On the highest Lyubeznik number of a local ring

Abstract

Let A be a d-dimensional local ring containing a field. We will prove that the highest Lyubeznik number λ d,d (A) is equal to the number of connected components of the Hochster-Huneke graph associated to B, where B = Âsh is the completion of the strict Henselization of the completion of A. This was proven by Lyubeznik in characteristic p > 0. Our statement and proof are characteristic-free.

Lyubeznik numbers for nonsingular projective varieties

“…In [Lyu02, p. 133], Lyubeznik asked whether λ i,j (A) depend only on V , i, and j, and not on the embedding V ֒→ P n k (or, for that matter, on n). Zhang settled this question in the affirmative in the case of the "top" Lyubeznik number λ r+1,r+1 (A), in any characteristic, in [Zha07]; he went on to give an affirmative answer for all λ i,j (A) in the char(k) = p > 0 case in [Zha11]. In [Zha11], several preliminary results are established in a characteristic-free setting, but the main line of argument makes crucial use of the Frobenius morphism.…”

Lyubeznik Table of Sequentially Cohen–Macaulay Rings

“…In this direction we recall the following notion developed by P. Schenzel in [24]: We say that R/I is canonically Cohen-Macaulay (CCM for short) if the canonical module For a general description of the highest Lyubeznik number we refer to [21], [28] where Proposition 4.2. Let (R, m) be a regular local ring of dimension n containing a field k and ϕ : R−→R a flat local endomorphism satisfying ( * ) for an ideal I ⊆ R such that R/I is unmixed and depth…”

“…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]. However very little is known about the possible configurations of Lyubeznik tables except for low dimension cases [14], [27] or the just mentioned 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]. However very little is known about the possible configurations of Lyubeznik tables except for low dimension cases [12], [24] or the just mentioned case of isolated singularities.…”