  In this paper we present a new approach to Grothendieck duality over
commutative rings. Our approach is based on the idea of rigid dualizing
complexes, which was introduced by Van den Bergh in the context of
noncommutative algebraic geometry. The method of rigidity was modified to work
over general commutative base rings in our paper [YZ5]. In the present paper we
obtain many of the important local features of Grothendieck duality, yet manage
to avoid lengthy and difficult compatibility verifications. Our results apply
to essentially finite type algebras over a regular noetherian finite
dimensional base ring, and hence are suitable for arithmetic rings. In the
sequel paper [Ye4] these results will be used to construct and study rigid
dualizing complexes on schemes.
