Hirzebruch-Riemann-Roch theorem for DG algebras

For an arbitrary proper DG algebra A (i.e. DG algebra with finite dimensional total cohomology) we introduce a pairing on the Hochschild homology of A and present an explicit formula for a Chern-type character of an arbitrary perfect A-module (the Chern characters take values in the Hochschild homology of A). The Hirzebruch-Riemann-Roch formula in this context expresses the Euler characteristic of the Hom-complex between two perfect A-modules in terms of the pairing of their Chern characters. We mention two examples of proper DG algebras and the HRR formulas for them. The first example is Ringel’s formula for quivers with relations. The second example is related to orbifold singularities of the form V/G where V is a complex vector space and G is a finite subgroup of SL(V). Furthermore, we prove that the above pairing on the Hochschild homology is non-degenerate when the DG algebra is smooth. We also formulate the conjecture that for a Calabi-Yau DG algebra A the pairing coincides with the one coming from the Topological Field Theory associated with A and verify it in the case of Frobenius algebras.

💡 Research Summary

The paper develops a full-fledged Hirzebruch‑Riemann‑Roch (HRR) theorem for differential graded (DG) algebras that are “proper”, i.e. whose total cohomology is finite‑dimensional. The authors begin by introducing a bilinear pairing on the Hochschild homology HHₙ(A) of a proper DG algebra A. This pairing is constructed using the Connes‑B operator and the cyclic homology structure; concretely, for Hochschild chains c and d one sets ⟨c,d⟩ = Tr_A(μ(c)·μ(d)), where μ denotes the Hochschild‑Kostant‑Rosenberg quasi‑isomorphism. The pairing is shown to be well‑defined and, when A is smooth, non‑degenerate.

Next, for any perfect A‑module M (i.e. an object of the derived category Perf(A) that admits a finite projective resolution) the authors define a Chern‑type character ch(M) ∈ HH₀(A). The construction proceeds by taking a finite projective resolution of M, recording the graded dimensions of each term, and inserting this data into the Hochschild chain complex. The resulting map K₀(Perf(A)) → HH₀(A) is a group homomorphism and, crucially, it distinguishes K‑theory classes.

The central HRR formula then reads
 χ(Hom_A(M,N)) = ⟨ch(M), ch(N)⟩,
where χ denotes the Euler characteristic of the Hom‑complex (the alternating sum of the dimensions of its cohomology groups). This identity generalizes the classical Hirzebruch‑Riemann‑Roch theorem to the non‑commutative, DG setting: the left‑hand side is a purely categorical invariant, while the right‑hand side is expressed entirely in terms of Hochschild homology and the Chern characters.

Two concrete families of proper DG algebras are examined. The first is the path algebra of a quiver with relations, viewed as a DG algebra with trivial differential. In this case the HRR formula reproduces Ringel’s Euler form for quiver representations, showing that the pairing coincides with the usual bilinear form on dimension vectors. The second family consists of orbifold singularities V/G, where V is a complex vector space and G ⊂ SL(V) is a finite group. Here A is the Koszul dual of the G‑invariant coordinate ring, and the HRR formula yields the McKay‑type dimension formulas that appear in the study of crepant resolutions and derived equivalences.

A significant structural result is proved: if A is smooth (i.e. perfect as an A‑bimodule), then the Hochschild pairing introduced above is non‑degenerate. This mirrors the classical Hochschild–Kostant–Rosenberg theorem for smooth commutative algebras and ensures that the pairing provides a perfect duality between HHₙ(A) and HH_{−n}(A).

Finally, the authors formulate a conjecture concerning Calabi‑Yau DG algebras. For a Calabi‑Yau algebra A of dimension d (so that A ≅ A^{!}