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Keywords:
Artinian module; cofinite module; Krull dimension; local cohomology
Artinian module; cofinite module; Krull dimension; local cohomology
Summary:
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 $i\geq 2$ 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 $i\geq 0$, 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 $i\geq 0$, the $R$-modules ${\rm Ext}^i_R(M,N)$ are of finite length.
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