n-Star modules and n-tilting modules

Abstract

We give some characterizations of (not necessarily selfsmall) n-star modules and prove that (not necessarily finitely generated) n-tilting modules are precisely (not necessarily selfsmall) n-star modules n-presenting all the injectives.  2004 Elsevier Inc. All rights reserved.

Relative T-injective modules and relative T-flat modules

“…In this section, we give basic notions and results and we recall some relevant background in tilting theory from [2,4,8,9]. First let us recall the following definition of (not necessarily finitely generated) tilting modules (see [2]).…”

“…T is a tilting module, then it is a 1-star module by [9,Theorem 4.3], and hence it is 1-quasi-projective by […”

“…Many subjects in homological algebra are based on the properties of tilting and cotilting modules (see [1,2,4,8,9] for instance). Throughout this paper, R is an associative ring with non-zero identity, all modules are unitary R-modules and T is a fixed R-module.…”

n-Star modules over ring extensions

“…In particular, it was proved that quasi-progenerators are just classical star modules generating all of their submodules while classical tilting modules are just classical star modules generating all the injectives [9,10]. The theory was later extended to n-star modules [23,26]. Moreover, it was shown that n-tilting modules in the sense of Angeleri-Hügel and Coelho [2] (respectively, in the sense of Miyashita [19]) are just n-star modules which n-present all the injectives (respectively, which admit finitely generated projective resolutions and n-present all the injectives) [4,23,26].…”

“…For convenience we also denote by Pres 0 A U the category of all left A-modules. Recall that an A-module A U is said to be n-quasi-projective provided every exact sequence of left A-modules 0 → L → U → M → 0 stays exact under the functor Hom A (U, −), where A L ∈ Pres n−1 A U and A U ∈ Add A U (see for instance [23,Definition 2.1]). An A-module A U is called an n-star module if A U is (n + 1)-quasi-projective and Pres n A U = Pres n+1 A U .…”