Perfectoid space

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Perfectoid spaces and perfectoid bodies are special structures in algebra and number theory that have proven to be very powerful in solving problems in arithmetic algebraic geometry . A central property of perfectoid space is that it brings together the number spaces of the p-adic numbers with that of the Laurent series . In order to ensure equivalence, certain requirements are placed on the p-adic body that must be met. The terms “perfect body” and “perfect space” were coined by Peter Scholze in 2012 and immediately attracted a lot of attention from number theorists. Scholze received the Fields Medal in 2018 for his work on this topic .

definition

Perfectoid spaces are a certain type of adic spaces (introduced by Roland Huber ) that occur when investigating problems of “mixed characteristics ”, such as local fields of characteristic zero that have residual class fields with primary characteristic.

A perfect body is a complete topological body , the topology of which is induced by a non-discrete evaluation of rank 1, so that the Frobenius endomorphism is on surjective , where denotes the ring of constrained elements. ${\displaystyle K}$ ${\displaystyle \Phi }$ ${\displaystyle K^{\circ }/p}$ ${\displaystyle K^{\circ }}$

Applications

Perfectoid spaces serve the purpose of comparing situations of “mixed characteristics” with such purely finite characteristics. Technical aids for this precision are the tilting by Fontaine and the Almost purity theorem by Faltings.

history

The theory is essentially based on the basic formulation of arithmetic-algebraic geometry in the school of Alexander Grothendieck ( scheme concept, various cohomology theories, etc.), on the work of Jean-Marc Fontaine on p-adic geometry (Fontaine rings), on Gerd Faltings (almost mathematics, almost purity theorem) and the first attempts to construct p-adic geometry ( John T. Tate , rigid analytic spaces ).

literature

• Peter Scholze : Perfectoid spaces . In: Publications mathématiques de l'IHÉS . tape 116 , no. 1 . Springer, November 2012, p. 245-313 , doi : 10.1007 / s10240-012-0042-x , arxiv : 1111.4914 (English).
• Peter Scholze: Perfectoid spaces and their applications . In: Sun Young Jang (Ed.): Proceedings of the International Congress of Mathematicians. Seoul 2014 . tape 2 : Invited Lectures. Kyung Moon SA, Seoul 2014, ISBN 978-89-6105-805-6 , pp. 461–486 (English, uni-bonn.de [PDF]).

Web links

• Matthew Morrow: Foundations of perfectoid spaces.
• Michael Harris: The Perfectoid Concept: Test Case for an Absent Theory.

Individual evidence

1. ↑ Erica Klarreich: Algebraic Geometry: Peter Scholze - the mathematical clairvoyant . In: spectrum . August 1, 2018 ( Spektrum.de [accessed August 3, 2018]).