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 $S\ge 2$ and dim $S\ge 3$ using a second vanishing theorem in mixed characteristic proved in arXiv:1609.05846

arXiv commentUnfortunately, the main result contains a mistake. One a priori only gets a surjection $E_2^{n,n-d}\to E_{\infty }^{n,n0d}$ for the ss in question which is too weak to conclude