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.
The second vanishing theorem has a long history in the theory of local cohomology modules, which connects the vanishing of a complete regular local ring with a topological property of the punctured spectrum of the ring under some conditions. However, the case of complete ramified regular local rings is unresolved. In this paper, we give a partial answer to the second vanishing theorem in the ramified case. Our proof is inspired by the theory of surjective elements in the theory of local cohomology.
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