Perfectoid spaces
type: Item
instance of: scholarly article
title: Perfectoid spaces
zbMATH Open document ID: 1263.14022
author: Peter Scholze
published in: Publications Mathématiques
publication date: 2013-01-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/1111.4914
review text: We closely follow the very clear and instructive introduction to the paper. From the introduction: ``In commutative algebra and algebraic geometry, some of the most subtle problems arise in the context of mixed characteristic, i.e. over local fields such as \({\mathbb Q}_p\) which are of characteristic \(0\), but whose residue field \({\mathbb F}_p\) is of characteristic \(p\). The aim of this paper is to establish a general framework for reducing certain problems about mixed characteristic rings to problems about rings in characteristic \(p\). We will use this framework to establish a generalization of Faltings's almost purity theorem, and new results on Deligne's weight-monodromy conjecture.'' The point of departure is the following theorem of \textit{J.-M. Fontaine} and \textit{J.-P. Wintenberger} [C. R. Acad. Sci., Paris, Sér. A 288, 441--444 (1979; Zbl 0403.12018)]: Theorem. The absolute Galois groups of \({\mathbb Q}_p (p^{1/p^{\infty}})\) and \({\mathbb F}_p((t))\) are canonically isomorphic. The autzor defines a perfectoid field to be a complete topological field \(K\) whose topology is induced by a nondiscrete valuation of rank \(1\), such that the Frobenius is surjective on \(K^0/p\), where \(K^0\) denotes the subring of powerbounded elements in \(K\). Following the construction of Fontaine, to any such \(K\) one associates another perfectoid field \(K^{\flat}\) of characteristic \(p\). The above theorem is then generalized to: Theorem. The absolute Galois groups of \(K\) and \(K^{\flat}\) are canonically isomorphic. The objective is then to generalize this to a comparison of geometric objects over \(K\) with geometric objects over \(K^{\flat}\). In order to adequately formulate such a generalization in turns out that Huber's \textit{adic spaces} are a suitable framework. For example, any variety over \(K\) has an associated adic space \(X^{\text{ad}}\) over \(K\) with underlying topological space \(|X^{\text{m ad}}|\). Theorem. There is a homeomorphism of…
reviewed by: Elmar Große-Klönne
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 11S20
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14G22
Mathematics Subject Classification ID: 11G25
zbMATH DE Number: 6120994
zbMATH Keywords: perfectoid space
zbMATH Keywords: perfectoid algebra
zbMATH Keywords: perfectoid field
zbMATH Keywords: almost mathematics
zbMATH Keywords: adic spaces
zbMATH Keywords: purity theorem
zbMATH Keywords: weight-monodromy conjecture
MaRDI profile type: Publication
OpenAlex ID: W4255501032
Wikidata QID: Q56673229
arXiv ID: 1111.4914
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DOI: 10.1007/S10240-012-0042-X
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description: scientific article