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

Abstract

To initiate a systematic study on the applications of perfectoid method to Noetherian rings, we introduce the notions of perfectoid towers and their tilts, and examine their properties. Using these, we establish a comparison theorem on finiteness of étale cohomology groups of a perfectoid tower and of the tilt. We also specialize this to prove the finiteness of the prime-to-ptorsion subgroup of the divisor class group of a local log-regular ring that appears in logarithmic geometry in the mixed characteristic case.