The present paper deals with a relative version of the Riemann-Roch theorem for Hochschild cohomology. The approach proposed by the author leads to a proof of this theorem by a direct computation of the pairing on Hodge cohomology to which Caldararu’s pairing on Hochschild cohomology descends via the Hochschild-Konstant-Rosenberg (HKR) map multiplied by the square root of the Todd genus. Indeed, the HKR isomorphism fails to preserve products and then has be corrected. In N. Markarian [Poincaré-Birkhoff-Witt isomorphism, Hochschild homology and Riemann-Roch theorem, Preprint (2001), MPI M 2001-52; cf. also The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem, arXiv:math/0610553], a similar statement was incorrectly proved.
Let \(X\) be a proper smooth scheme over a field of characteristic zero. Then we can define the HKR map between the complex \(\widehat{C}^\bullet (X)\), representing the object \(\Delta^* {\mathcal O}_{\Delta}\), and the complex \(S^\bullet (\Omega_X[1])\). This map induces an isomorphism (the HKR isomorphism) between the Hochschild and the Hodge cohomology. A. Caldararu [“The Mukai pairing. I. The Hochschild structure”, arXiv:math.AG/0308079], defines a pairing on the Hochschild cohomology. In order to prove a Riemann-Roch theorem, the author defines a duality map for the Hochschild cohomology and shows the compatibility by explicit calculations.