Lyubeznik numbers in mixed characteristic

TL;DR

This paper introduces a new family of invariants called Lyubeznik numbers in mixed characteristic, extending the concept to local rings with mixed characteristic and exploring their properties and differences from equal characteristic cases.

Contribution

It defines Lyubeznik numbers in mixed characteristic for all local rings, establishing their properties and comparing them to existing invariants in equal characteristic.

Findings

Lyubeznik numbers in mixed characteristic are well-defined for all local rings.

These invariants share properties with their equal characteristic counterparts.

Examples show differences between mixed and equal characteristic Lyubeznik numbers.

Abstract

This manuscript defines a new family of invariants, analogous to the Lyubeznik numbers, associated to any local ring whose residue field has prime characteristic. In particular, as their nomenclature suggests, these "Lyubeznik numbers in mixed characteristic" are defined for all local rings of mixed characteristic. Some properties similar to those in equal characteristic hold for these new invariants. Notably, the "highest" Lyubeznik number in mixed characteristic is a well-defined notion. Although the Lyubeznik numbers in mixed characteristic and their equal-characteristic counterparts are the same for certain local rings of equal characteristic p>0, we also provide an example where they differ.