The Generating Condition for Coalgebras

For a ring $R$, the properties of being (left) selfinjective or being cogenerator for the left $R$-modules do not imply one another, and the two combined give rise to the important notion of PF-rings. For a coalgebra $C$, (left) self-projectivity implies that $C$ is generator for right comodules and the coalgebras with this property were called right quasi-co-Frobenius; however, whether the converse implication is true is an open question. We provide an extensive study of this problem. We show that this implication does not hold, by giving a large class of examples of coalgebras having the “generating property”. In fact, we show that any coalgebra $C$ can be embedded in a coalgebra $C_\infty$ that generates its right comodules, and if $C$ is local over an algebraically closed field, then $C_\infty$ can be chosen local as well. We also give some general conditions under which the implication “$C$-projective (left) $\Rightarrow C$ generator for right comodules” does work, and such conditions are when $C$ is right semiperfect or when $C$ has finite coradical filtration.

💡 Research Summary

The paper investigates the relationship between two fundamental properties of coalgebras that are dual to classical ring-theoretic notions: self‑projectivity (the coalgebra being projective as a left comodule) and the generating condition (the coalgebra generating all right comodules). In ring theory, self‑injectivity and cogeneration are independent, and their conjunction defines PF‑rings. Dually, a left self‑projective coalgebra is known to be a generator for right comodules; such coalgebras are called right quasi‑co‑Frobenius (right QcF). The open problem addressed here is whether the converse holds: does the right generating condition imply right QcF (equivalently, left self‑projectivity)?

The authors first establish that the converse fails in general. They construct, for any coalgebra (C), an enlarged coalgebra (C_{\infty}) that contains (C) as a subcoalgebra and has the right generating property. The construction proceeds by repeatedly adjoining finite‑dimensional right comodules as quotients, ensuring that every finite‑dimensional right comodule of (C) becomes a quotient of (C_{\infty}). Consequently, (C_{\infty}) generates all right comodules, yet it need not be left projective; in particular it is not right QcF unless additional hypotheses are imposed. Moreover, when (C) is local over an algebraically closed field, the authors can choose (C_{\infty}) to be local as well, showing that even local coalgebras with a one‑dimensional coradical (almost connected) do not automatically satisfy the QcF condition under the generating hypothesis.

The second part of the paper identifies natural conditions under which the generating condition does force the QcF property. Three main scenarios are examined:

1. Right semiperfect coalgebras – where every simple right comodule has a finite‑dimensional injective envelope. In this setting the authors prove that the right generating condition is equivalent to being right QcF.

2. Coalgebras with finite coradical filtration – i.e., the coradical series terminates after finitely many steps. Using Loewy series techniques, they show that if the coalgebra generates its right comodules, then each injective envelope has finite Loewy length, which forces the coalgebra to be right semiperfect and hence right QcF.

3. Coalgebras whose indecomposable injective left components have finite Loewy length – this condition is shown to be equivalent to right semiperfectness, and therefore also guarantees the QcF property.

Key technical tools include the analysis of Loewy series for comodules, the relationship between the Jacobson radical of the dual algebra (C^{*}) and the coradical filtration of (C), and a careful study of finite‑dimensional subcomodules of injective envelopes. Proposition 3.1 establishes that left QcF is equivalent to left semiperfectness together with the left generating condition. Propositions 3.2 and 3.3 connect finite Loewy length of injective envelopes with the existence of dualities between left and right injective envelopes, thereby linking the generating condition to QcF under the above structural assumptions.

The main theorem (implicitly presented) can be summarized as follows: Every coalgebra embeds in a coalgebra that satisfies the right generating condition, but without extra finiteness or semiperfectness hypotheses this larger coalgebra need not be QcF. Conversely, when the coalgebra is right semiperfect or has a finite coradical filtration, the right generating condition is equivalent to being right QcF.

In conclusion, the paper resolves the long‑standing open question by providing both a negative answer in full generality and a positive answer under natural finiteness conditions. It clarifies the precise interplay between generation and projectivity in the coalgebra setting, and it opens avenues for further exploration of coalgebras that lie between the two extremes.