Modules of finite homological dimension with respect to a semidualizing module

Abstract

Abstract. We prove versions of results of Foxby and Holm about modules of finite (Gorenstein) injective dimension and finite (Gorenstein) projective dimension with respect to a semidualizing module. We also verify special cases of a question of Takahashi and White. Mathematics Subject Classification (2000). 13C05, 13D05, 13H10.Keywords. Dualizing modules, Gorenstein homological dimensions, Gorenstein injective dimension, Gorenstein projective dimension, Gorenstein rings, Semidualizing modules. 0. Introduction. Let R be a commutative noetherian ring. It is well-known that, if R is Gorenstein and local, then every module with finite projective dimension has finite injective dimension. Conversely, Foxby [4,5] showed that, if R is local and admits a finitely generated module of finite projective dimension and finite injective dimension, then R is Gorenstein. More recently, Holm [12] proved that, if M is an R-module of finite projective dimension and finite Gorenstein injective dimension, then M has finite injective dimension, and so the localization R p is Gorenstein for each p ∈ Spec(R) with depth Rp (M p ) < ∞. See Sect. 1 for terminology and notation.In this paper, we prove analogues of these results for homological dimensions defined in terms of semidualizing R-modules. For instance, the following result is proved in (2.1). Other variants of this result are also given in Sect. 2. It should be noted that our proof of this result is different from Holm's proof for the special case C = R. In particular, this paper also provides a new proof of Holm's result.Theorem A. Let C be a semidualizing R-module, and let M be an R-module with P C -pd R (M ) < ∞ and Gid R (M ) < ∞. Then id R (M ) = Gid R (M ) < ∞

Tate cohomology with respect to semidualizing modules

“…Corollary 6.3 shows how it improves upon a result of Asadollahi and Salarian [1, (4.8)]. It also compliments work of Iacob [16,Theorem 2] and implies some of the main results of [20]; see Corollary 6.5.…”

“…The condition W ⊥ G(W) from Fact 1.14 shows that W ⊥ M, so the displayed sequence is Hom A (W, −)-exact. From this we recover some of the main results of [20]. …”

On reduced G-perfection and horizontal linkage relative to a semidualizing module

“…We are now prepared to prove (i)⇒(ii). By Theorem 3.8 and Proposition 4.2, we have 29), (30), (31) and (32), we get…”

“…It is worth mentioning that the notion of semidualizing module has become a central object in homological algebra since its inception independently by Foxby [11], Golod [16], and Vasconcelos [34]. Many authors have studied and justified the concept under different viewpoints and objectives; we refer, besides the references already mentioned and just to quote a few, to [7], [17], [24], [25], [26], [27], [29], [30], [31], [32], [36].…”

Using semidualizing complexes to detect Gorenstein rings

“…Partial answers to Takahashi and White's question is given by Araya and Takahashi [1] and Sather-Wagstaff and Yassemi [16]. We give a complete answer to this question in the following result; see Section 2 for background on complexes and the derived category, and Theorem 3.2 for the proof.…”