  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.
