Relative and Tate homology with respect to semidualizing modules

Abstract

We introduce and investigate in this paper a kind of Tate homology of modules over a commutative coherent ring based on Tate ℱC-resolutions, where C is a semidualizing module. We show firstly that the class of modules admitting a Tate ℱC-resolution is equal to the class of modules of finite 𝒢(ℱC)-projective dimension. Then an Avramov–Martsinkovsky type exact sequence is constructed to connect such Tate homology functors and relative homology functors. Finally, motivated by the idea of Sather–Wagstaff et al. [Comparison of relative cohomology theories with respect to semidualizing modules, Math. Z. 264 (2010) 571–600], we establish a balance result for such Tate homology over a Cohen–Macaulay ring with a dualizing module by using a good conclusion provided in [E. E. Enochs, S. E. Estrada and A. C. Iacob, Balance with unbounded complexes, Bull. London Math. Soc. 44 (2012) 439–442].