Artinian cofinite top local cohomology modules on derived category

Abstract

Let $(R,\mathfrak{m})$ be a local ring such that the going-up theorem holds for the extension $R\subseteq \widehat{R},$ I an ideal of R and $X\in {\mathcal{D}}_{\square }^f(R),X\cancel{\simeq }0$ such that $H_I^c(X)$ is Artinian I-cofinite, where $\text{cd}(I,X)=c.$ Then ${\text{Att}}_R{H}_I^c(X)=\{\mathfrak{p}\in {\text{Supp}}_{R}X|\dim R/\mathfrak{p}-\text{inf}{X}_{\mathfrak{p}}=c\mathrm{ }\text{and}\mathrm{ }\sqrt{I+\mathfrak{p}}=\mathfrak{m}\}.$ We also provide a generalization of Lichtenbaum-Hartshorne Vanishing Theorem on derived category.

Keywords:

• Artinian I-cofinite
• attached primes
• derived category
• local cohomology

2020 Mathematics Subject Classification:

• 13D45
• 13D09
• 13D07