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So, I have heard GAGA works for Rigid Analytic spaces. I know next to nothing about this, but it made me curious as to whether there are any other contexts in which GAGA "works". Of course, this is a very vague question. Here is a way to make it more formal:

When is there a functor $F$ from the category of varieties over some field K to the category of locally ringed (in $K$-algebras) spaces. Where, say, each $F(V)$ is a locally compact Hausdorff space, and F takes proper morphisms [in the sense of algebraic geometry] to proper maps [in the topological sense]. And also that for any variety $V$, there is a morphism of locally ringed spaces $F(V) \rightarrow V$, and if $V$ is a projective variety, than this morphism induces an isomorphism between the category of coherent sheaves on $V$ and the category of coherent sheaves on $F(V)$.

Of course, these are very strong conditions, I would also be interested in weaker cases.

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The rigid-analytic case doesn't fit into your proposed axiomatics since rigid-analytic spaces are not actually topological spaces (in the way complex-analytic spaces are), and moreover the GAGA for them works even when the ground field is not locally compact. Probably there's no useful meta-theorem of the sort you're wondering about. Instead, one has to do some hard work in each case and then use some aspects of Serre's method (depending on the situation). Formal GAGA for proper schemes over an adic noetherian ring is another important "example", and that too does not fit your axiomatics. –  BCnrd Oct 25 '10 at 5:12
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Another class of (counter)examples: GAGA for proper algebraic spaces over the complex numbers (as well as over non-archimedean fields, albeit with a rather more subtle analytification functor). Here there's an even worse "problem", namely that on the algebraic side one no longer has actual ringed spaces (even over the complex numbers, where on the analytic side one does have such things). So even a notion of "morphism" $F(V) \rightarrow V$ becomes a new difficulty. This can be fixed up by using henselian ringed topoi, but that's a whole other can of worms... –  BCnrd Oct 25 '10 at 5:59
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If you ARE interested in rigid and formal versions of GAGA, this text: math.upenn.edu/~harbater/patch35.pdf takes exactly that approach (you will see several theorems there stated as "GAGA for"). –  H. Hasson Oct 25 '10 at 13:21
    
awesome, thanks. –  Ian M. Oct 25 '10 at 21:06
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If the link answers the question, then it would be nice if H. Hasson posts it as an answer, and Ian accepts it, just so the question disappears from the unanswered questions link. (Yes, I procrastinate from real work by scrolling through the unanswered questions list. How did you figure it out?) –  arsmath Nov 1 '10 at 10:03
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