Analytification, localization and homotopy epimorphisms

TL;DR

This paper investigates the properties of analytification functors in derived analytic geometry, demonstrating that several key types are homotopy epimorphisms and establishing their implications for Hochschild homology and the cotangent complex.

Contribution

It proves that important analytification processes are homotopy epimorphisms and shows how Hochschild homology and the cotangent complex can be computed in this setting.

Findings

Analytification functors like Tate, overconvergent, Stein are homotopy epimorphisms.

Hochschild homology and cotangent complex are computable for analytic rings.

Hochschild homology commutes with localizations, analytifications, and completions.

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 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 arises from Weierstrass, Laurent and rational localisations 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.