Frobenius Extensions: Fundamental Concepts

• Frobenius extensions are ring or category extensions defined by symmetric duality, nondegenerate trace maps, and naturally equivalent induction and coinduction functors.
• They provide powerful tools in modular representation theory, categorification, and quantum algebra, enabling decomposition and classification of invariants.
• Applications span link homology, fusion categories, and topology, bridging local subgroup structures with global algebraic insights.

A Frobenius extension is a pivotal concept integrating module theory, homological algebra, representation theory, category theory, quantum algebra, and geometry. It generalizes the notion of Frobenius algebras to ring and category extensions, characterized by symmetric duality properties between induction and coinduction functors, the existence of nondegenerate trace maps, and the preservation of exactness and self-duality across extensions. Frobenius extensions play a central role in modular representation theory, categorification, quantum group theory, and topology, providing both structural insights and practical tools for decomposing, transferring, and classifying invariants.

1. Core Definition and Equivalent Formulations

A ring extension $R \subset S$ is called a Frobenius extension if one of the following equivalent properties holds:

• Module duality: $S$ is finitely generated and projective as an $R$-module and there is an isomorphism of $S$-$R$-bimodules $S \cong \operatorname{Hom}_R(S, R)$ [see (Ren, 2017, Bao et al., 18 Sep 2024)].
• Functorial symmetry: The induction and coinduction functors between module categories, $S \otimes_R -$ and $\operatorname{Hom}_R(S, -)$, are naturally equivalent (Zhao, 2023).
• Bimodule trace: There exists an $R$-$R$-bimodule homomorphism (Frobenius trace) $t: S \to R$ along with elements $\{x_i, y_i\} \subset S$ so that for all $a \in S$, $\sum_i t(x_i a) y_i = a$ and $\sum_i x_i t(a y_i) = a$ (Ren, 2017).

These conditions ensure a perfect duality akin to that in Frobenius algebras and naturally extend to various categorical settings, such as monoidal categories (Bulacu et al., 2013), Hopf algebras (Flake et al., 19 Dec 2024), and category theory (Sánchez et al., 2018).

2. Categorical and Diagrammatic Perspectives

Frobenius extensions admit rich interpretations within categorical frameworks:

• Frobenius Monoidal Functors: An induction functor arising from a Frobenius extension between Hopf algebras is a Frobenius monoidal functor, i.e., it admits both a strong monoidal and an opmonoidal structure that are compatible via a Frobenius condition (Flake et al., 19 Dec 2024).
• Diagrammatic Calculus: In the theory of Soergel bimodules and link homology, multiple compatible Frobenius extensions are encoded in hypercube diagrams. Here, induction and restriction functors correspond to oriented 1-manifolds in the plane, with adjunction units and counits depicted as cups and caps, and natural transformations (such as induction isomorphisms) visualized via colored crossings satisfying categorical coherence (Reidemeister moves) (Elias et al., 2013, Khovanov et al., 2020).
• Monoidal Extensions: In sovereign monoidal categories, Frobenius (and separable) extensions are characterized in terms of compatibility with the Nakayama automorphisms and underlying monoidal structure, with forgetful functors reflecting the separable Frobenius property precisely when the algebra $R$ is Frobenius separable (Bulacu et al., 2013).

3. Homological and Representation-Theoretic Invariants

Frobenius extensions have far-reaching consequences for module-theoretic and homological invariants:

• Gorenstein Projectivity: A module is Gorenstein projective over the extension ring if and only if its restriction is Gorenstein projective over the base, and for separable or left-Gorenstein extensions, the converse also holds (Ren, 2017, Zhibing, 2017). This transfers directly to graded settings and to complexes.
• k-Torsionfreeness: $n$-torsionfreeness (the vanishing of certain $\operatorname{Ext}$-groups of transpose modules) is preserved under Frobenius extensions. In particular, $R$-modules are $k$-torsionfree if and only if their underlying $S$-modules are, enabling the construction of new quasi $k$-Gorenstein rings (Zhao, 2023).
• (Generalized) G-dimensions: Frobenius extensions preserve generalized Gorenstein dimensions, including those relative to Wakamatsu tilting modules (Bao et al., 18 Sep 2024).
• Dominant Dimensions: For Artin algebras and finite-dimensional algebras linked by Frobenius bimodules, flat-dominant dimensions satisfy precise inequalities, and, for split Frobenius extensions, they are equal. Universal enveloping algebras of semisimple Lie algebras and their quantum counterparts are shown to be $QF$-$3$ rings via this framework (Xi, 2019).

