Some results concerning the ideals with cohomological dimension one
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Let $I$ be a proper ideal of the commutative Noetherian ring $R$. One of the problems in local cohomology is to study the conditions under which ${\rm cd}(I,R)=1$. In this paper, we introduce a class of ideals, say $\Delta(R)$, that for each ideal $I\in \Delta(R)$, ${\rm cd}(I,R)=1 $. We show that if there exists a finitely generated $R$-module $N$ with ${\rm Supp}\ N=V(I)$ and ${\rm pd}_R N\leq 1$, then ${\rm cd}(I,R)\leq 1$. Also, for any $I\in \Delta(R)$ and $R$-module $M$ with ${\rm Supp} M\subseteq V(I)$, $M$ is $I$-cofinite if and only if ${\rm Ex}^i_R(R/I,M)$ is finitely generated for $i=0,1$. Finally, $I$-transform functors enable us to give a necessary and sufficient condition for ${\rm cd}(I,R)=1 $.
Abstract
Let $I$ be a proper ideal of the commutative Noetherian ring $R$. One of the problems in local cohomology is to study the conditions under which ${\rm cd}(I,R)=1$. In this paper, we introduce a class of ideals, say $\Delta(R)$, that for each ideal $I\in \Delta(R)$, ${\rm cd}(I,R)=1 $. We show that if there exists a finitely generated $R$-module $N$ with ${\rm Supp}\ N=V(I)$ and ${\rm pd}_R N\leq 1$, then ${\rm cd}(I,R)\leq 1$. Also, for any $I\in \Delta(R)$ and $R$-module $M$ with ${\rm Supp} M\subseteq V(I)$, $M$ is $I$-cofinite if and only if ${\rm Ex}^i_R(R/I,M)$ is finitely generated for $i=0,1$. Finally, $I$-transform functors enable us to give a necessary and sufficient condition for ${\rm cd}(I,R)=1 $.