
we obtain the following long exact sequence:
. . . - \rightarrow Hd - 1
I (\Gamma J(M))
f1 - \rightarrow Hd - 1
I (M)
f2 - \rightarrow Hd - 1
I (M/\Gamma J(M))
f3 - \rightarrow Hd
I (\Gamma J(M)) - \rightarrow . . . .
For all n \geq 0, the R-module Hn
I (\Gamma J(M)) is Artinian and I -cofinite, so by the above long exact
sequence, \mathrm{I}\mathrm{m}f3 and \mathrm{I}\mathrm{m}f2 are I -cofinite. Therefore, from the exact sequence
0 - \rightarrow \mathrm{I}\mathrm{m}f1 - \rightarrow Hd - 1
I (M) - \rightarrow \mathrm{I}\mathrm{m}f2 - \rightarrow 0,
we deduce that Hd - 1
I (M) is also I -cofinite.
References
1. K. Bahmanpour, R. Naghipour, M. Sedghi, On the finiteness of Bass numbers of local cohomology modules and
cominimaxness, Houston J. Math., 40, № 2, 319 – 337 (2014).
2. M. P. Brodmann, R. Y. Sharp, Local cohomology; an algebraic introduction with geometric applications, Cambridge
Univ. Press, Cambridge (1998).
3. W. Bruns, J. Herzog, Cohen Macualay rings, Cambridge Stud. Adv. Math. (1997).
4. A. Grothendieck, Local cohomology, Notes by R. Hartshorne, Lect. Notes Math., 862, Springer, New York (1966).
5. Y. Irani, K. Bahmanpour, Cominimaxness of local cohomology modules for ideals of dimension one, preprint.
6. H. Matsumura, Commutative ring theory, Cambridge Univ. Press, Cambridge, UK (1986).
7. T. Marley, The associated primes of local cohomology modules over rings of small dimension, Manuscripta Math.,
104, 519 – 525 (2001).
8. L. Melkersson, Modules cofinite with respect to an ideal, J. Algebra, 285, 649 – 668 (2005).
9. R. Naghipour, K. Bahmanpour, I. Khalili Gorji, Cofiniteness of torsion functors of cofinite modules, Colloq. Math.,
136, 221 – 230 (2014).
10. P. Schenzel, Proregular sequences, local cohomology, and completion, Math. Scand., 92, 161 – 180 (2003).
11. P. Schenzel, Einige Anwendungen der lokalen Dualitat und verallgemeinerte Cohen – Macaulay – Moduln, Math.
Nachr., 69, 227 – 242 (1975).
Received 04.06.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
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| id | umjimathkievua-article-127 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:02Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f8/7190eb7f1fef3b19e0f591aebc5a07f8.pdf |
| spelling | umjimathkievua-article-1272025-03-31T08:48:28Z Local cohomology modules and their properties Local cohomology modules and their properties Azami, J. Hasanzad, M. Azami, J. Hasanzad, M. UDC 512.5 Let $(R, \mathfrak{m})$ be a complete Noetherian local ring and let $M$ be a generalized Cohen-Macaulay $R$-module of dimension $d \geq 2.$We show that$$D \left(H_{\mathfrak{m}}^d\Big(D \big(H_{\mathfrak{m}}^d(D_{\mathfrak{m}}(M))\big)\Big)\right) \approx D_{\mathfrak{m}} (M),$$where $D = {\rm Hom} (-, E)$ and $D_{\mathfrak{m}} (-)$ is the ideal transform functor. Also, assuming that $I$ is a proper ideal of a local ring $R$, we obtain some results on the finiteness of Bass numbers, cofinitness, and cominimaxness of local cohomology modules with respect to $I.$ УДК 512.5 Локальні когомологічні модулі та їхні властивості Нехай $(R, \mathfrak{m})$ - повне локальне нетерове кільце, а $M$ - узагальнений $R$-модуль Коена-Маколея, що має розмірність $d\geq 2.$ Доведено, що$$D \left(H_{\mathfrak{m}}^d\Big(D\big(H_{\mathfrak{m}}^d(D_{\mathfrak{m}}(M))\big)\Big)\right) \approx D_{\mathfrak{m}} (M),$$де $D = {\rm Hom} (-, E)$ і $D_{\mathfrak{m}} (-)$ - функтор перетворення ідеалу. Також якщо $I$ є нетривіальним ідеалом локального кільця $R$, отримано деякі результати щодо фінітності чисел Басса, кофінітності та комінімаксності локальних модулів когомології відносно $I.$ Institute of Mathematics, NAS of Ukraine 2021-02-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/127 10.37863/umzh.v73i2.127 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 2 (2021); 268 - 274 Український математичний журнал; Том 73 № 2 (2021); 268 - 274 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/127/8950 Copyright (c) 2021 Jafar Azami, Masoumeh Hasanzad |
| spellingShingle | Azami, J. Hasanzad, M. Azami, J. Hasanzad, M. Local cohomology modules and their properties |
| title | Local cohomology modules and their properties |
| title_alt | Local cohomology modules and their properties |
| title_full | Local cohomology modules and their properties |
| title_fullStr | Local cohomology modules and their properties |
| title_full_unstemmed | Local cohomology modules and their properties |
| title_short | Local cohomology modules and their properties |
| title_sort | local cohomology modules and their properties |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/127 |
| work_keys_str_mv | AT azamij localcohomologymodulesandtheirproperties AT hasanzadm localcohomologymodulesandtheirproperties AT azamij localcohomologymodulesandtheirproperties AT hasanzadm localcohomologymodulesandtheirproperties |