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 $k=\mathbb{C}$.



Does GAGA hold over other topological fields?
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..*](http://eudml.org/doc/91922) 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).
> 


