A Study of Some Generalizations of Local Homology

Abstract

: Tate local cohomology and Gorenstein local cohomology theory, which are important generalizations of the classical local cohomology, has been investigated. It has been found that they have such vanishing properties and long exact sequences. However, for local homology, what about the duality? In this paper we are concerned with Tate local homology and Gorenstein local homology. In the first part of the paper we generalize local homology as Tate local homology, and study such vanishing properties, artinianness and some exact sequence of Tate local homology modules. We find that for an artianian R -module M and a finitely generated R -module N with finite Gorenstein projective dimension, the Tate local homology module of M and N with respect to an ideal I is also an artinian module. In the second part of the paper we consider Gorenstein local homology modules as Gorenstein version. We discuss vanishing properties and some exact sequences of Gorenstein local homology modules and obtain an exact sequence connecting Gorenstein, Tate and generalized local homology. Finally, as an applicaton of the exact sequence connecting these local homology modules, we find that for finitely generated R -modules with finite projective dimension and admitting Gorenstein projective proper resolution respectively, Gorenstein local homology coincides with generalized local homology in certain cases.