Non-archimedean analytification of algebraic spaces

Details Non-archimedean analytification of algebraic spaces Non-archimedean analytification of algebraic spaces

2007-06-25

arxiv.org/abs/0706.3441v1

Mathematics

Algebraic Geometry

34 pages

Scientific paper

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.

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