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

On de Rham Cohomology of Linear Categories