Perfectoid towers and their tilts : with an application to the \'etale cohomology groups of local log-regular rings

TL;DR

This paper introduces perfectoid towers and their tilts to connect perfectoid methods with Noetherian rings, demonstrating preservation of invariants and applying these to étale cohomology and divisor class groups in logarithmic geometry.

Contribution

It systematically develops the theory of perfectoid towers and tilts, showing their invariance properties and applying them to problems in étale cohomology and divisor class groups.

Findings

Tilting preserves homological invariants.

Comparison of étale cohomology groups under tilting.

Finiteness of certain divisor class groups in mixed characteristic.

Abstract

To initiate a systematic study on the applications of perfectoid methods to Noetherian rings, we introduce the notions of perfectoid towers and their tilts. We mainly show that the tilting operation preserves several homological invariants and finiteness properties. Using this, we also provide a comparison result on \'etale cohomology groups under the tilting. As an application, we prove finiteness of the prime-to-$p$-torsion subgroup of the divisor class group of a local log-regular ring that appears in logarithmic geometry in the mixed characteristic case.