For affinoid spaces the definition is similar to algebraic geometry, what about general analytic spaces? I can't find a reference about it. If yes then is the analytification of a variety of pure dimension still of pure dimension?
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$\begingroup$ Briefly, make a massive extension of the ground field so that the affinoid space becomes strictly analytic; then the usual definition from commutative algebra works well (and is invariant under further extension of the ground field, so is independent of the initial massive extension, hence is intrinsic). See papers of Antoine Ducros for a full development of a robust dimension theory (and especially good related properties) in the context of Berkovich spaces. $\endgroup$– nfdc23Commented Apr 19, 2016 at 16:40
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$\begingroup$ To complete nfdc23's comment, Antoine Ducros discusses the dimension of Berkovich spaces in his paper Variation de la dimension relative en géométrie analytique $p$-adique, Section 1, journals.cambridge.org/… $\endgroup$– ACLCommented Apr 19, 2016 at 17:15
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2 Answers
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Yes, there is a good notion of dimension, due to Berkovich and developed in my article, as mentioned in the two answers above.
Concerning your question about GAGA principle for pure dimension, the answer is positive. I do not know whether it is explicitly written down in the litterature (I did not write it in my paper, for instance), but I can sketch the proof here. Since everything is invariant under ground field extension, you may assume that your ground field is not trivially valued. Now in order to prove that a strict analytic space is purely d-dimensional, it is sufficient to check that the dimension of O_X,x is equal to d for every rigid point x. But this can be checked after completion, and at a rigid point the analytic and algebraic local rings have the same completions, QED.
Remark: there is certainly also GAGA for irreducible components (an irreducible scheme has an irreducible analytification) but it is not so easy to prove to my knowledge. Using normalization it can be reduced to GAGA for connectedness, for which I do not know any easy proof: one can compactify and use projective GAGA+Hartog's theorem, or use Berkovich's comparison theorem for étale cohomology at the H^0 level (!). Maybe there is a simpler one using Noether's normalization?
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By the way, my above proof shows in fact the following: if k is a non-trivially valued field, if A is a strictly affinoid k-algebra and if X is a finitely generated A-scheme of pure (Krull) dimension d, then X^an is also of pure dimension d.