On top dimensional Lyubeznik numbers in mixed characteristic

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