Relative T-injective modules and relative T-flat modules

Abstract

Let T be a Wakamatsu tilting module. A module M is called (n, T )-copure injective (resp. (n, T )-copure flat) if E 1 T (N, M ) = 0 (resp. Γ T 1 (N, M) = 0) for any module N with T -injective dimension at most n (see Definition 2.2). In this paper, it is shown that M is (n, T )-copure injective if and only if M is the kernel of an In(T )-precover f : A → B with A ∈ Prod T . Also, some results on Prod T -syzygies are presented. For instance, it is shown that every nth Prod T -syzygy of every module, generated by T , is (n, T )-copure injective.

Gorenstein σ[T]-injectivity on T-coherent rings

“…Note that for any tilting module M , if M ∈ Gen T , then [3,Theorem 3.11] implies that Gen T = Pres ∞ T and T is a 1-star module (see [16,Definition 3.1]). This shows that any module generated by T has an Add T -resolution, see also [12,Proposition 2.1]. The Add T -resolutions and the relative homological dimension were studied by Nikmehr and Shaveisi in [12].…”

“…This shows that any module generated by T has an Add T -resolution, see also [12,Proposition 2.1]. The Add T -resolutions and the relative homological dimension were studied by Nikmehr and Shaveisi in [12]. The right C-resolution and right C-dimension of modules are defined, similarly.…”

“…is an Add T -resolution of M and δ n * = Hom(δ n , id B ), for every i ≥ 0, see [12] for more details. Let M ∈ Gen T and N be two modules.…”

“…A similar proof to that of [13,Lemma 2.11] shows that E 0 A similar proof to that of [13,Proposition 2.3] shows that the definition of Γ T n (M, B) (respectively, E n T (C, M )) is independent from the choice of left Add Tresolutions. For unexplained concepts and notations, we refer the reader to [1,7,12,14].…”

GORENSTEIN $\pi[T]$-PROJECTIVITY WITH RESPECT TO A TILTING MODULE

“…ProdT -resolution of M and δ n * = Hom(id B , δ n ), for every i ≥ 0. see [6,9] for more details. (8) Let M ∈ CogenT and N be two modules.…”