On Lyubeznik's invariants and endomorphisms of local cohomology modules

Details On Lyubeznik's invariants and endomorphisms of local cohomology modules On Lyubeznik's invariants and endomorphisms of local cohomology modules

2007-04-16

arxiv.org/abs/0704.2007v3

Mathematics

Commutative Algebra

Revised, extended and corrected version

Scientific paper

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.

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