  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.
