Foundations of Logarithmic Adic Spaces

TL;DR

This paper develops the theory of logarithmic adic spaces, introduces their topologies, and proves finiteness results for cohomology of certain local systems, advancing the understanding of their geometric and cohomological properties.

Contribution

It establishes foundational definitions, analyzes key topologies, and proves cohomological finiteness for log adic spaces, providing new tools for their study.

Findings

Log adic spaces are locally 'log affinoid perfectoid' in the pro-Kummer étale topology.

Finiteness of cohomologies for Kummer étale $ extbf{F}_p$-local systems on proper log smooth adic spaces.

Development of the basic theory and topological properties of log adic spaces.

Abstract

The main objects of study are adic spaces with logarithmic structures. After establishing the basic definitions, we analyze the Kummer \'etale and pro-Kummer \'etale topologies on log adic spaces. In particular, we show that log adic spaces are locally "log affinoid perfectoid" in the pro-Kummer \'etale topology. As an application, we prove finiteness of cohomologies for Kummer \'etale $\mathbb{F}_p$-local systems on proper log smooth adic spaces.