Symmetrical Separable Equivalences

• Symmetrical separable equivalences are defined as two-sided categorical equivalences induced by bimodules and exact functors that split identities under projectivity conditions.
• They generalize classical notions like Morita equivalence by using direct summand conditions to transfer homological and representation-theoretic invariants.
• These equivalences facilitate rigorous analysis in representation theory and derived settings by preserving key invariants such as Gorenstein categories and tilting modules.

Symmetrical separable equivalences denote a class of categorical and algebraic equivalence relations between rings, algebras, or categories, achieved via bimodules or exact functors that satisfy both projectivity and splitting conditions, typically augmented with adjunction (Frobenius) properties. The symmetry requires the equivalence operations to be two-sided and mutually adjoint, thus allowing robust transfer of homological, representation-theoretic, and categorical structures. This framework encompasses and generalizes Morita equivalence, stable equivalence of Morita type, and separable equivalence, but is fundamentally weaker than Morita equivalence, emphasizing direct summand conditions instead of isomorphism. Throughout contemporary algebra, representation theory, and homological algebra, symmetrical separable equivalences facilitate the paper of invariants, module categories, tilting theory, Gorenstein categories, and derived structures, ensuring preservation of key properties and enabling constructions between extended families of algebras and their endomorphism rings.

1. Definitions and Bimodule Constructions

Symmetrical separable equivalence between Artin algebras (or more generally, rings) Λ and Γ is achieved via pairs of bimodules $_\Lambda M_\Gamma$ and $_\Gamma N_\Lambda$, each projective as left and right modules. The central conditions are the existence of bimodule isomorphisms: $$M \otimes_\Gamma N \simeq \Lambda \oplus X, \qquad N \otimes_\Lambda M \simeq \Gamma \oplus Y,$$ for some bimodules $X$ and $Y$, together with the requirement that the associated tensor functors $T_N = N \otimes_\Lambda -$ and $T_M = M \otimes_\Gamma -$ are exact, send projective modules to projectives, and form adjoint pairs in both directions. In this symmetrical context, both algebras serve as direct summands of the appropriate bimodule tensors, but not necessarily as direct summands in a way that is Morita equivalent. The functors also split the identity: $F \circ G \cong \operatorname{Id}_A \oplus ...$ and $G \circ F \cong \operatorname{Id}_B \oplus ...$.

This paradigm extends to category equivalences, where exact additive functors $F: \mathcal{A} \to \mathcal{B}$, $G: \mathcal{B} \to \mathcal{A}$ are required to preserve projectives and split the identity as direct summands when composed, with adjunctions representing additional symmetry (Peacock, 2017).

2. Invariance of Homological and Representation-theoretic Properties

A central property of symmetrical separable equivalence is preservation of extensive homological invariants, such as the (one-sided) Gorenstein categories, Wakamatsu tilting modules, Frobenius-finite type, rigidity dimension, and representation type. For example, if $C$ is a Wakamatsu tilting $\Lambda$-module, then $N \otimes_\Lambda C$ is a Wakamatsu tilting $\Gamma$-module; similarly, Gorenstein categories $\mathcal{G}(C)$ and relative Gorenstein projective/injective objects are preserved under the transfer induced by equivalence functors (Zhao et al., 29 Jun 2025). Rigidity dimension, representation dimension, Auslander-type conditions, the (strong) Nakayama conjecture, and the Auslander-Gorenstein conjecture remain invariant under symmetrical separable equivalence provided suitable centrally projective and additive conditions are met (Sun et al., 20 Aug 2025).

Numerical invariants such as the Krull dimension of cohomology, the Rouquier dimension of stable categories, and representation dimension demonstrate sharp inequalities and equalities in symmetric tensor categories, e.g.,

$$\operatorname{Kdim}\,\mathscr{C} = \operatorname{dim}(\operatorname{stab}\,\mathscr{C}) + 1 = \operatorname{repdim}\,\mathscr{C} - 1$$

for finite symmetric tensor categories over an algebraically closed field of characteristic zero (Bergh, 2021).

3. Endomorphism Algebras and Lifting Equivalences

Symmetrical separable equivalence extends naturally to endomorphism algebras of suitable modules. If $N \otimes_\Lambda U \in \operatorname{add}(V)$ and $M \otimes_\Gamma V \in \operatorname{add}(U)$ for modules $U$, $V$ over $\Lambda$, $\Gamma$ respectively, then a symmetrical separable equivalence is induced between $E_\Lambda(U)$ and $E_\Gamma(V)$ (Sun et al., 20 Aug 2025). Methods based on Yoneda’s lemma and functorial isomorphisms are used to construct module category equivalences for these endomorphism algebras.

Applications include the Auslander algebra of a representation-finite algebra, Frobenius parts of algebras (preserving Frobenius-finite type), and tilted algebras. For instance, given a stable equivalence of adjoint type and an $m$-tilting module $T$, one obtains a new symmetrical separable equivalence between $\operatorname{End}_\Lambda(T)$ and $\operatorname{End}_\Gamma(N\otimes_\Lambda T)$, thus showing the class of tilted algebras is preserved under these equivalences (Sun et al., 20 Aug 2025).

