On de Rham Cohomology of Linear Categories
Andrei Chite\c{s}, M\u{a}d\u{a}lin Ciungu, Drago\c{s} \c{S}tefan

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.
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Taxonomy
TopicsCommutative Algebra and Its Applications · Advanced Algebra and Logic · Topological and Geometric Data Analysis
