Cohomological dimension, Lyubeznik numbers, and connectedness properties in mixed characteristic

TL;DR

This paper proves a second vanishing theorem linking connectedness of spectra with local cohomology vanishing over regular rings in mixed characteristic, and explores Lyubeznik numbers and connectedness properties of certain rings.

Contribution

It introduces a new vanishing theorem in mixed characteristic and provides novel insights into Lyubeznik numbers and spectral connectedness in this setting.

Findings

Established a second vanishing theorem for local cohomology in mixed characteristic.

Derived new results on Lyubeznik numbers in mixed characteristic.

Analyzed connectedness properties of spectra for specific classes of rings.

Abstract

We establish a "second vanishing theorem" for local cohomology modules over regular rings of unramified mixed characteristic, which relates the connectedness of the spectrum of a ring with the vanishing of local cohomology. Applying this, and new results on the mixed characteristic Lyubeznik numbers, we further study connectedness properties of the spectra of a certain class of rings.