  In this paper we present a new approach to Grothendieck duality on schemes.
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. We obtain most of the important features of Grothendieck duality, yet
manage to avoid lengthy and difficult compatibility verifications. Our results
apply to finite type schemes over a regular noetherian finite dimensional base
ring, and hence are suitable for arithmetic geometry.
