Graded almost noetherian rings and applications to coalgebras

Cuadra, Juan; Năstăsescu, C.; Oystaeyen, Freddy Van

doi:10.1016/s0021-8693(02)00099-6

Cited by 17 publications

(25 citation statements)

References 16 publications

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“…By hypothesis, there is a surjective C * -module map f : (1) c (2) ) for all c ∈ C. It is easy to check that g * = f . Hence, g is a…”

Section: Proof (I) (Ii) It Is Just To Take Into Account That C E(i)mentioning

confidence: 92%

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Extensions of rational modules

Cuadra

2003

International Journal of Mathematics and Mathematical Sciences

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For a coalgebraC, the rational functorRat (−):ℳC∗→ℳC∗is a left exact preradical whose associated linear topology is the familyℱC, consisting of all closed and cofinite right ideals ofC∗. It was proved by Radford (1973) that ifCis rightℱ-Noetherian (which means that everyI∈ℱCis finitely generated), thenRat (−)is a radical. We show that the converse follows ifC1, the second term of the coradical filtration, is rightℱ-Noetherian. This is a consequence of our main result onℱ-Noetherian coalgebras which states that the following assertions are equivalent: (i)Cis rightℱ-Noetherian; (ii)Cnis rightℱ-Noetherian for alln∈ℕ; and (iii)ℱCis closed under products andC1is rightℱ-Noetherian. New examples of rightℱ-Noetherian coalgebras are provided.

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“…By hypothesis, there is a surjective C * -module map f : (1) c (2) ) for all c ∈ C. It is easy to check that g * = f . Hence, g is a…”

Section: Proof (I) (Ii) It Is Just To Take Into Account That C E(i)mentioning

confidence: 92%

“…is a torsion class (closed under extensions) has been studied in [5,6,8,10] and, recently, in [2]. In this paper, we continue the study of this problem and relate it to a finiteness condition on Ᏺ C , that every I ∈ Ᏺ C is finitely generated.…”

mentioning

confidence: 91%

Extensions of rational modules

Cuadra

2003

International Journal of Mathematics and Mathematical Sciences

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“…We study this problem for a class of coalgebras which we introduce by using a strong semilattice of semigroups. In particular, we give a negative answer to the conjecture raised in [3,Remark 2.13]: There exists a coalgebra with the rational functor a radical, but the filter associated to this radical does not have a basis of finitely generated left ideals.…”

Section: Introductionmentioning

confidence: 89%

“…In general, it is easy to see that if L has a cofinal subset of finitely generated left ideals, then L is a Gabriel filter. In [3], it was asked if the converse is true. In Section 3 we answer this question in the negative.…”

Section: Preliminariesmentioning

confidence: 99%

“…Moreover if C is locally projective over K, then there exists a functor, the rational functor, from the category of left C * -modules to the category of right C-comodules. Several authors (see [3,6] and their references) have considered the problem of when this functor is a radical functor. We study this problem for a class of coalgebras which we introduce by using a strong semilattice of semigroups.…”

Section: Introductionmentioning

confidence: 99%

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Coalgebras with a radical rational functor

Teply

Torrecillas

2005

Journal of Algebra

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Comodules and Contramodules Over Graded Rings

Positselski

2021

Relative Nonhomogeneous Koszul Duality

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This paper is a follow-up to [38]. We consider two algebraic settings of comodules over a coring and contramodules over a topological ring with a countable base of two-sided ideals. These correspond to two (noncommutative) algebraic geometry settings of certain kind of stacks and ind-affine ind-schemes. In the context of a coring C over a noncommutative ring A such that C is a flat right A-module, we show that all A-flat left C-comodules are < ℵ 1 -direct limits of A-countably presented A-flat C-comodules. In the context of a complete, separated topological ring R with a countable base of neighborhoods of zero consisting of two-sided ideals, we prove that all flat R-contramodules are < ℵ 1 -direct limits of countably presented flat R-contramodules. We also describe short exact sequences and pure acyclic complexes of A-flat C-comodules and flat R-contramodules as certain < ℵ 1 -direct limits. Applications to cotorsion periodicity in the respective settings are discussed. In particular, in any acyclic complex of cotorsion R-contramodules, all the contramodules of cocycles are cotorsion.

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Graded almost noetherian rings and applications to coalgebras

Cited by 17 publications

References 16 publications

Extensions of rational modules

Extensions of rational modules

Coalgebras with a radical rational functor

Comodules and Contramodules Over Graded Rings

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