Vanishing and Artinianness of graded generalized local cohomology
UDC 512.5 Let $R=\oplus_{j\geq 0}R_j$ be a homogeneous Noetherian ring with semilocal base ring $R_0.$ Let $R_+=\oplus_{j\geq 1}R_j$ be the irrelevant ideal of $R.$ For two finitely generated graded $R$-modules $M$ and $N,$ several results on the vanishing, Artiniannes and tameness property of the g...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512226859810816 |
|---|---|
| author | Azari, A. Khojali, A. Zamani , N. A. A. N. Azari, A. Khojali, A. Zamani , N. |
| author_facet | Azari, A. Khojali, A. Zamani , N. A. A. N. Azari, A. Khojali, A. Zamani , N. |
| author_sort | Azari, A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:49:43Z |
| description | UDC 512.5
Let $R=\oplus_{j\geq 0}R_j$ be a homogeneous Noetherian ring with semilocal base ring $R_0.$ Let $R_+=\oplus_{j\geq 1}R_j$ be the irrelevant ideal of $R.$ For two finitely generated graded $R$-modules $M$ and $N,$ several results on the vanishing, Artiniannes and tameness property of the graded $R$-modules $H^i_{R_+}(M, N)$ will be investigated.
 
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| doi_str_mv | 10.37863/umzh.v72i10.6026 |
| first_indexed | 2026-03-24T03:25:26Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i10.6026
UDC 512.5
A. Azari, A. Khojali, N. Zamani (Univ. Mohaghegh Ardabili, Ardabil, Iran)
VANISHING AND ARTINIANNESS OF GRADED GENERALIZED
LOCAL COHOMOLOGY
ЗНИКНЕННЯ ТА АРТIНОВIСТЬ ГРАДУЙОВАНОЇ УЗАГАЛЬНЕНОЇ
ЛОКАЛЬНОЇ КОГОМОЛОГIЇ
Let R = \oplus j\geq 0Rj be a homogeneous Noetherian ring with semilocal base ring R0. Let R+ = \oplus j\geq 1Rj be the irrelevant
ideal of R. For two finitely generated graded R-modules M and N, several results on the vanishing, Artiniannes and
tameness property of the graded R-modules Hi
R+
(M,N) will be investigated.
Нехай R = \oplus j\geq 0Rj — однорiднe ньотерове кiльце з напiвлокальним базовим кiльцем R0. Нехай також R+ =
= \oplus j\geq 1Rj є iррелевантним iдеалом R. Для двох скiнченнопороджeних градуйованих R-модулiв M i N наведено
деякi результати щодо властивостей зникнення, артiновостi та приборкання градуйованих R-модулiв Hi
R+
(M,N).
1. Introduction. Throughout this paper R = \oplus n\geq 0Rn is a homogeneous graded (Noetherian) ring
with semilocal base ring R0, so that R0 is a Noetherian ring and R, as an R0-algebra is generated
by finitely many homogeneous elements of degree one. Let R+ = \oplus n>0Rn be the irrelevant
ideal of R and m
(1)
0 , . . . ,m
(t)
0 be the maximal ideals of R0. Assume that M = \oplus n\in \BbbZ Mn and
N = \oplus n\in \BbbZ Nn are two finitely generated \BbbZ -graded R-modules. For any graded ideal I of R, the ith
generalized local cohomology module H i
I(M,N) has a natural graded structure, such that the long
exact sequences induced from suitable short exact sequences (in both component) in the category of
finitely generated graded R-modules and homogeneous homomorphisms is an exact sequence in this
category. Furthermore, with I = R+, it is well-known that the R0-module H i
R+
(M,N)n is finitely
generated for all n \in \BbbZ and is zero for all n \gg 0 (see [12]). For more results on the graded modules
H i
R+
(M,N) see [13].
