The relative Riemann-Roch theorem from Hochschild homology

摘要

This paper attempts to clarify a preprint of Markarian (2001). Markarian's preprint proves the relative Riemann-Roch theoremusing a result describing how the HKR map fails to respectcomultiplication. This paper elaborates on the core computations inMarkarian's preprint. These computations show that the HKR map twisted by the squareroot of the Todd genus "almost preserves" the Mukai pairing. Thissettles a part of a conjecture of Caldararu, 2005. The relativeRiemann-Roch theorem follows from this and a result of Caldararu,preprint, 2003.