Local Duality with Dualizing Complexes and Other Dualities

Abstract

In this chapter we present a version of Grothendieck local duality for a Noetherian local ring admitting a dualizing complex. We derive it from a more general result involving the local cohomology with respect to an arbitrary ideal of a Noetherian ring admitting a dualizing complex, originally proved by Hartshorne for a regular ring of finite Krull dimension. We also extend Hartshorne’s affine duality stated for regular rings of finite Krull dimension to any Noetherian ring with a dualizing complex and provide a counterpart in local homology. Among other things we provide duality results involving both local homology and local cohomology, a recurrent theme in this monograph. We also investigate the local homology of a complex with Artinian homology, more generally with mini-max homology. We end the chapter with a short approach to Greenlees’ Warwick duality.