Used to compare mixed characteristic situations with purely finite characteristic ones
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In mathematics, perfectoid spaces are adic spaces of special kind, which occur in the study of problems of "mixed characteristic", such as local fields of characteristic zero which have residue fields of characteristic prime p.
A perfectoid field is a complete topological field K whose topology is induced by a nondiscrete valuation of rank 1, such that the Frobenius endomorphism Φ is surjective on K°/p where K° denotes the ring of power-bounded elements.
Perfectoid spaces may be used to (and were invented in order to) compare mixed characteristic situations with purely finite characteristic ones. Technical tools for making this precise are the tilting equivalence and the almost purity theorem. The notions were introduced in 2012 by Peter Scholze.[1]
In mathematics, perfectoidspaces are adic spaces of special kind, which occur in the study of problems of "mixed characteristic", such as local fields...
Geometric Langlands program. Peter Scholze (2012) Peter Scholze introduced Perfectoidspace. Leonhard Euler (1748) The eminent historian of mathematics Carl Boyer...
Jean-Marc Fontaine and later by Kiran Kedlaya. His PhD thesis on perfectoidspaces yields the solution to a special case of the weight-monodromy conjecture...
certain Shimura varieties. In the 2010s, Peter Scholze developed perfectoidspaces and new cohomology theories in arithmetic geometry over p-adic fields...
R-module. The conjecture was proven by Yves André using a theory of perfectoidspaces. The Canonical Element Conjecture. Let x1,…,xd{\displaystyle x_{1}...
particularly in the development and applications of the theory of perfectoidspaces" 2013 Rahul Pandharipande "For his recent outstanding work in enumerative...
essential dimension of groups 2015 Peter Scholze for his work on perfectoidspaces which has led to a solution of an important special case of the weight-monodromy...
interacting particle systems" Peter Scholze "for his invention of perfectoidspaces and their application to fundamental problems in algebraic geometry...
algebraic geometry over p-adic fields through his introduction of perfectoidspaces, with application to Galois representations, and for the development...
Fontaine, and Scholze" (PDF). bourbaki.ens.fr. "Geometric Langlands, PerfectoidSpaces, and the Fargues-Fontaine Curve" (PDF). Michael Rapoport (28 December...
after the first book, and especially an extended theory of perfectoid rings and perfectoidspaces which generalizes the recent work of Peter Scholze. These...
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