4. Frobenius Bimodules and Structural Characterization

Symmetrical separable equivalence is intimately connected to the theory of Frobenius bimodules. If $A$ and $B$ are symmetric separably equivalent rings, the linking bimodule $_A P_B$ is Frobenius, characterized by isomorphisms: $$B \operatorname{Hom}(P_B, B_B) \cong \operatorname{Hom}(_A P, {_A}A)$$ and the ring extension $A \to \operatorname{End}(P_B)$ is split, separable, Frobenius (Kadison, 2017). This splitting is realized by evaluation maps (e.g., $e: P \otimes_B \operatorname{Hom}(_A P, A) \to A$) and associated multiplicative structures on $P \otimes_B Q$. In symmetric algebras, any finite projective bimodule is automatically Frobenius (possibly twisted by the Nakayama automorphism), thus for symmetric algebras, separable equivalence and symmetrical separable equivalence coincide.

5. Symmetry and Adjunction in Equivalence Constructions

The symmetry of separable equivalence is encoded via adjunctions: the functor pairs associated with bimodules (or, categorically, with exact functors) are adjoint in both directions (i.e., $(F,G)$ and $(G,F)$ are adjoint pairs), leading to Frobenius contexts. This property ensures that all transferred homological or categorical invariants are preserved on both sides. In symmetric contexts, this also gives rise to the equivalence between separable equivalence and symmetrical separable equivalence due to the self-duality of the regular bimodule (Peacock, 2017, Kadison, 2017).

6. Extensions to Homotopical Algebra and Derived Settings

In the context of symmetric monoidal stable $\infty$-categories, separable objects are defined via the existence of a separability idempotent (splitting of multiplication), and the concept of "ind-separability" is introduced for colimits of separable algebras (Ramzi, 2023). Descent phenomena in Hochschild and topological Hochschild homology are tied to absolute separability: if $B$ is absolutely separable over $A$, then the relative Hochschild homology $HH_C(A;B)$ is linear in $B$ and equivalent to $HH_C(B)$; consequently, descent along Galois extensions is governed by these separation properties. For example, in Galois extensions, the "norm" map splits, and THH descent holds along such extensions. Ind-separability elucidates the uniqueness of higher structures, as in the Goerss–Hopkins–Miller theorem for Morava $E$-theory.

Center and Azumaya algebra results reduce to Auslander–Goldman theory; for instance, under finiteness assumptions, the center $Z(A)$ of a separable algebra $A$ is also separable, and separable central algebras are Azumaya when the unit splits (Ramzi, 2023).

7. Examples, Applications, and Future Directions

Key applications include:

• Representation-finite algebras: their Auslander algebras are symmetrically separably equivalent (Sun et al., 20 Aug 2025).
• Matrix rings $M_n(\Lambda)$ symmetrically separably equivalent to $\Lambda$.
• Skew group algebras $\Lambda G$ symmetrically separably equivalent to $\Lambda$ (when $|G|$ is invertible in $k$).
• Frobenius-finite type, rigidity dimension, and representation type are preserved under symmetry.
• Transfer of Gorenstein categories, Wakamatsu tilting modules, and relative homological constructs between algebras.

Open avenues include exploration of further derived invariants, extension to non-Noetherian or more general rings, characterization of intertwining phenomena between symmetrical separable and other equivalences (e.g., derived or singular equivalence), and advancement toward resolving homological conjectures across larger classes of algebras via known symmetrical separable equivalences.

Table: Key Properties and Invariants Preserved under Symmetrical Separable Equivalence

Preserved Feature	Reference	Context/Condition
Gorenstein Category	(Zhao et al., 29 Jun 2025)	All separably equivalent pairs
Wakamatsu Tilting Module	(Zhao et al., 29 Jun 2025, Sun et al., 20 Aug 2025)	Functors send tilting modules to tilting modules
Frobenius-finite Type	(Sun et al., 20 Aug 2025)	Via Frobenius parts
Representation Dimension	(Bergh, 2021)	In symmetric tensor categories
Auslander-type Condition	(Sun et al., 20 Aug 2025)	Invariance established

Conclusion

Symmetrical separable equivalence constitutes a unified framework for linking rings, algebras, and categories through precisely constructed bimodules and functors that are both projective and split the identity, often with adjunctions ensuring symmetry. This equivalence preserves a wide array of categorical, homological, and representation-theoretic properties including tilting structures, Gorenstein categories, rigidity/representation dimension, and conjectural properties, with robust applications to derived categories, stable categories, and homotopical algebra. The approach yields methods for transferring properties, constructing new equivalences, and addressing deep conjectures in algebra, representation theory, and homological settings, with continued expansion into derived, topological, and stable $\infty$-category contexts.