**Title:** Hirzebruch-Riemann-Roch Theorem for DG algebras

**Speaker:** Dmytro Shklyarov

**Speaker Info:** Kansas State University

**Brief Description:**

**Special Note**: **Special Session**

**Abstract:**

Due to results of A. Bondal, M. Van den Bergh, B. Keller, any reasonable scheme is affine in the dg (differential graded) sense. This means that the derived category of perfect complexes on the scheme is equivalent to the derived category of perfect modules over a dg algebra. The most popular example of such an equivalence is the celebrated result of A. Beilinson that describes the derived categories of the projective spaces in terms of certain quiver algebras with relations. These results have prompted the idea that Noncommutative Algebraic Geometry should incorporate the study of dg algebras, dg categories and various homological invariants thereof.The talk is devoted to a Hirzebruch-Riemann-Roch type formula for proper dg algebras (such algebras are though of as proper schemes in the dg world). Namely, for any proper dg algebra A, I'll describe an explicit pairing on the Hochschild homology of A and for an arbitrary perfect A-module I'll present an explicit formula for its Chern class with the value in the Hochschild homology of A. The Hirzebruch-Riemann -Roch formula in this setting expresses the Euler characteristic of the Hom-complex between any two perfect A-modules in terms of the pairing of their Chern classes.

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