The Dickson Subcategory Splitting Conjecture for Pseudocompact Algebras

Let $A$ be a pseudocompact (or profinite) algebra, so $A=C^*$ where $C$ is a coalgebra. We show that the if the semiartinian part (the “Dickson” part) of every $A$-module $M$ splits off in $M$, then $A$ is semiartinian, also giving a positive answer in the case of algebras arising as dual of coalgebras (pseudocompact algebras), to a well known conjecture of Faith.

💡 Research Summary

The paper addresses a long‑standing conjecture in ring theory, often attributed to Faith, which asks whether a ring whose Dickson subcategory (the smallest localizing subcategory containing all simple modules, equivalently the class of semi‑artinian modules) splits off in every left module must itself be semi‑artinian. While counter‑examples are known in full generality, the authors prove the conjecture for the important class of pseudocompact (or profinite) algebras, i.e. algebras of the form (A=C^{*}) where (C) is a coalgebra over a field (k).

The paper proceeds in several stages:

1. Preliminaries on torsion theory and the Dickson subcategory.
   The authors recall that a closed subcategory (\mathcal{A}) of a Grothendieck category (\mathcal{C}) yields a preradical (t) (the torsion functor) given by the sum of all subobjects belonging to (\mathcal{A}). The Dickson subcategory is the smallest localizing subcategory containing all simples; its objects are precisely the semi‑artinian modules. The “splitting property” means that for every left (A)-module (M) the torsion submodule (t(M)) is a direct summand of (M).

2. Translation to coalgebra language.
   For a coalgebra (C) with dual algebra (A=C^{}), the category of left pseudocompact (A)-modules is dual to the category of left (C)-comodules. The authors use the standard orthogonal correspondence (X^{\perp}={f\in C^{}\mid f|_{X}=0}) and ((I)^{\perp}) for subspaces/ideals, which identifies left coideals of (C) with left ideals of (A). This duality allows them to transport module‑theoretic statements into comodule language.

3. First structural results.
   Let (\mathcal{S}) be a set of representatives of simple left (C)-comodules and (A_{S}=C^{*}{S}) the corresponding simple subalgebras of (A/J) (where (J) is the Jacobson radical). The authors prove that the sum (\Sigma=\bigoplus{S\in\mathcal{S}}A_{S}) coincides with the Dickson torsion of the semisimple algebra (A/J) and is its essential socle. From the splitting hypothesis applied to the cyclic module (A/J) they deduce that (\mathcal{S}) must be finite, hence the coradical (C_{0}) is finite‑dimensional.

4. Stability under subcoalgebras.
   If (C) enjoys the Dickson splitting property, then any subcoalgebra (D\subseteq C) also has it (Proposition 1.3). The proof uses the exact sequence (0\to D^{\perp}\to C^{}\to D^{}\to0) and shows that the semi‑artinian part of a (D^{})-module, viewed as a (C^{})-module, remains a direct summand.

5. Localization technique.
   The authors develop a localization framework using idempotents (e\in A). For an idempotent (e) they consider the exact functor (T_{e}=eA\otimes_{A}-) and its right adjoint (G_{e}=\operatorname{Hom}{eAe}(eA,-)). Several technical lemmas (Propositions 1.5–1.7) establish that essential socles are preserved under this adjunction. The main localization theorem (Theorem 1.8) states: if (A=\bigoplus{i\in F}Ae_{i}) with orthogonal idempotents (e_{i}) and each localized ring (e_{i}Ae_{i}) is left semi‑artinian, then (A) itself is left semi‑artinian. This reduces the global problem to a finite family of local rings.

6. Domain case.
   When (C^{}) is a (local) domain, Lemma 2.2 shows that every non‑zero comodule endomorphism of (C) is surjective. Using this, the authors prove that the coradical (C_{0}) must be simple, which forces (C) to have finite Loewy length; consequently (C^{}) is a division algebra and therefore semi‑artinian.

7. General case.
   Combining the previous steps, the authors first prove that a coalgebra with the Dickson splitting property must be “almost connected”, i.e. its coradical is finite‑dimensional (Corollary 2.5). Then they apply the localization theorem to the decomposition of (A) into a finite direct sum of local components (obtained via the finite set (\mathcal{S}) from step 3). Each component satisfies the splitting hypothesis, hence is semi‑artinian; Theorem 1.8 then yields that the whole algebra (A=C^{}) is semi‑artinian. As a corollary, any coalgebra whose rational submodule of every left (C^{})-module splits off must be finite‑dimensional, recovering results from earlier works