In this paper we shall study the vanishing, Artinianness and tameness behavior of the graded
R-modules H i
R+
(M,N), in case that R0 is a semilocal ring and the projective dimension of M
(denoted by pd(M)) is finite. In Section 2, among some preliminaries, a vanishing theorem on these
modules will be proved which improves [11] (Lemma 3.1) and [13] (Theorem 3.2) in this graded case.
More precisely, it will be shown that H i
R+
(M,N) = 0 for all i > pd(M)+dim(N/\Gamma J0R(N)), where
J0 = \cap t
i=1m
(i)
0 is the Jacobson radical of R0, \Gamma J0R(N)= \{ x \in N | \exists n \in \BbbN such that Jn
0 x = 0\} is
the J0R-torsion submodule of N and \mathrm{d}\mathrm{i}\mathrm{m} stands for the Krull dimension of an R-module. Section
c\bigcirc A. AZARI, A. KHOJALI, N. ZAMANI, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10 1315
1316 A. AZARI, A. KHOJALI, N. ZAMANI
3 deals with Artinianness and tameness properties of the modules H i
R+
(M,N). One of the results
in this section states as follows: Let R0 be a semilocal ring with Jacobson radical J0. Let M,N be
two finitely generated graded R-modules with p = pd(M) < \infty . Set s = dim(N/J0N +\Gamma J0R(N)).
Then H i
R+
(M,N) is Artinian for i > p + s and is tame for i = p + s. It is well-known that over
a complete semilocal ring any Artinian module is Matlis reflexive. So, it is natural to ask that when
generalized local cohomology modules are Matlis reflexive. Concerning this question we refer to [8].
2. Vanishing theorem. Our aim in this section is to prove a theorem on vanishing of the graded
modules H i
R+
(M,N). Recall that these modules was defined in [6], as the direct limits of some
Ext-modules; that is, for two R-modules M,N,
H i
R+
(M,N) = \mathrm{l}\mathrm{i}\mathrm{m} - \rightarrow
n\in \BbbN
ExtiR(M/(R+)
nM,N).
One can observe that each element of H i
R+
(M,N) is annihilated by a power of R+ and so
H i
R+
(M,N) is an R+-torsion module. Other approaches of these modules can be found in [11]
and [1]. To name one of them in a special case for which the first component is finitely generated,
we have
H i
R+
(M,N) \sim = H i(\Gamma R+(HomR(M, \bfI N ))) \sim = H i(HomR(M,\Gamma R+(\bfI
N ))), (2.1)
where \bfI N is an injective resolution of N. From this fact and using [3] (Corollary 2.1.6), it is con-
cluded that whenever M is finitely generated and \Gamma R+(N) = N. Then H i
R+
(M,N) = ExtiR(M,N),
and if in addition p = pd(M) < \infty , then H i
R+
(M,N) = 0 for all i > p. This fact will be used
several times in this paper.
We continue with the following key lemma. This lemma, appeared in [2] in the case that R0 is a
local ring, has been proved using a theorem of Kirby [7]. Here we give another proof, whenever R0
is semilocal.
Lemma 2.1. Let R be a homogeneous Noetherian ring with semilocal base ring R0 and N =
= \oplus i\in \BbbZ Ni be a finitely generated graded R-module. Let J0 be the Jacobson radical of R0 and
d = \mathrm{d}\mathrm{i}\mathrm{m}(N/J0N). Then \Gamma R+(N) = N if and only if d \leq 0.
Proof. One direction is clear. If \Gamma R+(N) = N, then Nn = 0 for all n \gg 0. This gives that
Nn/J0Nn = 0 for all n \gg 0 and so dim(N/J0N) \leq 0 as desired.
