On de Rham Cohomology of Linear Categories

TL;DR

This paper introduces a Chern map linking the Grothendieck group of a linear category to its de Rham cohomology, utilizing connections on modules and curvature traces to establish cohomology classes.

Contribution

It defines the notion of connection on a C-module and constructs a Chern map to de Rham cohomology, extending classical concepts to linear categories.

Findings

The trace of the curvature of a connection yields a de Rham cocycle.

The cohomology class of this cocycle is independent of the connection choice.

A new framework for de Rham cohomology in linear categories is established.

Abstract

We define the Chern map from the Grothendieck group of a linear category C to the de Rham cohomology of C with coefficients in a DG-category. In order to achieve our goal, we define the notion of connection on a C-module, and we show that the trace of the curvature of a connection is a de Rham cocycle, whose cohomology class does not depend on the choice of the connection.