A Riemann-Roch Theorem for dg Algebras

TL;DR

This paper establishes a Riemann-Roch type theorem for smooth proper dg-algebras, connecting Hochschild classes of modules and endomorphisms with Hochschild homology through a convolution formula.

Contribution

It introduces a novel Riemann-Roch formula for dg-algebras that relates Hochschild classes of modules and endomorphisms to Hochschild homology.

Findings

Defines Hochschild class for pairs (M,f) in dg-algebras

Derives a convolution-based Riemann-Roch formula

Extends classical Riemann-Roch to derived algebraic setting

Abstract

Given a smooth proper dg-algebra $A$, a perfect dg $A$-module $M$, and an endomorphism $f$ of $M$, we define the Hochschild class of the pair $(M,f)$ with values in the Hochschild homology of $A$. Our main result is a Riemann-Roch type formula involving the convolution of two such Hochschild classes.