2007/06/25 by Conrad, Brian, Temkin, Michael · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.0706.3441
It is now a classical result that an algebraic space locally of finite type over 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.