Explicit computations around the Lichtenbaum-Hartshorne vanishing theorem
The Lichtenbaum-Hartshorne vanishing theorem says that \(H^ d_ I(A)=0\) for an ideal \(I\) in a Noetherian local ring \((A,{\mathfrak m})\) such that \(\dim \widehat A/(I \widehat A + {\mathfrak p})>0\) for all prime ideals \({\mathfrak p} \in \text{Ass} \widehat A\) with \(\dim \widehat A/{\mathfrak p} = d\) \((d = \dim A)\). The objective of this paper is a precise description of (the Matlis dual of) \(H^ d_ I (A)\) and, more generally, of \(H^ n_ I (X)\) for a finitely generated \(A\)-module \(X\) of dimension \(n\). An essential tool for the description of \(H^ d_ I (X)\) is the canonical module \(K_ X\) of \(X\), which was introduced by the author [``Dualisierende Komplexe in der lokalen Algebra und Buchsbaum-Ringe'', Lect. Notes Math. 907 (1982; Zbl 0484.13016)]. The construction of \(K_ X\) requires only the existence of a dualizing complex for \(A\), so that \(K_{\widehat X}\) always exists. -- The main result of the paper is the following: (a) \(Y = \Hom_ A (H^ n_ I (X),E)\) is isomorphic to the intersection of those primary components of 0 in \(K_{\widehat X}\) whose associated prime ideals \({\mathfrak q}\) satisfy the condition \(\dim \widehat A/(I \widehat A + {\mathfrak q})>0\); (b) \({\mathfrak p} \in \text{Spec} \widehat A\) is associated to the \(\widehat A\)-module \(Y\) if and only if \({\mathfrak p} \in \text{Ass} \widehat X\), \(\dim \widehat A/(I \widehat A + {\mathfrak p})=0\), and \(\dim \widehat A/{\mathfrak p} = n\); (c) \(H^ n_ I (X) = 0\) if and only if the topology defined by the family of submodules \(U_ n = \{x \in K_{\widehat X} : {\mathfrak m}^ k x \in I^ nK_{\widehat X}\) for some \(k \geq 0\}\), \(n \in \mathbb{N}\), is equivalent to the \(I\)-adic topology on \(K_{\widehat X}\). An essential point of the proof is that the topologies in (c) are equivalent if and only if \(\bigcap U_ n = 0\).
Lichtenbaum-Hartshorne vanishing theorem
scientific article; zbMATH DE number 490669
scientific article