Now let d \leq 0. As in the introduction we assume that m
(1)
0 , . . . ,m
(t)
0 are the maximal ideals
of R0. If d < 0 there is nothing to prove. So assume that d = 0. In this case the only minimal
prime ideals of N/J0N are among the graded maximal ideals m
(1)
0 + R+, . . . ,m
(t)
0 + R+ and so
there exists n \in \BbbN such that (\cap t
j=1(m
(j)
0 +R+))
n \subseteq (0 :R N/J0N). This, in turn, gives that Rm
+ \subseteq
\subseteq (0 :R N/J0N) and so RmN \subseteq J0N for m \geq n. Therefore, we conclude that \oplus i\geq d1Ni+m \subseteq
\subseteq \oplus i\geq d1J0Ni for m \geq n, where d1 = min\{ i \in \BbbZ | Ni \not = 0\} is the beginning of N. From this, using
NAK lemma, we obtain that Nm = 0 for m \geq n+ d1 and, so, \Gamma R+(N) = N as desired.
Lemma 2.1 is proved.
The next theorem improves [11] (Lemma 3.1) and [13] (Theorem 3.2).
Theorem 2.1. Let R be a homogeneous Noetherian ring with semilocal base ring R0. Let
M,N be two finitely generated graded R-modules such that p = \mathrm{p}\mathrm{d}(M) is finite. Assume that
d = \mathrm{d}\mathrm{i}\mathrm{m}(N/J0N). Then H i
R+
(M,N) = 0 for all i > p+ d.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
VANISHING AND ARTINIANNESS OF GRADED GENERALIZED LOCAL COHOMOLOGY 1317
Proof. We prove this by induction on d. If d \leq 0, then by Lemma 2.1, \Gamma R+(N) = N and so
H i
R+
(M,N) = ExtiR(M,N) = 0 for all i > p.
So, assume that d > 0 and the result has been proved for d - 1. Put \=N = N/\Gamma R+(N). Since
H i
R+
(M,\Gamma R+(N)) = ExtiR(M,\Gamma R+(N)) = 0 for all i > p, the short exact sequence
0 \rightarrow \Gamma R+(N) \rightarrow N \rightarrow \=N \rightarrow 0
gives rise to the isomorphism H i
R+
(M,N) \sim = H i
R+
(M,N/\Gamma R+(N)) in the category of graded R-
modules and R-morphisms (i.e., homogeneous R-homomorphisms) for all i > p. Since \Gamma R+(N)
has only finitely many non-zero components and since d > 0, then dim( \=N/J0 \=N) = dim(N/J0N).
Therefore, we can replace N by \=N and may assume that \Gamma R+(N) = 0. So, by [3] (Lemma 2.1.1(ii)),
R+ \nsubseteq ZR(N) = \cup p\in AssR(N)p, where ZR(N) denotes the set of all zero divisors of N in R. On the
other hand, since d > 0, we see that for each minimal member p of the set AssR(N/J0N), R+ \nsubseteq p.
So,
R+ \nsubseteq
\bigcup
p\in AssR(N)
p \cup
\bigcup
p\in MinAssR(N/J0N)
p
and by [2] (Lemma 15.10), there exists a homogeneous element x \in R+ which is a non-zero divisor
on N and at the same time
dim((N/xN)/J0(N/xN)) = \mathrm{d}\mathrm{i}\mathrm{m}((N/J0N)/x(N/J0N)) = d - 1.
Considering the short exact sequence 0 \rightarrow N
x\rightarrow N \rightarrow N/xN \rightarrow 0 and using the induction
hypothesis we get the isomorphisms
H i
R+
(M,N)
x\sim = H i
R+
(M,N)
for all i > p + d. Now, as H i
R+
(M,N) is R+-torsion we conclude that H i
R+
(M,N) = 0 for each
i > p+ d.
Theorem 2.1 is proved.
The top non-vanishing problem of generalized local cohomology seems to be more subtle. While
there is a partial answer for this problem in some special cases in [11], until now we were not able to
formulate ordinary local cohomology non-vanishing counterparts in generalized local cohomology.
3. Artinian and tame properties. In this section, we will draw several results concerning the
Artinian property and tameness of the modules H i
R+
(M,N). Following [2], a graded R-module T
is said to be tame if there exists m \in \BbbZ such that Tn = 0 for all n \leq m or Tn \not = 0 for all n \leq m.
