Analytic Hochschild-Kostant-Rosenberg Theorem

TL;DR

This paper extends the Hochschild-Kostant-Rosenberg theorem to derived analytic geometry using a framework of derived algebra, relating Hochschild homology, de Rham complexes, and derived loop stacks.

Contribution

It introduces a derived algebraic framework for bornological modules and proves a generalized HKR theorem with geometric interpretations in derived analytic geometry.

Findings

Category of chain complexes forms a derived algebraic context.

Hochschild-Kostant-Rosenberg theorem is extended to derived analytic stacks.

Derived loop stack is equivalent to the shifted tangent stack.

Abstract

Let $R$ be a Banach ring. We prove that the category of chain complexes of complete bornological $R$-modules (and several related categories) is a derived algebraic context in the sense of Raksit. We then use the framework of derived algebra to prove a version of the Hochschild-Kostant-Rosenberg Theorem, which relates the circle action on the Hochschild algebra to the de Rham-differential-enriched-de Rham algebra of a simplicial, commutative, complete bornological algebra. This has a geometric interpretation in the language of derived analytic geometry, namely, the derived loop stack of a derived analytic stack is equivalent to the shifted tangent stack. Using this geometric interpretation we extend our results to derived schemes.