Artinian Cofinite Modules and Going-up for R⊆R̂
Let ( R , 𝔪 ) be a commutative Noetherian local ring. In this paper, it is shown that the going-up theorem holds for R ⊆ R ̂ if and only if Rad ( I + Ann R A ) = 𝔪 for any proper ideal I of R and any non-zero Artinian I -cofinite module A . Furthermore, using the main result of Zöschinger, Arch. Mat...
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Published in | Acta mathematica vietnamica Vol. 42; no. 4; pp. 605 - 613 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Singapore
Springer Singapore
2017
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | Let
(
R
,
𝔪
)
be a commutative Noetherian local ring. In this paper, it is shown that the going-up theorem holds for
R
⊆
R
̂
if and only if
Rad
(
I
+
Ann
R
A
)
=
𝔪
for any proper ideal
I
of
R
and any non-zero Artinian
I
-cofinite module
A
. Furthermore, using the main result of Zöschinger, Arch. Math.
95
, 225–231 (
2010
), it is shown that these equivalent conditions are equivalent to
R
being formal catenary with
α
(
R
) = 0 and to
Att
R
H
I
dim
M
(
M
)
=
{
𝔭
∈
Assh
R
(
M
)
:
Rad
(
𝔭
+
I
)
=
𝔪
}
for any ideal
I
of
R
and any non-zero finitely generated
R
-module
M
. |
---|---|
ISSN: | 0251-4184 2315-4144 |
DOI: | 10.1007/s40306-017-0203-6 |