For ease in access we collect some known facts on generalized local cohomology in the frame of the
following theorem.
Theorem 3.1. Let a be an (not necessarily graded) ideal of R and let X and Y be two finitely
generated R-modules.
(i) If R/a is Artinian, then for each i \in \BbbN 0 the R-module H i
a(X,Y ) is Artinian [13] (Theo-
rem 2.2).
(ii) H i
a(X,Y ) \sim = H i\surd
a
(X,Y ), for each i \in \BbbN 0 [4] (Lemma 2.1 (i)).
(iii) Let x \in R. Then there is a natural long exact sequence
. . . \rightarrow H i
a+(x)(X,Y ) \rightarrow H i
a(X,Y ) \rightarrow H i
aRx
(X,Y ) \rightarrow H i+1
a+(x)(X,Y ) \rightarrow . . .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1318 A. AZARI, A. KHOJALI, N. ZAMANI
of generalized local cohomology modules. Furthermore, if R, X, Y and a are graded and x is a
homogeneous element of R, then all the maps in this exact sequence are homogeneous, so that for
each n \in \BbbZ , there exists the long exact sequence
. . . \rightarrow H i
a+(x)(X,Y )n \rightarrow H i
a(X,Y )n \rightarrow H i
aRx
(X,Y )n \rightarrow H i+1
a+(x)(X,Y )n \rightarrow . . .
of R0-modules [5] (Lemma 3.1).
(iv) If R\prime is another commutative Noetherian ring and f : R \rightarrow R\prime is a flat ring homomorphism,
then, for each ideal a of R,
H i
a(X,Y )\otimes R R\prime \sim = H i
aR\prime (X \otimes R R\prime , Y \otimes R R\prime ).
Thus for a multiplicatively closed subset S of R,
S - 1H i
a(X,Y ) \sim = H i
S - 1a(S
- 1X,S - 1Y ).
If R, X, Y and a are graded and S \subseteq R0, then, for each n \in \BbbZ ,
S - 1(H i
a(X,Y )n) \sim = [H i
S - 1a(S
- 1X,S - 1Y )]n,
as R0-modules. In particular, for each p0 \in Spec(R0) and each n \in \BbbZ ,
(H i
a(X,Y )n)p0
\sim = H i
aRp0
(Xp0 , Yp0)n.
Theorem 3.2. Let R be a homogeneous Noetherian ring with semilocal base ring R0 and
J0 be the Jacobson radical of R0. Let M,N be two finitely generated graded R-modules with
p = \mathrm{p}\mathrm{d}(\mathrm{M}) < \infty . Put d = \mathrm{d}\mathrm{i}\mathrm{m}(N/J0N). Then:
(1) The R-module Q = R0/J0 \otimes R0 H
p+d
R+
(M,N) is Artinian (see [13], Theorem 3.3).
(2) For each i \geq 0, the R-module H i
R+
(M,\Gamma J0R(N)) is Artinian (see [13], Lemma 3.5).
(3) If \mathrm{d}\mathrm{i}\mathrm{m}(R0) \leq 1, then \Gamma J0R(H
i
R+
(M,N)), H1
J0R
(H i
R+
(M,N)) and (0 :Hi
R+
(M,N) J0) are
Artinian.
(4) The R-module H i
R+
(M,N) is Artinian for i > p + s and is tame for i = p + s, where
s = \mathrm{d}\mathrm{i}\mathrm{m}(N/J0N + \Gamma J0R(N)).
(5) For each i \in \BbbN 0, if R0/J0 \otimes R0 H i
R+
(M,N/\Gamma J0R(N)) is Artinian, then
R0/J0 \otimes R0H
i
R+
(M,N) is Artinian too.
