Analytification, localization and homotopy epimorphisms

Abstract

We study the interaction between various analytification functors, and a class of morphisms of rings, called homotopy epimorphisms. An analytification functor assigns to a simplicial commutative algebra over a ring R, along with a choice of Banach structure on R, a commutative monoid in the symmetric monoidal model category of simplicial ind-Banach R-modules. We show that several analytifications relevant to analytic geometry - such as Tate, overconvergent, Stein analytification, and formal completion - are homotopy epimorphisms. Another class of examples of homotopy epimorphisms arises from Weierstrass, Laurent and rational localizations in derived analytic geometry. As applications of this result, we prove that Hochschild homology and the cotangent complex are computable for analytic rings, and the computation relies only on known computations of Hochschild homology for polynomial rings. We show that in various senses, Hochschild homology as we define it commutes with localizations, analytifications and completions.