Non-archimedean analytification of algebraic spaces

TL;DR

This paper extends the theory of analytification to non-archimedean algebraic spaces, showing that separatedness ensures analytifiability in rigid and analytic categories, but local separatedness alone is not sufficient.

Contribution

It constructs quotients for non-archimedean analytic equivalence relations and proves separated algebraic spaces are analytifiable over non-archimedean fields.

Findings

Separated algebraic spaces are analytifiable over non-archimedean fields.

Examples of non-analytifiable locally separated spaces are provided.

Analytifiability depends on more than local separatedness.

Abstract

It is now a classical result that an algebraic space locally of finite type over $\mathbf{C}$ is analytifiable if and only if it is locally separated. In this paper we study non-archimedean analytifications of algebraic spaces. We construct a quotient for any etale non-archimedean analytic equivalence relation whose diagonal is a closed immersion, and deduce that any separated algebraic space locally of finite type over any non-archimedean field $k$ is analytifiable in both the category of rigid spaces and the category of analytic spaces over $k$. Also, though local separatedness remains a necessary condition for analytifiability in either of these categories, we present many surprising examples of non-analytifiable locally separated smooth algebraic spaces over $k$ that can even be defined over the prime field.