Proof. (1) We prove this by induction on d. If d \leq 0, then, by using Lemma 2.1, we see that
Hp
R+
(M,N) = ExtpR(M,N) vanishes by a power of R+. Thus, SuppR(Q) \subseteq \{ m(1)
0 +R+, . . . ,m
(t)
0 +
+R+\} where as usual m(1)
0 , . . . ,m
(t)
0 are the maximal ideals of R0. So, we deduce that Q is Artinian.
For d > 0 as in the proof of Theorem 2.1 we can find a homogeneous element x \in R+ which is
a non-zero divisor on N and dim((N/xN)/J0(N/xN)) = d - 1. Therefore, by using Theorem 2.1,
we can obtain the exact sequence
Hp+d - 1
R+
(M,N/xN)
\Delta - \rightarrow Hp+d
R+
(M,N)
x - \rightarrow Hp+d
R+
(M,N) \rightarrow 0.
By induction hypothesis the R-module R0/J0\otimes R0H
p+d - 1
R+
(M,N/xN) is Artinian. Thus R0/J0\otimes R0
Im(\Delta ) is Artinian too. Now, considering the exact sequence
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VANISHING AND ARTINIANNESS OF GRADED GENERALIZED LOCAL COHOMOLOGY 1319
R0/J0 \otimes R0 Im(\Delta ) \rightarrow R0/J0 \otimes R0 H
p+d
R+
(M,N)
x - \rightarrow R0/J0 \otimes R0 H
p+d
R+
(M,N) \rightarrow 0
gives that (0 :
R0/J0\otimes R0
Hp+d
R+
(M,N)
x) as a homomorphic image of R0/J0 \otimes R0 Im(\Delta ) is Artinian.
The result now follows by [10] (Theorem 1.3).
(2) By [3] (Corollary 2.1.6), there exists an injective resolution \bfI of \Gamma J0R(N) at which each term
is a J0R-torsion R-module. Let Ii be its ith term. Hence there exists a family (p\lambda ) of prime ideals
of R such that J0R \subseteq p\lambda for each \lambda and
Ii = \oplus \lambda E(R/p\lambda )
\mu i ,
where ER( - ) stands for the injective hull and \mu i = \mu i(p\lambda ,\Gamma J0R(N)) is the ith Bass number of
\Gamma J0R(N) with respect to p\lambda . We conclude that for each \lambda there exists 0 \leq j \leq t such that m(j)
0 \subseteq p\lambda
and \Gamma R+(E(R/p)) would be E(R/m
(j)
0 + R+) if R+ \subseteq p\lambda and it is zero if R+ \nsubseteq p\lambda . Therefore,
since E(R/m
(j)
0 + R+) is an Artinian R-module, the module HomR(M, Ii)) which is \mu i copies
of HomR(M,E(R/m
(j)
0 + R+) will be Artinian. Now by (2.1) we see that H i
R+
(M,\Gamma J0R(N)) =
= \mathrm{E}\mathrm{x}\mathrm{t}iR(M,\Gamma R+(N)) as a subquotient of an Artinian module is Artinian.
(3) When dim(R0) = 0, by Theorem 3.1(i), H i
R+
(M,N) is Artinian and the claim holds in this
case. So, assume that dim(R0) = 1. By the proof of [9] (Theorem 13.6), there exists a0 \in J0 such
that
\surd
a0R0 = J0. Thus, by using Theorem 3.1(iii),(iv), there exists an exact sequence
H i - 1
R+
(M,N)
f i - 1
a0 - - - \rightarrow H i - 1
R+
(M,N)a0 \rightarrow H i
(R+,a0)
(M,N) \rightarrow H i
R+
(M,N)
f i
a0 - - \rightarrow H i
R+
(M,N)a0
of graded generalized local cohomology modules at which f i - 1
a0 and f i
a0 are natural homomorphisms.
