Certain endomorphism rings of local cohomology modules and Lyubeznik numbers. (English) Zbl 1525.13022
Let \((R,\mathfrak{m})\) be a local Noetherian commutative ring with identity, and let \(I\) be an ideal of \(R\). Let \(n=\dim R\). It is known that \(\text{H}_{I}^i(R)=0\) for all \(i>n\) and \(\text{H}_{\mathfrak{m}}^n(R)\neq 0\). The paper under review investigates the local cohomology module \(\text{H}_{I}^{n-1}(R)\). Among other things, the authors prove the following result:
{Theorem.}
{Theorem.}
- (i)
- Assume that \(R\) is Cohen-Macaulay, \(\dim R/I=1\) and \(\text{H}_{I}^n(R)=0\). Then \[\text{H}_{I}^i((\text{H}_{I}^{n-1}(R))= \begin{cases} \text{H}_{\mathfrak{m}}^n(R) \hspace{.3cm} \text{if} \hspace{.5cm} i=1 \\ 0 \hspace{1.33cm} \text{if} \hspace{.5cm} i\neq 1. \end{cases} \] In particular, \(\text{H}_{I}^{n-1}(R)\) is not Artinian.
- (ii)
- Assume that \(R\) is regular, containing a field, and \(\dim R/I=2\). Then \(\text{H}_{I}^{n-1}(R)\) is a direct sum of finitely many copies \(\text{E}_R(R/{\mathfrak{m}})\).
- (iii)
- Let \(V\) be a complete unramified DVR of mixed characteristic \((0,p)\), and let \(R=V[[x_1,x_2,\dots, x_n]]\). Assume that \(\dim R/I=2\). Then \(\text{H}_{I}^{n-1}(R)\) is injective if and only the map \(\text{H}_{I}^{n-1}(R) \overset{p}\longrightarrow \text{H}_{I}^{n-1}(R)\) is surjective.
Reviewer: Kamran Divaani-Aazar (Tehran)
Keywords:
local cohomology; ring endomorphisms; annihilators; Lyubeznik numbers; cohomological dimension; sequentially Cohen-Macaulay ringsReferences:
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