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September 2011 Bivariance, Grothendieck duality and Hochschild homology I: Construction of a bivariant theory
Leovigildo Alonso Tarrío, Ana Jeremías López, Joseph Lipman
Asian J. Math. 15(3): 451-498 (September 2011).

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.

Citation

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Leovigildo Alonso Tarrío. Ana Jeremías López. Joseph Lipman. "Bivariance, Grothendieck duality and Hochschild homology I: Construction of a bivariant theory." Asian J. Math. 15 (3) 451 - 498, September 2011.

Information

Published: September 2011
First available in Project Euclid: 28 February 2012

zbMATH: 1251.14010
MathSciNet: MR2838216

Subjects:
Primary: 14F99

Rights: Copyright © 2011 International Press of Boston

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Vol.15 • No. 3 • September 2011
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