GAGA

One of the most strong “evidence” that the G-topologies defined on rigid analytic spaces is right, I guess, is GAGA. It is well known that there is some kind of connection between complex geometry and scheme theory on the complex field, called GAGA. What rigid geometry tried to do is it’s p-adic analogue and it succeeded. More precisely, the category of coherent sheaves on a projective scheme over a CDVF is categorically equivalent to that of those on the corresponding projective rigid analytic space via functor called GAGA (standing for Geometry analytique et geometry algébrique). The correspondence itself looks very natural because we just regard schemes as rigid analytic spaces just as we regard schemes over complex field as complex manifolds by changing its sheaf structure with taking maximal ideals as points. This can be possible by the well known fact that there is 1 to 1 correspondence between points on complex field and maximal ideals of polynomial rings and the same thing happens in the non-Archimedean case, for which we think GAGA on non-Archimedean fields.

According to the textbook a proof can be done exactly the same way as original GAGA, though I read another proof by means of GFGA and relation between formal geometry and rigid one here. I’m not good at scheme theory and I’m still not sure of a proposition about spectral sequences, but it looks sophisticated. Since rigid geometry looks like algebraic rather than analytic regardless of the purpose of this, “analytic geometry over non-Archimedean fields”, the proof looks like algebraic, too.