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

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

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Published 11 April 2012