Bivariance, Grothendieck duality and Hochschild homology

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 fixed noetherian scheme S. The theory takes values in the category of symmetric graded modules over the graded-commutative ring \oplus_i H^i(S,O_S). In degree i, the cohomology and homology H^0(S,O_S)-modules thereby associated to such an x: X -> S, with Hochschild complex H_x, are Ext^i(H_x, H_x) and Ext^{-i}(H_x, x^!O_S). 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.

No reviews yet.

No comments yet.