Pure injectivity of n-cotilting modules: the Pr�fer and the countable case

We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.

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Cosilting complexes and AIR-cotilting modules

Journal of Algebra ◽

10.1016/j.jalgebra.2017.07.022 ◽

2017 ◽

Vol 491 ◽

pp. 1-31 ◽

Cited By ~ 4

Author(s):

Peiyu Zhang ◽

Jiaqun Wei

Keyword(s):

Cotilting Modules

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A Note on Cotilting Modules and Generalized Morita Duality

Rings, Modules, Algebras, and Abelian Groups - Lecture Notes in Pure and Applied Mathematics ◽

10.1201/9780824750817.ch7 ◽

2004 ◽

Author(s):

Kent Fuller ◽

Riccardo Colpi ◽

Robert Colby

Keyword(s):

Cotilting Modules

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A Theorem on Pure Submodules

Canadian Journal of Mathematics ◽

10.4153/cjm-1960-042-5 ◽

1960 ◽

Vol 12 ◽

pp. 483-487

Author(s):

George Kolettis

Keyword(s):

Free Abelian Group ◽

Chain Condition ◽

Affirmative Answer ◽

Principal Ideal Ring ◽

Torsion Free Abelian Group ◽

Direct Sum ◽

Rank One ◽

Pure Submodules ◽

Torsion Free ◽

Countable Case

In (1) Baer studied the following problem: If a torsion-free abelian group G is a direct sum of groups of rank one, is every direct summand of G also a direct sum of groups of rank one? For groups satisfying a certain chain condition, Baer gave a solution. Kulikov, in (3), supplied an affirmative answer, assuming only that G is countable. In a recent paper (2), Kaplansky settles the issue by reducing the general case to the countable case where Kulikov's solution is applicable. As usual, the result extends to modules over a principal ideal ring R (commutative with unit, no divisors of zero, every ideal principal).The object of this paper is to carry out a similar investigation for pure submodules, a somewhat larger class of submodules than the class of direct summands. We ask: if the torsion-free i?-module M is a direct sum of modules of rank one, is every pure submodule N of M also a direct sum of modules of rank one? Unlike the situation for direct summands, here the answer depends heavily on the ring R.

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