HirzebruchRiemannRoch 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 Cherntype character of an arbitrary perfect Amodule (the Chern characters take values in the Hochschild homology of A). The HirzebruchRiemannRoch formula in this context expresses the Euler characteristic of the Homcomplex between two perfect Amodules 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 nondegenerate when the DG algebra is smooth. We also formulate the conjecture that for a CalabiYau 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2007
 arXiv:
 arXiv:0710.1937
 Bibcode:
 2007arXiv0710.1937S
 Keywords:

 Mathematics  KTheory and Homology;
 Mathematics  Algebraic Geometry;
 16E45;
 18F99;
 14C40
 EPrint:
 References added