Balance in Generalized Tate Cohomology #

Abstract

ABSTRACT We consider two preenveloping classes of left R-modules ℐ, ℰ such that Inj ⊂ ℐ ⊂ ℰ, and a left R-module N. For any left R-module M and n ≥ 1 we define the relative extension modules (M, N) and prove the existence of an exact sequence connecting these modules and the modules (M, N) and (M, N). We show that there is a long exact sequence of (M, −) associated with a Hom(−, ℰ) exact sequence 0 → N′ → N → N′′ → 0 and a long exact sequence of (−, N) associated with a Hom(−, ℰ) exact sequence 0 → M′ → M → M′′ → 0. Using these properties we prove that for two complete hereditary cotorsion theories (𝒞, ℒ), (ℒ, ℰ) we have (M, N) for any left R modules M, N and for any n ≥ 1, where (M, N) are the generalized Tate cohomology modules (see Section 1 for the definition). So in this case we have an occurrence of balance, i.e. the generalized Tate cohomology can be computed either using a left 𝒞-resolution and a projective resolution of M or using a right ℰ-resolution and an injective resolution of N.