Local cohomology multiplicities in terms of étale cohomology

Abstract

In this paper we give an interpretation of the invariants λa,i(A) introduced by Lyubeznik in [Lyu93] for a reasonably general class of singularities. In positive characteristic it is the newly introduced class of close to F -rational varieties and the invariants are described in terms of étale cohomology with Z/pZ-coefficients. This result presents the first application of Emerton and Kisin's Riemann-Hilbert type correspondence to local algebra. In fact our proof works in characteristic zero as well so that we obtain generalizations of results on these invariants which were previously obtained for isolated singularities by analytic techniques.

Nilpotence of Frobenius action and the Hodge filtration on local cohomology

“…Thus, we obtain the assertion. 2 Blickle and Bondu introduced in [6] the notion of rings close to F -rational in order to study Lyubeznik numbers in terms ofétale cohomology. In this paper, we use the term "F -nilpotent rings" for the same notion to emphasize the nilpotence of the Frobenius actions on the local cohomology modules.…”

“…(2) This is the statement of [6,Proposition 4.2]. (3) We use the characterization of F -nilpotent rings given in Lemma 2.3.…”

“…Then, by a result of Smith [32], there exists a unique maximal proper submodule 0 * It is easy to see that R is F -rational if and only if it is F -injective and F -nilpotent. To the best of authors' knowledge, the notion of F -nilpotence first appeared in [6] 1 in order to describe some invariants of singularities in positive characteristic, the so-called Lyubeznik numbers, in terms ofétale cohomology. In this paper, we pursue a geometric interpretation of F -nilpotence in the case of isolated singularities.…”

Lyubeznik Numbers of Local Rings and Linear Strands of Graded Ideals

“…For instance, in the case of isolated singularities, Lyubeznik numbers can be described in terms of certain singular cohomology groups in characteristic zero (see [6]) or étale cohomology groups in positive characteristic (see [5], [4]). The highest Lyubeznik number λ d,d (A) can be described using the so-called Hochster-Huneke graph as it has been proved in [15], [31].…”

Lyubeznik Table of Sequentially Cohen–Macaulay Rings

“…For instance, in the case of isolated singularities, Lyubeznik numbers can be described in terms of certain singular cohomology groups in characteristic zero (see [11]) orétale cohomology groups in positive characteristic (see [6], [5]). The highest Lyubeznik number λ d,d (A) can be described using the so-called Hochster-Huneke graph as it has been proved in [21], [28].…”