Homological aspects of semidualizing modules

Abstract

We investigate the notion of the C-projective dimension of a module, where C is a semidualizing module. When C = R, this recovers the standard projective dimension. We show that three natural definitions of finite Cprojective dimension agree, and investigate the relationship between relative cohomology modules and absolute cohomology modules in this setting. Finally, we prove several results that demonstrate the deep connections between modules of finite projective dimension and modules of finite C-projective dimension. In parallel, we develop the dual theory for injective dimension and C-injective dimension.

Local cohomology modules and Gorenstein injectivity with respect to a semidualizing module

“…First, in view of [18,Proposition 3.4] we see that C-id R M ′′ is finite. Therefore, by [18, Corollary 2.9], M ′′ ∈ A C (R) and hence Tor R 1 (C, M ′′ ) = 0.…”

“…The reader is referred to [1], [15] and [18] for some basic results about those classes. Definition 2.7.…”

Characterizations of some rings with C-projective, C-(FP)-injective and C-flat modules

“…Thus {C ⊗ R F λ } ∼ = C ⊗ R F for some flat module F . Therefore we have the following isomorphisms, which shows that F λ is flat: [20,Proposition 5.3] proved that a commutative ring R is noetherian if and only if every direct sum of C-injective modules is C-injective. We include it in our theorem.…”

“…Takahashi and White [20] investigated the C-projective dimension of a module, and their results showed that three natural definitions of the finite C-projective dimension agree.…”

Balance for Relative Homology With Respect to Semidualizing Modules

“…For more details about semidualizing modules and their related categories, we refer the reader to [3,11,12,13,18,20,21,22].…”