The Mukai pairing and integral transforms in Hochschild homology. (English) Zbl 1208.14013
The Hochschild homology of any quasi-compact separated scheme over a field of characteristic zero can be endowed with two different pairings:
–The Shklyarov pairing, introduced in the DG-framework in the preprint [Hirzebruch-Riemann-Roch theorems for DG-algebras, arXiv:math0710.1937].
–The Mukai pairing, defined by A. Căldăraru and S. Willerton in [New York J. Math. 16, 61–98 (2010; Zbl 1214.14013)] via Serre duality.
The main result in the paper is that the two parings are the same (up to some sign modifications). This is proved by identifying the action on Hochschild homology given by kernels, that is integral transforms, in Shklyarov and Căldăraru theories.
–The Shklyarov pairing, introduced in the DG-framework in the preprint [Hirzebruch-Riemann-Roch theorems for DG-algebras, arXiv:math0710.1937].
–The Mukai pairing, defined by A. Căldăraru and S. Willerton in [New York J. Math. 16, 61–98 (2010; Zbl 1214.14013)] via Serre duality.
The main result in the paper is that the two parings are the same (up to some sign modifications). This is proved by identifying the action on Hochschild homology given by kernels, that is integral transforms, in Shklyarov and Căldăraru theories.
Reviewer: Julien Grivaux (Marseille)
MSC:
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 19L10 | Riemann-Roch theorems, Chern characters |
