# nLab perfectoid space

Contents

### Context

#### Analytic geometry

analytic geometry (complex, rigid, global)

## Basic concepts

analytic function

analytification

GAGA

# Contents

## Idea

Perfectoid spaces are a variant of Huber spaces in analytic geometry. The concept was introduced (Scholze 11) in order to generalize the classical theorem of (Fontaine-Winterberger 79) (see also at function field analogy).

This theorem establishes an isomorphism between the absolute Galois groups of an extension of the p-adic numbers and of the perfection of the field of Laurent series of the finite field $\mathbb{F}_p$. In (Scholze 11) this is generalized by the statement that the perfectoid spaces over fields related this way are equivalent. (See Bhatt 14 for a review).

## References

Exposition includes

• Michael Harris, The perfectoid concept: Test case for an absent theory (pdf)

The concept is due to

motivated by

Review includes