## Reference for rigid analytic GAGA

I'm looking for a reference for the following result.

Theorem. Let $K$ be a complete, non-archimedean field, and let $X/K$ be a projective scheme, with analytification $X^\mathrm{an}$. Then the analytification functor from coherent $\mathcal{O}_X$-modules to coherent ${\mathcal{O}}_{X^\mathrm{an}}$-modules is an equivalence of categories.

While I've seen this sort of statement in a lot of introductory notes on rigid analytic geometry (most attributing it to Keihl), none of them seem to give a published reference. Any help would be much appreciated.

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Have you looked at the papers\notes of Brian Conrad? – Keenan Kidwell Feb 8 at 14:08
His notes on rigid geometry didn't seem to have a reference. Rooting around some of his papers has done the job though, thanks! – ChrisLazda Feb 8 at 14:28

I am quite surprised by the attribution to Kiehl that you saw. Anyway, I think the result is due to Ursula Köpf (not only over a field $K$ but actually over an affinoid space): "Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen", Schriftenreihe Univ. Münster, 2 Serie, Heft 7 (1974).

Brian Conrad gave another proof as an application of his results of relative ampleness in the rigid analytic setting (see "Relative ampleness in rigid geometry", Ann. Inst. Fourier (Grenoble) 56 (2006), n° 4).

I also learned a proof from Antoine Ducros in the setting of Berkovich spaces. I wrote in down in an appendix to my paper "Raccord sur les espaces de Berkovich", Algebra & Number Theory 4 (2010), n° 3). It is very close to Serre's proof in the complex analytic setting and probably very close to Köpf's proof too, but I cannot say for sure since I never saw her paper.

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Great, thanks. It was Köpf's paper that eventually came up after digging through Conrad's papers. – ChrisLazda Feb 8 at 22:45
Perhaps the attribution to Kiehl (which I agree is misplaced) is due to the fact that in the rigid-analytic case the main "new" ingredient is setting up a robust theory of coherent sheaves mixing features of the scheme-theoretic and complex-analytic cases, and that the creation of that theory of coherence in the rigid-analytic case is unquestionably due to Kiehl? Indeed, once that is available, the argument of Serre more-or-less carries over as Jerome says (with a bit of care due to non-rational points). Of course, Kopf goes a bit further, with a result over affinoid bases. – nosr Feb 9 at 23:17
One other clarification (as J\'erome is certainly well aware): it is K\"opf's generalization of GAGA over an affinoid base that is proved in another way in the relative ampleness paper (the case over a field is taken as input). Over a field, as in the original question above, then probably there is only one proof, namely transporting Serre's method using Kiehl's theory of coherence. (One could make another proof using formal schemes and formal GAGA when the base field is discretely-valued, but for a non-noetherian valuation ring it would be a much harder proof than via Serre's method.) – nosr Feb 9 at 23:24
Thanks for the clarifications, nosr! – Jérôme Poineau Feb 10 at 20:52
Could I ask whether this paper of Köpf answers my question mathoverflow.net/questions/121881/…? It doesn't seem to be so easy to obtain a copy. – Simon Wadsley Feb 20 at 22:08