Graded almost Noetherian rings and applications to coalgebras
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 Hiroshima Math. J. 6, 297-304 (1976; Zbl 0332.16013)]. The proof of Theorem 2.8 heavily rests upon the results given in Section 1 for graded almost Noetherian algebras, which might be of independent interest. Some other interesting results are Proposition 2.5 (left strongly coreflexive coalgebras are precisely coreflexive and \(\mathcal F\)-Noetherian coalgebras), or Theorem 2.12 (the coalgebra is right semiperfect if and only if \(\mathcal F\) has a basis of principal left ideals generated by an idempotent). This last result has been recently used by the reviewer, \textit{C. Manu} and \textit{C. Năstăsescu} [in Commun. Algebra 31, No. 10, 5169-5177 (2003)] to characterize quasi-co-Frobenius coalgebras as the \(\mathcal F\)-Noetherian coalgebras such that \(C^*\) is a self-injective ring. The paper is completed with a study of some properties of coalgebras such that \(C^*\) is Noetherian, including Artin-Rees property, stability of the subcategory of rational \(C^*\)-modules, or the rational character of the Poincaré series associated to the coradical filtration in the co-commutative case.
scientific article; zbMATH DE number 1868023