Graded almost Noetherian rings and applications to coalgebras
type: Item
instance of: scholarly article
title: Graded almost Noetherian rings and applications to coalgebras
zbMATH Open document ID: 1019.16031
DOI: 10.1016/S0021-8693(02)00099-6
author: Constantin Năstăsescu
author: Juan Cuadra
author: Fred Van Oystaeyen
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: The convolution or dual algebra \(C^*=\Hom_k(C,k)\) of a \(k\)-coalgebra \(C\) (\(k\) is a commutative field) can be used to investigate the coalgebra from a topological point of view (using the weak \(*\)-topology on \(C^*\)) or a categorical one (using the properties of the embedding of the category \({\mathcal M}^C\) of all right comodules in the category \(_{C^*}{\mathcal M}\) of all left \(C^*\)-modules). The link between both settings is the left linear topology \(\mathcal F\) consisting of those closed left ideals \(I\) of \(C^*\) which are cofinite, i.e., \(C^*/I\) is a finite-dimensional \(k\)-vector space. Thus \({\mathcal M}^C\) is identified with the full subcategory \(\text{Rat}(_{C^*}{\mathcal M})\) of \(_{C^*}{\mathcal M}\) consisting of those left \(C^*\)-modules \(M\) such that the annihilator left ideal of every element of \(M\) belongs to \(\mathcal F\). Since \textit{D. E. Radford} proved [in J. Algebra 26, 512-535 (1973; Zbl 0272.16012)] that if every left ideal in \(\mathcal F\) is finitely generated (\(C\) is then said to be \(\mathcal F\)-Noetherian), then \(\text{Rat}(_{C^*}{\mathcal M})\) is closed under extensions (\(C\) is said to have a torsion rat functor), this type of conditions gained interest. This paper introduces techniques of graded algebras in this framework. So, from the coradical filtration on \(C\), a filtration is canonically considered on \(C^*\), which leads to its associated graded algebra \(\text{gr}(C^*)\). The authors prove (Theorem 2.8) that for an almost connected coalgebra \(C\) (that is, with finite dimensional coradical), several conditions are equivalent, among them: \(C\) has a torsion rat functor; each term of the coradical filtration is finite-dimensonal; \(C^*\) is left almost Noetherian (that is, cofinite left ideals are finitely generated). This result completes those by \textit{R. G. Heyneman} and \textit{D. E. Radford} [in J. Algebra 28, 215-246 (1974; Zbl 0291.16008)] and by \textit{T. Shudo} [in Hiroshi…
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 1868023
zbMATH Keywords: almost Noetherian graded algebras
zbMATH Keywords: coalgebras with torsion rat functors
zbMATH Keywords: Noetherian coalgebras
zbMATH Keywords: categories of comodules
zbMATH Keywords: strongly coreflexive coalgebras
zbMATH Keywords: Artin-Rees property
zbMATH Keywords: left linear topologies
zbMATH Keywords: coradical filtrations
reviewed by: José Gómez-Torrecillas
MaRDI profile type: Publication
cites work: Q5563439
cites work: IDEMPOTENTS AND MORITA-TAKEUCHI THEORY
cites work: Des catégories abéliennes
cites work: Locally finite representations of algebras∗
cites work: Locally finite representations
cites work: Reflexivity and coalgebras of finite type
cites work: Semiperfect coalgebras
cites work: Morita's Theorem for Coalgebras
cites work: Products of torsion theories and applications to coalgebras
cites work: Quasi-co-Frobenius coalgebras
cites work: Q3957087
cites work: Hereditary coalgebras
cites work: Coreflexive coalgebras
cites work: On the Structure of Ideals of the Dual Algebra of a Coalgebra
cites work: A note on coalgebras and rational modules
cites work: Q4049223
cites work: Q5590089
full work available at URL: s0021-8693(02)00099-6
OpenAlex ID: W2009925934
description: scientific article; zbMATH DE number 1868023