Cohomological dimension and top local cohomology modules
Let \(R\) be a commutative Noetherian ring with identity. Let \(I\) be an ideal of \(R\) and \(M\) a finitely generated \(R\)-module. The authors prove some interesting results concerning the notion of cohomological dimension. The \textit{cohomological dimension} of \(M\) with respect to \(I\) is defined as \[ \text{cd}(I,M):=\sup\{i\in \mathbb{N}_0\mid \text{H}_{I}^i(M)\neq 0\}, \] where the \(i\)-th local cohomology module of \(M\) with respect to \(I\) is defined by \[ \text{H}_I^i(M):=\varinjlim_n \text{Ext}^i_R(R/I^n,M). \] In Theorem 2.2, they provide a lower bound for \(\text{cd}(I,M)\). They use this to show that if \(R\) is a non-catenary domain, then it possesses a prime ideal that is not set-theoretic complete intersection. Then they study the Artinianness of the top local cohomology module \(\text{H}_I^{\text{cd}(I,M)}(M)\). Their related results could be summarized as below: Theorem. Assume that \(R\) is local with the unique maximal ideal \(\mathfrak m\) and \(n:=\dim R\). \begin{itemize} \item[(i)] Suppose that \(R\) is an UFD and \(n\leq 3\). Then \(\text{H}_I^{\text{cd}(I,R)}(R)\) is Artinian if and only if \(\text{cd}(I,R)=\dim R\). \item[(ii)] Suppose that \(\text{depth}(I,R)>0\) and \(c:=\text{cd}(I,R)\leq n-2\). Then there exists an ideal \(J\subsetneqq I\) of \(R\) such that \(\text{cd}(J,R)=\text{cd}(I,R)=c\) and \(\text{H}_J^c(R)\) is non-Artinian. \item[(iii)] Suppose that \(\dim_R(M/IM)=1\). Then \(\text{H}_I^{\text{cd}(I,M)}(M)\) is Artinian if and only if \(\text{cd}(I,M)=\dim_R M\). \item[(iv)] Suppose that \(\dim_R(M/IM)=2\). If \(\text{H}_I^{\text{cd}(I,M)}(M)\) is Artinian, then either \(\text{cd}(I,M)=\dim_R M\); or \(\text{cd}(I,M)=\dim_R M-1\) and \(\text{H}_{\mathfrak m}^2(\text{H}_I^{\dim_R M-2}(M))\neq 0\). \end{itemize}
scientific article; zbMATH DE number 7136581
scientific article