Some results on Artinian cofinite top local cohomology modules
Let \(R\) be a commutative Noetherian local ring with maximal ideal \(\mathfrak{m}\), \(I\) an ideal of \(R\), and \(M\) a finitely generated \(R\)--module such that \(\text{H}^{t}_{I}\left(M\right)\) is Artinian and \(I\)-cofinite, where \(t= \text{cd}\left(I, M\right)\). Recall that an arbitrary \(R\)-module \(X\) is said to be \(I\)--cofinite if \(\text{Supp}_R\left(X\right)\subseteq \{ \mathfrak{p}\in \text{Spec}\left(R\right) : \mathfrak{p}\supseteq I\}\) and \(\text{Ext}^i_R\left(R/I, X\right)\) is finitely generated for all \(i\). Recall also that the cohomological dimension of \(M\) with respect to \(I\) is defined as \(\text{cd}\left(I, M\right)= \text{max}\{ i\in \mathbb{Z} : \text{H}^{i}_{I}\left(M\right)\neq 0\}\). Let \(0= \cap_{\mathfrak{p}\in \text{Ass}_R\left(M\right)} N\left(\mathfrak{p}\right)\) be a reduced primary decomposition of the zero submodule of \(M\) and set \(N= \cap_{\mathfrak{p}\in \text{Ass}_R\left(I, M\right)_t} N\left(\mathfrak{p}\right)\), where \(\text{Ass}_R\left(I, M\right)_t= \{ \mathfrak{p}\in \text{mAss}_R\left(M\right) : \dim\left(R/\mathfrak{p}\right)= t \;\text{and} \;\sqrt{I+ \mathfrak{p}}= \mathfrak{m} \}\). Here, \(\text{mAss}_R\left(M\right)\) is the set of minimal members of \(\text{Ass}_R\left(M\right)\). In this paper, the authors prove that if \(R\) is complete, then \(\text{Att}_R\left(\text{H}^{t}_{I}\left(M\right)\right)= \text{Ass}_R\left(I, M\right)_t\). They use this result to generalize Theorems 2.7 and 3.7 of [\textit{L. T. Nhan} and \textit{T. D. M. Chau}, J. Algebra 349, No. 1, 342--352 (2012; Zbl 1244.13013)] by showing that the following statements are equivalent: {\parindent=0.7cm\begin{itemize}\item[(i)] \(\text{Ann}_R\left(0:_{\text{H}^{t}_{I}\left(M\right)}\mathfrak{p}\right)= \mathfrak{p}\) for all prime ideals \(\mathfrak{p}\supseteq \text{Ann}_R\left(\text{H}^{t}_{I}\left(M\right)\right)\); \item[(ii)] The ring \(R/\text{Ann}_R\left(\text{H}^{t}_{I}\left(M\right)\right)\) is catenary and \(\sqrt{I+ \mathfrak{p}}= \mathfrak{m}\) for all \(\mathfrak{p}\in \text{Att}_R\left(\text{H}^{t}_{I}\left(M\right)\right)\); \item[(iii)] The ring \(R/\text{Ann}_R\left(\text{H}^{t}_{I}\left(M\right)\right)\) is catenary and \(\text{H}^{t}_{I}\left(M\right)\cong \text{H}^{t}_{\mathfrak{m}}\left(M/N\right)\); \item[(iv)] For all \(\mathfrak{p}\in \text{Spec}\left(R\right)\), \(\text{H}^{t- \dim\left(R/\mathfrak{p}\right)}_{\mathfrak{p}R_{\mathfrak{p}}}\left(M_{\mathfrak{p}}/N_{\mathfrak{p}}\right)\neq 0\) if and only if \(\mathfrak{p}\supseteq \text{Ann}_R\left(\text{H}^{t}_{I}\left(M\right)\right)\). \end{itemize}} It is shown that if \(\text{H}^{t}_{I}\left(M\right)\) satisfies the above equivalent conditions, then \(\text{Att}_R\left(\text{H}^{t}_{I}\left(M\right)\right)= \text{Ass}_R\left(I, M\right)_t= \{ \mathfrak{p}\in \text{Ass}_R\left(M\right) : \text{cd}\left(I, R/\mathfrak{p}\right)= t\}\). They also prove that when \(\text{H}^{t}_{I}\left(M\right)\) is Artinian (not necessarily \(I\)--cofinite) and \(R/\text{Ann}_R\left(\text{H}^{t}_{I}\left(M/N\right)\right)\) is catenary, then \(\text{H}^{t}_{I}\left(M\right)\) is not \(\mathfrak{p}\)-cofinite and \(\text{Ann}_R\left(\text{Hom}_R\left(R/\mathfrak{p}, \text{H}^{t}_{I}\left(M\right)\right)\right)= \mathfrak{p}\) for all prime ideals \(\mathfrak{p}\supseteq \text{Ann}_R\left(\text{H}^{t}_{I}\left(M/N\right)\right)\) with \(\dim(R/\mathfrak{p})= 1\).
scientific article; zbMATH DE number 6982271
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