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Hirzebruch-Riemann-Roch theorem for DG algebras

D. Shklyarov

Abstract

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.

Hirzebruch-Riemann-Roch theorem for DG algebras

Abstract

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.

Paper Structure

This paper contains 24 sections, 30 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Geometry of DG categories
  3. A categorical version of the Hirzebruch-Riemann-Roch theorem
  4. Main results
  5. Other viewpoints on the noncommutative HRR theorem
  6. Acknowledgements
  7. Hochschild homology of DG algebras and DG categories
  8. DG algebras, DG categories, and DG functors
  9. Perfect modules
  10. Hochschild homology
  11. Künneth isomorphism
  12. Hirzebruch-Riemann-Roch theorem
  13. Euler character
  14. Hirzebruch-Riemann-Roch theorem
  15. On the computational aspect of the HRR theorem
  16. ...and 9 more sections

Key Result

Proposition 2.1

The DG category ${\mathsf{Perf}} A$ is closed under passing to homotopically equivalent modules, taking shifts and cones of degree 0 morphisms, and taking homotopy direct summands.

Theorems & Definitions (30)

  • Proposition 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Proposition 2.9
  • Proposition 2.10
  • ...and 20 more