# Does GAGA hold over other topological fields?

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 $k=\mathbb{C}$.

• Oct 12, 2016 at 14:08

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