Algebraization Techniques and Rigid-Analytic Artin-Grothendieck Vanishing

Abstract

First, we prove an algebraization result for rig-smooth algebras over a general noetherian ring; this positively answers the question raised in [Sta24, Tag 0GAX]. Then we prove a general partial algebraization result in non-archimedean geometry. The result says that we can always algebraize a geometrically reduced affinoid rigid-analytic space in "one direction" in an appropriate sense. As an application of this result, we show the remaining cases of the Artin-Grothendieck Vanishing for affinoid algebras, which were previously conjectured in [BM21, {\S}7]. This allows us to deduce a stronger version of the rigid-analytic Artin-Grothendieck Vanishing Conjecture (see [Han20, Conj. 1.2]) over a field of characteristic 0. Using a completely different set of ideas, we also obtain a weaker version of this conjecture over a field of characteristic p>0.