On top dimensional Lyubeznik numbers in mixed characteristic

TL;DR

This paper proves that the top mixed characteristic Lyubeznik number of certain quotient rings is 1 under specific depth and dimension conditions, using a new vanishing theorem in mixed characteristic.

Contribution

It establishes a new result on the value of top Lyubeznik numbers in mixed characteristic, extending previous understanding with a novel vanishing theorem.

Findings

Top mixed characteristic Lyubeznik number equals 1 under given conditions

Uses a second vanishing theorem in mixed characteristic

Extends Lyubeznik number theory to mixed characteristic settings

Abstract

We prove that the top mixed characteristic Lyubeznik number of a ring $S$ that is a quotient of a complete unramified regular local ring of mixed characteristc with algebraically closed residue field is $1$ provided that depth $S \geq 2$ and dim $S \geq 3$ using a second vanishing theorem in mixed characteristic proved in arXiv:1609.05846