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# Asian Journal of Mathematics

## Volume 15 (2011)

### Number 3

### Bivariance, Grothendieck duality and Hochschild homology I: Construction of a bivariant theory

Pages: 451 – 498

DOI: http://dx.doi.org/10.4310/AJM.2011.v15.n3.a7

#### Authors

#### Abstract

A procedure for constructing bivariant theories by means of Grothendieck duality is developed. This produces, in particular, a bivariant theory of Hochschild (co)homology on the category of schemes that are flat, separated and essentially of finite type over a noetherian scheme $S$. The theory takes values in the category of symmetric graded modules over the graded-commutative ring $\oplus_i \mathrm{H}^i(S,\mathcal{O}_S)$. In degree $i$, the cohomology and homology $\mathrm{H}^0(S,\mathcal{O}_S)$-modules thereby associated to such an $x: X \to S$, with Hochschild complex $\mathcal{H}_x$, are $\mathrm{Ext}^i_{\mathcal{O}_X} (\mathcal{H}_x,\mathcal{H}_x)$ and $\mathrm{Ext}^{−i}_{\mathcal{O}_X} (\mathcal{H}_x, x^!\mathcal{O}_S) (i \in \mathbb{Z})$. This lays the foundation for a sequel that will treat orientations in bivariant Hochschild theory through canonical relative fundamental class maps, unifying and generalizing previously known manifestations, via differential forms, of such maps.

#### Keywords

Hochschild homology; bivariant; Grothendieck duality; fundamental class

#### 2010 Mathematics Subject Classification

14F99