On Picard groups of perfectoid covers of toric varieties

Abstract

Let X be a proper smooth toric variety over a perfectoid field of prime residue characteristic p. We study the perfectoid space which covers X constructed by Scholze, showing that is canonically isomorphic to $$\textrm{Pic}(X)[p^{-1}]$$ . We also compute the cohomology of line bundles on and establish analogs of Demazure and Batyrev–Borisov vanishing. This generalizes the first author’s analogous results for projectivoid space.