By [3] (Corollary 2.2.18), we have
Coker(f i - 1
a0 ) \sim = H1
a0R(H
i - 1
R+
(M,N)) = H1
J0R(H
i - 1
R+
(M,N))
and
Ker(f i
a0) = \Gamma a0R(H
i
R+
(M,N)) = \Gamma J0R(H
i
R+
(M,N)),
which gives the short exact sequence
0 \rightarrow H1
J0R(H
i - 1
R+
(M,N)) \rightarrow H i
(R+,a0)
(M,N) \rightarrow \Gamma J0R(H
i
R+
(M,N)) \rightarrow 0.
Now, by Theorem 3.1(ii), we get H i
(a0R,R+)(M,N) = H i
J0R+R+
(M,N) and H i
(a0R,R+)(M,N) is
Artinian by Theorem 3.1(i). This proves the claim if i runs through \BbbN 0. Finally, the R-module
(0 :Hi
R+
(M,N) J0) as a submodule of the Artinian module \Gamma J0R(H
i
R+
(M,N)) is Artinian.
(4) Consider the short exact sequence
0 \rightarrow \Gamma J0R(N) \rightarrow N \rightarrow N/\Gamma J0R(N) \rightarrow 0
to obtain the exact sequence
H i
R+
(M,\Gamma J0R(N))
ui - \rightarrow H i
R+
(M,N) \rightarrow H i
R+
(M,N/\Gamma J0R(N))
\Delta i - \rightarrow H i+1
R+
(M,\Gamma J0R(N))
of generalized local cohomology modules. By part (2), the left- and right-hand sides of this long
exact sequence are Artinian for each i \geq 0. Hence, for each i \geq 0, we get the exact sequence
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1320 A. AZARI, A. KHOJALI, N. ZAMANI
0 \rightarrow Ui \rightarrow H i
R+
(M,N) \rightarrow H i
R+
(M,N/\Gamma J0R(N))
\Delta i - \rightarrow Vi \rightarrow 0, (3.1)
where Ui = ker(ui) and Vi = Im(\Delta i) are Artinian. Now, by Theorem 2.1,
H i
R+
(M,N/\Gamma J0R(N)) = 0
for all i > p+ s and, hence, H i
R+
(M,N) \sim = Ui is Artinian for all i > p+ s.
For tameness at p+ s, using the exact sequence (3.1), we get the exact sequence
0 \rightarrow K = ker(\Delta p+s) \rightarrow Hp+s
R+
(M,N/\Gamma J0R(N)) \rightarrow Vp+s \rightarrow 0.
This gives us the exact sequence
TorR1 (R/J0R, Vp+s) \rightarrow R/J0R\otimes R K \rightarrow R/J0R\otimes R Hp+s
R+
(M,N/\Gamma J0R(N))
of graded R-modules. Since Vp+s is Artinian, the left-hand side module in this exact sequence is
Artinian, while
R/J0R\otimes R Hp+s
R+
(M,N/\Gamma J0R(N)) \sim = R0/J0 \otimes R0 H
p+s
R+
(M,N/\Gamma J0R(N))
is Artinian by (1). So R/J0R\otimes R K is Artinian. Now, from the short exact sequence
0 \rightarrow Up+s \rightarrow Hp+s
R+
(M,N) \rightarrow K \rightarrow 0
we obtain the exact sequence
R/J0R\otimes R Up+s \rightarrow R/J0R\otimes Hp+s
R+
(M,N) \rightarrow R/J0R\otimes R K
at which the left and right most modules are Artinian. So R/J0R \otimes Hp+s
R+
(M,N) as an R-Artinian
module is tame and, hence, Hp+s
R+
(M,N) is tame.
