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For an arbitrary differential graded algebra  A  with finite‐dimensional total cohomology we introduce a pairing on the Hochschild homology of  A , derive an explicit formula for the Chern characters of perfect  A ‐modules (the Chern characters take values in the Hochschild homology of  A ), and prove a Hirzebruch–Riemann–Roch‐type formula which expresses the Euler characteristic of the Hom‐complex between two perfect  A ‐modules in terms of the pairing of their Chern characters.

Hirzebruch–Riemann–Roch‐type formula for DG algebras