EXTENSION FUNCTORS OF LOCAL COHOMOLOGY MODULES AND SERRE CATEGORIES OF MODULES
Let (R, m) be a complete Noetherian local ring,Ia proper ideal of R andM, Ntwo finitely generated R-modules such that Supp(N) ⊆V(I). Lett≥ 0 be an integer such that for each 0 ≤i≤t, theR-module H I i ( M ) is in dimension <n. Then we show that each elementLof the set 𝔍, which is defined as: { Ext...
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Published in | Taiwanese journal of mathematics Vol. 19; no. 1; pp. 211 - 220 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Mathematical Society of the Republic of China
01.02.2015
|
Subjects | |
Online Access | Get full text |
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Summary: | Let (R, m) be a complete Noetherian local ring,Ia proper ideal of R andM, Ntwo finitely generated R-modules such that Supp(N) ⊆V(I). Lett≥ 0 be an integer such that for each 0 ≤i≤t, theR-module
H
I
i
(
M
)
is in dimension <n. Then we show that each elementLof the set 𝔍, which is defined as:
{
Ext
R
j
(
N
,
H
I
i
(
M
)
)
:
j
≥
0
and
0
≤
i
≤
t
}
∪
{
Hom
R
(
N
,
H
I
t
+
1
(
M
)
)
,
Ext
R
1
(
N
,
H
I
t
+
1
(
M
)
)
}
is in dimension <n− 2 and so as a consequence, it follows that the set
Ass
R
(
L
)
∩
{
𝔭
∈
Spec
(
R
)
:
dim
(
R
/
𝔭
)
≥
n
−
2
}
is finite. In particular, the set
Ass
R
(
⊕
i
=
0
t
+
1
H
I
i
(
R
)
)
∩
{
𝔭
∈
Spec
(
R
)
:
dim
(
R
/
𝔭
)
≥
n
−
2
}
is finite. Also, as an immediately consequence of this result it follows that theR-modules
Ext
R
j
(
N
,
H
I
i
(
M
)
)
are in dimension <n− 1, for all integersi, j≥ 0, whenever dim(M/IM) =n. These results generalizes the main results of Huneke-Koh [17], Delfino [10], Chiriacescu [9], Asadollahi-Naghipour [1], Quy [18], Brodmann-Lashgari [7], Bahmanpour-Naghipour [5] and Bahmanpour et al. [6] in the case of complete local rings.
2010Mathematics Subject Classification: 13D45, 14B15, 13E05.
Key words and phrases: Associated prime ideal, Cofinite module, Complete local ring, Krull dimension, Local cohomology. |
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ISSN: | 1027-5487 2224-6851 |
DOI: | 10.11650/tjm.19.2015.4315 |