General local cohomology modules with small dimensions (Q6189117)

Let \(R\) be a commutative Noetherian ring, \(\Phi\) a system of ideals of \(R\), \(M\) a finitely generated \(R\)-module, and \(n\) an integer. n this article, the authors introduce the notion \(n\)-\(\operatorname{depth}(\Phi, M)= \inf\{n\)-\(\operatorname{depth}(\mathfrak{a}, M): \mathfrak{a}\in \Phi\}\) and show that if \(-1\leq n\leq 1\), then \(n\)-\(\operatorname{depth}(\Phi, M)= \inf\{i: \dim\operatorname{Supp}_R(\operatorname{H}^i_\Phi(M))> n\}\). They prove that when \(R\) is a local ring and \(g= 1\)-\(\operatorname{depth}(\Phi, M)< \infty\), then there exists an ideal \(\mathfrak{a}\in \Phi\) such that the sets \(\{\mathfrak{p}\in \operatorname{Spec}(R) : \mathfrak{p}\supseteq \mathfrak{a}\}\cap \operatorname{Ass}_R(\operatorname{H}^g_\Phi(M))\) and \(\{\mathfrak{p}\in \operatorname{Spec}(R) : \mathfrak{p}\supseteq \mathfrak{a}\}\cap \operatorname{Supp}_R(\operatorname{H}^i_\Phi(M))\) are finite for all \(i< g\). They also show that there exists an ideal \(\mathfrak{b}\in \Phi\) such that \(\operatorname{H}^i_\mathfrak{b}(M)\cong \operatorname{H}^i_\Phi(M)\) for all \(i< n\) whenever \(n\) is a non-negative integer such that \(\operatorname{Supp}_R(\operatorname{H}^i_\Phi(M))\) is finite for all \(i< n\). Recall that the general local cohomology module \(\operatorname{H}^i_\Phi(M)\) is said to be \(\Phi\)-weakly cofinite if there exists an ideal \(\mathfrak{a}\in \Phi\) such that the set of associated prime ideals of any quotient module of \(\operatorname{Ext}^j_R(R/\mathfrak{a}, \operatorname{H}^i_\Phi(M))\) is finite for all \(j\geq 0\). t is proven that if \(R\) is a complete local ring and \(n\) is a non-negative integer such that \(\dim\operatorname{Supp}_R(\operatorname{H}^i_\Phi(M))\leq 2\) for all \(i< n\) (e.g., \(\dim_R(M)\leq 2\)), then \(\operatorname{H}^i_\Phi(M)\) is \(\Phi\)-weakly cofinite for all \(i< n\). Also, t is shown that when \(R\) is a local ring and \(n\) is a non-negative integer such that \(\dim\operatorname{Supp}_R(\operatorname{H}^i_\Phi(R))\leq 2\) for all \(i< n\) (e.g., \(\dim(R)\leq 2\)), then \(\operatorname{H}^i_\Phi(R)\) is \(\Phi\)-weakly cofinite for all \(i< n\).