On modules with trivial self-extensions

Wakamatsu, Takayoshi

doi:10.1016/0021-8693(88)90215-3

Cited by 95 publications

(61 citation statements)

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“…Semidualizing modules occur in the literature with several different names, e.g., in the work of Foxby [3], Golod [9], Mantese and Reiten [17], Vasconcelos [20] and Wakamatsu [21]. The prototypical semidualizing modules are the dualizing (or canonical) modules of Grothendieck and Hartshorne [10].…”

Section: − −− → · · ·mentioning

confidence: 99%

Modules of finite homological dimension with respect to a semidualizing module

Sather-Wagstaff

Yassemi

2009

Arch. Math.

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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 ) < ∞

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Section: − −− → · · ·mentioning

confidence: 99%

Modules of finite homological dimension with respect to a semidualizing module

Sather-Wagstaff

Yassemi

2009

Arch. Math.

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“…If R is a left Noetherian ring, S is a right Noetherian ring and R ω S is a faithfully balanced and n-selforthogonal bimodule for all n, then R ω is just a generalized tilting module with S = End( R ω) in the sense of Wakamatsu [4,5]. In this case, ω S is also a generalized tilting module with R = End(ω S ) by [5,Corollary 3.2].…”

Section: Theorem 11 ([3 Theorem A])mentioning

confidence: 99%

Relative syzygies and grade of modules

Liu

Huang

2012

Acta. Math. Sin.-English Ser.

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Recently, Takahashi established a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies the spherical approximation theorem due to Auslander and Bridger and the Cohen-Macaulay approximation theorem due to Auslander and Buchweitz. In this paper we generalize these results to much more general case over non-commutative rings. As an application, we establish a relation between the injective dimension of a generalized tilting module ω and the finitistic dimension with respect to ω.

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“…One crucial result with respect to approximations is the Wakamatsu's Lemma [62]. It says the following.…”

Section: !mentioning

confidence: 99%