Properties of Lyubeznik numbers under localization and polarization

TL;DR

This paper investigates the behavior of Lyubeznik numbers in prime characteristic rings, demonstrating bounds, invariance under polarization, and notable changes under localization, with implications for algebraic properties.

Contribution

It establishes a global bound for Lyubeznik numbers, proves their invariance under polarization for monomial ideals, and provides examples of their complex behavior under localization.

Findings

Lyubeznik numbers are bounded globally in prime characteristic rings

Lyubeznik numbers are invariant under polarization for monomial ideals

Localization can cause significant changes in Lyubeznik numbers

Abstract

We exhibit a global bound for the Lyubeznik numbers of a ring of prime characteristic. In addition, we show that for a monomial ideal, the Lyubeznik numbers of the quotient rings of its radical and its polarization are the same. Furthermore, we present examples that show striking behavior of the Lyubeznik numbers under localization. We also show related results for generalizations of the Lyubeznik numbers.