On the Hochschild Homology of Curved Algebras

TL;DR

This paper computes the Hochschild homology of curved modules over curved rings, generalizing previous results and introducing de Rham models, with a key proof based on a curved version of a classical theorem.

Contribution

It extends Hochschild homology computations to curved algebras and modules, providing new de Rham models and including a proof of a curved version of Hopkins-Neeman's theorem.

Findings

Hochschild homology of curved modules is explicitly computed.

Introduction of de Rham models for curved algebra modules.

Proof of a curved version of Hopkins-Neeman's theorem included.

Abstract

We compute the Hochschild homology of the differential graded category of perfect curved modules over suitable curved rings, giving what might be termed "de Rham models" for such. This represents a generalization of previous results by Dyckerhoff, Efimov, Polishchuk, and Positselski concerning the Hochschild homology of matrix factorizations. A key ingredient in the proof is a theorem due to B. Briggs, which represents a "curved version" of a celebrated theorem of Hopkins and Neeman. The proof of Briggs' Theorem is included in an appendix to this paper.