(5) Let \v N = N/\Gamma J0R(N). The short exact sequence
0 \rightarrow \Gamma J0R(N) \rightarrow N \rightarrow \v N \rightarrow 0,
gives rise to the exact sequence
H i
R+
(M,\Gamma J0R(N))
ui - \rightarrow H i
R+
(M,N)
vi - \rightarrow H i
R+
(M, \v N)
\Delta i - \rightarrow H i+1
R+
(M,\Gamma J0R(N)),
which in turn gives the following two exact sequences:
H i
R+
(M,\Gamma J0R(N))
ui - \rightarrow H i
R+
(M,N) \rightarrow H i
R+
(M,N)/Im(ui) \rightarrow 0 (3.2)
and
0 \rightarrow H i
R+
(M,N)/Im(ui) \rightarrow H i
R+
(M, \^N) \rightarrow H i
R+
(M, \v N)/Im(vi) \rightarrow 0, (3.3)
and the monomorphism
0 \rightarrow H i
R+
(M, \v N)/Im(vi)
\=\Delta i - \rightarrow H i+1
R+
(M,\Gamma J0R(N)).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
VANISHING AND ARTINIANNESS OF GRADED GENERALIZED LOCAL COHOMOLOGY 1321
Set Y := H i
R+
(M, \v N), U := H i
R+
(M,N)/Im(ui) and V := H i
R+
(M, \v N)/Im(vi). We note that,
by our assumption, R0/J0 \otimes R0 Y is Artinian. Also, as a submodule of H i+1
R+
(M,\Gamma J0R(N)) the
R-module V and, hence, TorR0
1 (R0/J0, V ) is Artinian. Therefore, the exact sequence
TorR0
1 (R0/J0, V ) \rightarrow R0 \otimes R0 U \rightarrow R0/J0 \otimes R0 Y,
which we obtain from (3.3), gives that the R-module R0/J0 \otimes R0 U is Artinian. On the other hand,
from (3.2) we deduce the exact sequence
R0/J0 \otimes R0 H
i
R+
(M,\Gamma J0R(N)) \rightarrow R0/J0 \otimes R0 H
i
R+
(M,N) \rightarrow R0/J0 \otimes R0 U.
Now, using part (2) the result follows.
Theorem 3.2 is proved.
In the next theorem our aim is to improve [14] (Theorem 2.8). The proof is almost the same, but
we present its proof for the reader’s convenience. To do so, we need the following notation. Let R0
be a semilocal ring and let X be an R-module. We put
cd(X) = sup\{ dimR(X/n0X)| n0 \in Max(R0)\}
and
\frakN 0 =
\prod
\{ n0| n0 \in Max(R0) and \mathrm{d}\mathrm{i}\mathrm{m}R(N/n0N) = cd(N)\} ,
where Max(R0) is the set of all maximal ideals of R0.
Theorem 3.3. Let R0 be a semilocal ring and M, N be two finitely generated graded R-
modules with \mathrm{p}\mathrm{d}(\mathrm{M}) < \infty . Set k = \mathrm{p}\mathrm{d}(M) + \mathrm{c}\mathrm{d}(N/\frakN 0N + \Gamma \frakN 0R(N)). Then, for each i > k, the
R-module H i
R+
(M,N) is Artinian.
Proof. We set N := N/\Gamma \frakN 0R(N) and
\scrC = \{ p0 \in Spec(R0)| dim(Np0/p0Np0) = cd(N)\} .
Note that, if p0 \in Spec(R0) \setminus \scrC , then Np0
\sim = Np0 and by Theorem 3.1(iv), for each i \geq 0, we have
the isomorphism
H i
R+
(M,N)p0
\sim = H i
(Rp0 )+
(Mp0 , Np0), (3.4)
of graded Rp0 -modules. Since, by Theorem 2.1, the right-hand side of (3.4) is zero for all i >
> pd(Mp0) + dim(Np0/J0Np0) =: \ell and \ell \leq k, we see that SuppR0
(H i
R+
(M,N)) \subseteq \scrC for all
i > k. Now let m0 \in \scrC . Since Nm0/\Gamma J0Rm0
(Nm0)
\sim = Nm0 , by applying Theorem 3.2(4) for the
graded Rm0 -modules Mm0 and Nm0 we conclude that H i
R+
(M,N)m0
\sim = H i
(Rm0 )+
(Mm0 , Nm0) is
Artinian for i > k. Since the set of maximal ideals in SuppR0
(H i
R+
(M,N)) is finite, this gives that
H i
R+
(M,N) is Artinian for i > s.