4. Grothendieck Groups, Block Theory, and Inverse-Limit Constructions

Puig’s theory establishes the construction of Grothendieck groups for Frobenius categories (often arising from fusion systems over $p$-groups) as inverse limits over chain categories with central $k^*$-extensions folded in:

• Inverse limits over chain categories: The Grothendieck group is defined as an inverse limit of Grothendieck groups of module categories associated to chains of self-centralizing subgroups, capturing both modular and ordinary characteristics and encoding extra central extension (Frobenius extension) data (Puig, 2010).
• Character-theoretic decompositions: The Grothendieck groups admit canonical decompositions indexed by character twists (roots of unity), providing effective methods to determine their ranks and structural invariants.
• Functorial restriction and cohomological vanishing: There exist natural restriction (decomposition) maps associated to centralizers ("general decomposition maps"), and higher cohomology vanishing results ensure Euler–Poincaré formulas for ranks.

This machinery bridges local subgroup data and global block invariants in modular representation theory, leveraging Frobenius extension structures on categories.

5. Twisted and Generalized Frobenius Extensions

Frobenius extension theory extends beyond the classical module and ring context:

• Twisted Frobenius extensions: For graded superrings, one introduces twists by automorphisms and degree shifts so that the induction functor is right adjoint to a suitably twisted restriction functor. These generalize untwisted Frobenius extensions and are essential in categorification, especially in Heisenberg categorification or for nilcoxeter algebra extensions (Pike et al., 2015).
• Frobenius morphisms of groupoids: Induction and coinduction functors between categories of representations of groupoids are naturally isomorphic (a "Frobenius morphism") if and only if certain fibers of associated bisets have finitely many orbits. This identifies the groupoid-level generalization of Frobenius extensions for algebras with enough orthogonal idempotents (Sánchez et al., 2018).
• Relative torsionfreeness and Wakamatsu modules: Relative $n$-torsionfreeness and generalized G-dimension with respect to Wakamatsu tilting modules are stable under centrally projective Frobenius extensions, and so is the Frobenius property of induced endomorphism rings (Bao et al., 18 Sep 2024).

6. Applications: Geometry, Topology, and Categorification

Frobenius extensions are fundamental in diverse applications:

• Fusion categories and modular invariants: Étale algebras (commutative separable algebras) in abelian tensor categories are automatically Frobenius algebras; induction functors along Frobenius extensions in Hopf algebras yield Frobenius monoidal functors, critical for the classification and construction of modular invariants and fusion categories (Flake et al., 19 Dec 2024).
• Link homology and TQFT: In $\mathfrak{sl}_2$ link homology and its equivariant variants, the algebraic operations (multiplication, comultiplication, involution) underlying TQFTs are governed by the structure of Frobenius extensions. Sufficiently non-degenerate base rings (inverting the discriminant) guarantee well-behaved state spaces and diagrammatic relations (neck-cutting, seam crossing) (Khovanov et al., 2020).
• Exotic nilCoxeter algebras: In the context of complex reflection groups $G(m,m,n)$, Frobenius extension theory is crucial for understanding the structure of the polynomial ring as a free Frobenius extension of its invariants, the presentation and graded dimension of exotic nilCoxeter algebras, and the explicit construction of Frobenius trace operators via Demazure divided difference operators. The emergence of roundabout relations at roots of unity controls the graded dimension and structure of these algebras (Elias et al., 24 Dec 2024).

7. Structures, Transfer, and Obstruction Results

Key structural theorems reveal transfer and obstruction principles:

• Transfer of Frobenius property via gradings and filtrations: A free-filtered extension is Frobenius if and only if the associated free-graded extension is Frobenius. The construction and analysis of Rees algebras mediate this transfer and provide effective criteria for examples and counterexamples among quantum and classical algebras (Launois et al., 2017).
• Invariance and failure of inverses: Properties such as Gorenstein projectivity, torsionfreeness, quasi-$k$-Gorenstein-ness, and G-dimension often transfer upward from the base ring to the extension via a Frobenius extension, but the converse generally fails, as demonstrated with extensions involving direct product rings (Zhao, 2023).
• Preservation of block-theoretic structure: Under separable Frobenius extensions, crucial invariants—CM-finiteness, CM-freeness, representation dimension—are preserved across extension and base algebras, enabling reductions and transference in representation theory (Zhibing, 2017).

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Frobenius extensions, in all their variants, provide a unifying algebraic and categorical infrastructure that supports the transfer, decomposition, and detailed analysis of module-theoretic, categorical, and geometric invariants. Their applications in block theory, categorification, quantum algebra, geometry, and topology stem from their robust duality properties, functorial adjunctions, and the existence of trace maps, offering an indispensable framework across modern algebra and representation theory.