Lyubeznik tables of $S_r$ and $CM_r$ rings

TL;DR

This paper characterizes the structure of Lyubeznik tables for certain classes of rings, including those satisfying Serre's condition and Cohen-Macaulay rings, in any characteristic, and demonstrates the optimality of these results.

Contribution

It provides a detailed description of Lyubeznik tables for $S_r$ and $CM_r$ rings across all characteristics, extending previous knowledge and proving the sharpness of these characterizations.

Findings

Lyubeznik tables are explicitly described for $S_r$ and $CM_r$ rings.

Results hold in positive characteristic and for Stanley-Reisner rings in any characteristic.

The theorems presented are shown to be sharp.

Abstract

We describe the shape of the Lyubeznik table of either rings in positive characteristic or Stanley-Reisner rings in any characteristic when they satisfy Serre's condition $S_r$ or they are Cohen-Macaulay in a given codimension, condition denoted by $CM_r$. Moreover we show that these results are sharp.