Theorem 3.3 is proved.
References
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ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1322 A. AZARI, A. KHOJALI, N. ZAMANI
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Received 04.09.17,
after revision — 20.05.18
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
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| id | umjimathkievua-article-6026 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:26Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/79/0da43ecac9568bc1e8c4a0c1c75af579.pdf |
| spelling | umjimathkievua-article-60262025-03-31T08:49:43Z Vanishing and Artinianness of graded generalized local cohomology Vanishing and Artinianness of graded generalized local cohomology Vanishing and Artinianness of graded generalized local cohomology Azari, A. Khojali, A. Zamani , N. A. A. N. Azari, A. Khojali, A. Zamani , N. Bass numbers Generalized local cohomology modules tameness Bass numbers Generalized local cohomology modules tameness UDC 512.5 Let $R=\oplus_{j\geq 0}R_j$ be a homogeneous Noetherian ring with semilocal base ring $R_0.$ Let $R_+=\oplus_{j\geq 1}R_j$ be the irrelevant ideal of $R.$ For two finitely generated graded $R$-modules $M$ and $N,$ several results on the vanishing, Artiniannes and tameness property of the graded $R$-modules $H^i_{R_+}(M, N)$ will be investigated. &nbsp; &nbsp; &nbsp;УДК 512.5 Зникнення та артiновiсть градуйованої узагальненої локальної когомологiї Нехай $R=\oplus_{j\geq 0}R_j$ --- одноріднe ньотерове кільце з напівлокальним базовим кільцем $R_0.$ Нехай також $R_+ =\oplus_{j\geq 1}R_j$ є іррелевантним ідеалом $R.$ Для двох скінченнопороджeних градуйованих $R$-модулів $M$ i $N$ наведено деякі результати щодо властивостей зникнення, артіновості та приборкання градуйованих $R$-модулів $H^i_{R_+}(M, N)$. Institute of Mathematics, NAS of Ukraine 2020-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6026 10.37863/umzh.v72i10.6026 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 10 (2020); 1315-1322 Український математичний журнал; Том 72 № 10 (2020); 1315-1322 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6026/8757 |
| spellingShingle | Azari, A. Khojali, A. Zamani , N. A. A. N. Azari, A. Khojali, A. Zamani , N. Vanishing and Artinianness of graded generalized local cohomology |
| title | Vanishing and Artinianness of graded generalized local cohomology |
| title_alt | Vanishing and Artinianness of graded generalized local cohomology Vanishing and Artinianness of graded generalized local cohomology |
| title_full | Vanishing and Artinianness of graded generalized local cohomology |
| title_fullStr | Vanishing and Artinianness of graded generalized local cohomology |
| title_full_unstemmed | Vanishing and Artinianness of graded generalized local cohomology |
| title_short | Vanishing and Artinianness of graded generalized local cohomology |
| title_sort | vanishing and artinianness of graded generalized local cohomology |
| topic_facet | Bass numbers Generalized local cohomology modules tameness Bass numbers Generalized local cohomology modules tameness |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6026 |
| work_keys_str_mv | AT azaria vanishingandartiniannessofgradedgeneralizedlocalcohomology AT khojalia vanishingandartiniannessofgradedgeneralizedlocalcohomology AT zamanin vanishingandartiniannessofgradedgeneralizedlocalcohomology AT a vanishingandartiniannessofgradedgeneralizedlocalcohomology AT a vanishingandartiniannessofgradedgeneralizedlocalcohomology AT n vanishingandartiniannessofgradedgeneralizedlocalcohomology AT azaria vanishingandartiniannessofgradedgeneralizedlocalcohomology AT khojalia vanishingandartiniannessofgradedgeneralizedlocalcohomology AT zamanin vanishingandartiniannessofgradedgeneralizedlocalcohomology |