Artinian cofinite modules over complete Noetherian local rings

Sadeghi, Behrouz, Bahmanpour, Kamal, and A'zami, Jafar. "Artinian cofinite modules over complete Noetherian local rings." Czechoslovak Mathematical Journal 63.4 (2013): 877-885. <http://eudml.org/doc/260836>.

@article{Sadeghi2013,
abstract = {Let $(R,\mathfrak \{m\})$ be a complete Noetherian local ring, $I$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module. In this paper it is shown that if $\mathfrak \{p\}$ is a prime ideal of $R$ such that $\dim R/\mathfrak \{p\}=1$ and $(0:_M\mathfrak \{p\})$ is not finitely generated and for each $i\ge 2$ the $R$-module $\{\rm Ext\}^i_R(M,R/\mathfrak \{p\})$ is of finite length, then the $R$-module $\{\rm Ext\}^1_R(M,R/\mathfrak \{p\})$ is not of finite length. Using this result, it is shown that for all finitely generated $R$-modules $N$ with $\operatorname\{Supp\}(N)\subseteq V(I)$ and for all integers $i\ge 0$, the $R$-modules $\{\rm Ext\}^i_R(N,M)$ are of finite length, if and only if, for all finitely generated $R$-modules $N$ with $\operatorname\{Supp\}(N)\subseteq V(I)$ and for all integers $i\ge 0$, the $R$-modules $\{\rm Ext\}^i_R(M,N)$ are of finite length.},
author = {Sadeghi, Behrouz, Bahmanpour, Kamal, A'zami, Jafar},
journal = {Czechoslovak Mathematical Journal},
keywords = {Artinian module; cofinite module; Krull dimension; local cohomology; Artinian module; cofinite module; local cohomology},
language = {eng},
number = {4},
pages = {877-885},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Artinian cofinite modules over complete Noetherian local rings},
url = {http://eudml.org/doc/260836},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Sadeghi, Behrouz
AU - Bahmanpour, Kamal
AU - A'zami, Jafar
TI - Artinian cofinite modules over complete Noetherian local rings
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 4
SP - 877
EP - 885
AB - Let $(R,\mathfrak {m})$ be a complete Noetherian local ring, $I$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module. In this paper it is shown that if $\mathfrak {p}$ is a prime ideal of $R$ such that $\dim R/\mathfrak {p}=1$ and $(0:_M\mathfrak {p})$ is not finitely generated and for each $i\ge 2$ the $R$-module ${\rm Ext}^i_R(M,R/\mathfrak {p})$ is of finite length, then the $R$-module ${\rm Ext}^1_R(M,R/\mathfrak {p})$ is not of finite length. Using this result, it is shown that for all finitely generated $R$-modules $N$ with $\operatorname{Supp}(N)\subseteq V(I)$ and for all integers $i\ge 0$, the $R$-modules ${\rm Ext}^i_R(N,M)$ are of finite length, if and only if, for all finitely generated $R$-modules $N$ with $\operatorname{Supp}(N)\subseteq V(I)$ and for all integers $i\ge 0$, the $R$-modules ${\rm Ext}^i_R(M,N)$ are of finite length.
LA - eng
KW - Artinian module; cofinite module; Krull dimension; local cohomology; Artinian module; cofinite module; local cohomology
UR - http://eudml.org/doc/260836
ER -