Foundations of Logarithmic Adic Spaces

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.