ATTACHED PRIMES OF LOCAL COHOMOLOGY MODULES OF COMPLEXES

Abstract

Let $(R,\mathfrak{𝔪})$ be a local ring, $\mathcal{𝒵}$ a specialization closed subset of Spec $R$ and $X$ an $R$-complex with finitely generated homology and finite dimension. We show that

$${\text{Att}}_R{H}_{\mathcal{𝒵}}^{\text{dim}X}(X)=\{\mathfrak{𝔭}\in {\text{Supp}}_{R}X\mid \text{cd}(\mathcal{𝒵},R∕\mathfrak{𝔭})-\text{inf}{X}_{\mathfrak{𝔭}}={\text{dim}}_{R}X\}.$$

We also present a generalization of the Lichtenbaum–Hartshorne vanishing theorem for complexes of $R$-modules.