Separable functors and firm modules

Technical Summary: Separable Functors and Firm Modules

Problem Statement
The paper addresses the extension of classical ring-theoretic concepts—specifically separable ring extensions and separable functors—from the unital setting to the context of nonunital rings. While separable extensions are well-understood for unital rings (where they provide a categorical explanation for Maschke's theorem and the inheritance of semisimplicity), the literature on nonunital rings is sparse. Existing results often rely on restrictive assumptions, such as rings having "enough idempotents." The central problem is to determine whether the machinery of separable functors can be systematically extended to appropriate categories of modules over general nonunital rings, specifically utilizing the framework of firm modules introduced by Quillen.

Methodology
The author, Patrik Lundström, adopts a categorical approach grounded in the theory of firm modules.

1. Categorical Framework: The paper begins by establishing a systematic account of separable functors between arbitrary categories. It proves that separable functors reflect key structural properties, including limits, colimits, monomorphisms, epimorphisms, sections, retractions, and isomorphisms. Crucially, it demonstrates that separability reflects properties of objects such as subobject/quotient simplicity, semisimplicity, and the existence of zero objects.
2. Module Categories: The study focuses on nonunital associative rings. The author distinguishes between various module types: unital, unitary, s-unital, and firm. A left $A$-module $M$ is firm if the multiplication map $\mu_{A,M}: A \otimes_A M \to M$ is an isomorphism. The paper clarifies that for rings that are left s-unital (where every element $m$ satisfies $m \in Am$), the categories of unitary, s-unital, and firm modules coincide.
3. Functorial Analysis: The core methodology involves analyzing the restriction functor $\text{Res}_f: A\text{FMod} \to B\text{FMod}$ and the induction functor $\text{Ind}_f: B\text{FMod} \to A\text{FMod}$ associated with a ring homomorphism $f: B \to A$. The author investigates the conditions under which these functors are separable, linking the separability of the functor to the separability of the ring extension $A/B$.

Key Contributions and Results
The paper establishes several nonunital analogues of classical theorems:

• Theorem 3 (Main Equivalence): Let $A$ be a firm ring and $f: B \to A$ a left firm ring homomorphism. The extension $A/B$ is separable if and only if the restriction functor $\text{Res}_f: A\text{FMod} \to B\text{FMod}$ is separable. This generalizes a classical result by Nastasescu, Van den Bergh, and Van Oystaeyen to the firm module setting.
• Theorem 4 (Semisimplicity Inheritance): If $A$ and $B$ are left s-unital rings, $B$ is left semisimple, and $f: B \to A$ is a left s-unital homomorphism such that $A/B$ is separable, then $A$ is left semisimple. This result relies on the fact that separable functors reflect semisimplicity in the category of firm modules.
• Theorem 5 (Separability of Group Rings): Let $G$ be a finite group and $B$ a ring with local units satisfying $|G|B = B$. The group ring $A = B[G]$ forms a separable extension over $B$. The proof constructs an explicit $A$-bimodule splitting for the multiplication map using local units.
• Theorem 6 (Locally Unital Maschke's Theorem): Combining Theorems 4 and 5, the paper proves that if $B$ is a semisimple ring with local units satisfying $|G|B = B$, then the group ring $A = B[G]$ is semisimple. This provides a locally unital version of Maschke's theorem.
• Theorem 45 (Hereditary Property): The paper extends the results to hereditary rings, showing that under similar separability and preservation conditions, the hereditary property is inherited by the extension.

Significance and Claims
The paper claims to provide a "locally unital version of Maschke's theorem" and to extend classical semisimplicity results to a broad nonunital setting. By utilizing firm modules, the author generalizes previous work that required rings to have "enough idempotents." The significance lies in demonstrating that the conceptual link between separable ring extensions and the inheritance of semisimplicity holds even in the absence of a global unit, provided the rings and modules satisfy the firmness condition.

The author notes that while few results exist on separability for nonunital rings, this work bridges that gap by adapting the functorial machinery of separability. The results are presented as direct analogues of classical theorems, recovering known results (like Maschke's theorem) as special cases while applying them to rings that arise from infinite direct sums or local constructions, which are common in nonunital algebra but lack global units. The paper does not propose new experimental applications but rather solidifies the theoretical foundation for studying semisimple structures in nonunital contexts.