On Lyubeznik's invariants and endomorphisms of local cohomology modules

Abstract

Let $(R,\mathfrak{m})$ denote an $n$-dimensional Gorenstein ring. For an ideal $I\subset R$ of height $c$ we are interested in the endomorphism ring $B = \Hom_R(H^c_I(R), H^c_I(R)).$ It turns out that $B$ is a commutative ring. In the case of $(R,\mathfrak{m})$ a regular local ring containing a field $B$ is a Cohen-Macaulay ring. Its properties are related to the highest Lyubeznik number $l = \dim_k \Ext_R^d(k,H^c_I(R)).$ In particular $R\simeq B$ if and only if $l=1.$ Moreover, we show that the natural homomorphism $\Ext_R^d(k, H^c_I(R)) \to k$ is non-zero.

Revised, extended and corrected version

Cited by in corpus (1)

• On connectedness and indecomposibility of local cohomology modules