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# Mathematics > Commutative Algebra

# Title: Finite étale extension of Tate rings and decompletion of perfectoid algebras

(Submitted on 10 Feb 2020 (v1), last revised 22 May 2021 (this version, v3))

Abstract: In this paper, we examine the behavior of ideal-adic separatedness and completeness under certain ring extensions using trace map. Then we prove that adic completeness of a base ring is hereditary to its ring extension under reasonable conditions. We aim to give many results on ascent and descent of certain ring theoretic properties under completion. As an application, we give conceptual details to the proof of the almost purity theorem for Witt-perfect rings by Davis and Kedlaya. Witt-perfect rings have the advantage that one does not need to assume that the rings are complete and separated.

## Submission history

From: Kazuma Shimomoto Mr. [view email]**[v1]**Mon, 10 Feb 2020 03:10:14 GMT (34kb)

**[v2]**Sun, 15 Mar 2020 03:52:06 GMT (33kb)

**[v3]**Sat, 22 May 2021 10:02:08 GMT (38kb)

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