  This paper brings together two theories in algebra that have had been
extensively developed in recent years. First is the study of various
homological dimensions and what information such invariants can give about a
ring and its modules. A collection of intermediate C-Gorenstein dimensions is
defined and this allows generalizations of results concerning C-Gorenstein
dimension and certain Serre-like conditions. Second is the theory of linkage
first introduced by Peskine and Szpiro and generalized to modules by
Martinskovsky and Strooker. Using the further generalization of module linkage
of Nagel, results are proven connecting linkage with these homological
dimensions and Serre-like conditions.
