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If k is a non-discrete topological field, we can define an analytic space over k just like complex analytic spaces over $\mathbb{C}$. If you replace "complex analytic space" and "complex algebraic variety" with "analytic space over $k$" and "algebraic variety over $k$", respectively, under what conditions on $k$ does GAGA, or weaker similar results, hold? Presumably $k$ must be algebraically closed, but I'm wondering whether this is enough, or whether more conditions must be added, or whether this really only works for $k=\mathbb{C}$.

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If $k$ is a field that is complete with respect to some ultrametric valuation, then there is the "GAGR" (i.e. géométrie algébrique et géométrie rigide) theorem. A succinct explanation (in French, without proof) is:

Jarraud, Pierre. À propos de G.A.G.R.. Groupe de travail d'analyse ultramétrique 11 (1983-1984): 1-4.

The original source for the result, listed as $[4]$ in Jarraud's note, seems to be:

Köpf, Ursula. Über eigentliche Familien algebraischer Varietäten über affinoïden Räumen, Schriftenreihe Univ. Münster, 2. Serie, Heft 7 (1974).
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    $\begingroup$ All this time I thought it was "est" instead of "et". $\endgroup$
    – arsmath
    Oct 12, 2016 at 10:25

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