# Is there a notion of pure dimension for Berkovich analytic space?

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?

• 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. – nfdc23 Apr 19 '16 at 16:40
• 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/… – ACL Apr 19 '16 at 17:15