Extensions of rational modules

Abstract

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.

Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors

“…When is the essential image of Υ closed under extensions in Mod-C * ? There is a vast body of literature on this topic, including the papers [32,34,13,4,3,36,8]. In this paper we discuss further questions going in this direction, under an additional assumption.…”

Comodules and Contramodules Over Graded Rings

“…More generally, given a dense subring R ⊂ C * , is the essential image of the inclusion functor C-Comod −→ R-Mod closed under extensions? There is a vast body of literature on this question, including such papers as [15,16,5,2,1,18,4]. In particular, for a conilpotent coalgebra C (called "pointed irreducible" in the traditional terminology of [17,6]), the full subcategory C-Comod is closed under extensions in C * -Mod if and only if the full subcategory Comod-C is closed under extensions in Mod-C * , and if and only if the coalgebra C is finitely cogenerated [ In Section 6 of this paper, we show that the full subcategory Comod-C E is closed under extensions in Mod-E for any left strictly locally finite k-linear category E. This observation is not really new, but it complements the main results of this paper nicely; so we include it for the benefit of the reader.…”

Complete Path Algebras and Rational Modules

“…One problem of a particular interest is deciding when is this subcategory of rational modules closed under extensions. This has been considered before by many authors [4,6,11,14,10,19,20,26,28,33]. In general, given an abelian or Grothendieck category A and a closed subcategory B of A, one can consider the trace functor T with respect to B, which is defined as T (M ) =the sum of all subobjects of M which belong to B.…”

“…Since all the classes of coalgebras with torsion rational functor seemed to be F-Noetherian, the authors asked in [6] if this is perhaps an equivalent characterization; this motivated the research of [33], where a counterexample was produced. However, in certain situations this property can be characterized equivalently by F-Noetherianity; for example, if the coradical C 0 is finite dimensional, then C has a left rational torsion functor if and only if C * is left F-Noetherian, and equivalently, all the terms of the coradical filtration are finite dimensional (see [4,6,11]). In this situation, the notion becomes left-right symmetric.…”