Perfect body

A perfect body is a mathematical term from the branch of arithmetic algebraic geometry . It is a body in the algebra sense , which is used to understand perfectoid spaces . The theory of perfectoid spaces was developed by Peter Scholze , who received the Fields Medal for it in 2018 .

definition

A perfect field is an algebraic field that also fulfills the following properties: ${\displaystyle K}$

• It has characteristic 0.
• An amount function that induces a metric is defined on it .
• In terms of metric, the body is a complete body .
• The amount defined on it is induced by a non-discrete rating of rank 1. This means that the distance between two points can come as close as desired to any positive real number (non-discrete) and every pair of numbers from the field (except 0) has natural numbers , so that (rank 1). ${\displaystyle (a,b)}$ ${\displaystyle (m,n)}$ ${\displaystyle |a|^{n}=|b|^{m}}$
• The amount is not Archimedean .
• Viewed locally (in terms of metric) the perfect body is a ring with mixed characteristics . ${\displaystyle (0,p)}$
• The set of elements with norm , i.e. all elements with a maximum distance of 1 from the 0 in the metric, is a ring. ${\displaystyle K^{\circ }}$ ${\displaystyle \leq 1}$
• The Frobenius endomorphism , i.e. the mapping , is surjective on the remainder class field . ${\displaystyle x\mapsto x^{p}}$ ${\displaystyle K^{\circ }/{pK^{\circ }}}$

Examples

If you consider the p-adic numbers , the equation cannot be solved in this number range. Thus the body of the p-adic numbers by Add all these zeros can be of polynomials expand . The body thus obtained is Whose analytical completion is a perfectoid body. ${\displaystyle \mathbb {Q} _{p},}$ ${\displaystyle X^{(p^{n})}=p}$ ${\displaystyle \mathbb {Q} _{p}(p^{1/p^{\infty }}).}$

use

In algebraic geometry , perfectoid bodies serve to be able to understand number-theoretic problems geometrically and in this way to solve previously unsolved mathematical problems in the future. One of the properties of perfectoid space is to make the mixed characteristics easier to understand.

So far (as of October 2018) there should be only a few people worldwide who have really understood the concept of perfectoid spaces, which also include perfectoid bodies.

See also

• Perfectoid space

Individual evidence

1. ↑ ^{a b} What are "perfectoid spaces"?, Mathoverflow.net. Retrieved October 3, 2018 .  . With a presentation of the subject by Peter Scholze.
2. ↑ Jörg Sauerwein: Bonn mathematician Peter Scholze is considered an exceptional talent worldwide . August 1, 2018 ( wdr.de [accessed October 3, 2018]).