Stable Equivalence of Morita Type

• Stable Equivalence of Morita Type is a relationship between finite-dimensional algebras that generalizes classical Morita equivalence by focusing on stable module categories modulo projective components.
• It is realized through pairs of bimodules that induce mutually quasi-inverse functors via standard induction and restriction, linking the algebras' stable structures.
• The framework leverages Frobenius extension and coring structures to unify induction-restriction adjunctions, facilitating extensions to derived and graded contexts and deeper categorical insights.

A stable equivalence of Morita type is a categorical relationship between finite-dimensional algebras that generalizes Morita equivalence to the context of their stable module categories, focusing on relationships modulo projective summands. This equivalence is tightly connected to deep structural aspects such as Frobenius extensions, coring structures, and adjunctions akin to those governing induction and restriction functors. The following exposition synthesizes the full landscape of stable equivalences of Morita type and their realization in terms of Frobenius extensions and corings, following Dugas–Martínez-Villa and Beattie–Caenepeel–Raianu (Beattie et al., 2012).

1. Fundamental Definition and Equivalence Data

Let $\Lambda$ and $\Gamma$ be finite-dimensional $k$-algebras over a fixed field $k$, without semisimple blocks. A pair of bimodules, ${}_\Lambda M_\Gamma$ and ${}_\Gamma N_\Lambda$, is said to induce a stable equivalence of Morita type between $\Lambda$ and $\Gamma$ if:

• Both bimodules are projective as left and as right modules.
• The following stable isomorphisms hold in the corresponding module categories modulo projectives:

$$M \otimes_\Gamma N \cong \Lambda \oplus P\quad\text{in }\underline{\mathrm{mod}\text{-}\Lambda}, \qquad N \otimes_\Lambda M \cong \Gamma \oplus Q\quad\text{in }\underline{\mathrm{mod}\text{-}\Gamma},$$

where $P$ (resp. $Q$) is projective as a $\Lambda$–$\Lambda$-bimodule (resp. $\Gamma$–$\Gamma$-bimodule).

These data ensure that the functors

$$\mathrm{Ind} = -\otimes_\Gamma M : \mathrm{mod}\text{-}\Gamma \to \mathrm{mod}\text{-}\Lambda,\qquad \mathrm{Res} = -\otimes_\Lambda N : \mathrm{mod}\text{-}\Lambda \to \mathrm{mod}\text{-}\Gamma$$

descend to the stable categories and induce mutually quasi-inverse equivalences: $$\underline{\mathrm{mod}\text{-}\Gamma} \;\leftrightarrows\; \underline{\mathrm{mod}\text{-}\Lambda}.$$

2. Realization via Inclusion in a Morita-Equivalent Algebra

A core result, due to Dugas–Martínez-Villa, is that for any stable equivalence of Morita type, one can replace $\Lambda$ by a Morita-equivalent algebra $\Delta$ such that $\Gamma$ embeds as a subalgebra of $\Delta$, and the stable equivalence is realized through standard induction and restriction functors:

• There exists a $k$-algebra $\Delta$, Morita equivalent to $\Lambda$, and an injective ring homomorphism $\iota : \Gamma \hookrightarrow \Delta$.
• The induction and restriction functors associated to this inclusion,

$$\mathrm{Res}_\Gamma^\Delta : \mathrm{mod}\text{-}\Delta \to \mathrm{mod}\text{-}\Gamma, \qquad \mathrm{Ind}_\Gamma^\Delta = -\otimes_\Gamma \Delta : \mathrm{mod}\text{-}\Gamma \to \mathrm{mod}\text{-}\Delta,$$

are exact and induce inverse equivalences on stable module categories.

The constructive mechanism is $\Delta = \mathrm{End}_\Lambda(M)$, with $\Gamma$ mapping into $\Delta$ via its right-action on $M$. The usual induction and restriction become biadjoint functors realizing the same equivalence as $M \otimes_\Gamma -$ and $N \otimes_\Lambda -$ in the original context.

3. Frobenius Extension Structure and Corings

Beattie–Caenepeel–Raianu establish that the extension $\iota: \Gamma \to \Delta$ is not only inclusion but is indeed a Frobenius extension:

• Frobenius extension: $\Delta$ is finitely generated projective as a right $\Gamma$-module and $\mathrm{Hom}_\Gamma(\Delta, \Gamma) \cong \Delta$ as $(\Gamma,\Delta)$-bimodules.
• Dual bases formulation: There exists a Frobenius system $(e, \epsilon)$ with $e \in \Delta \otimes_\Gamma \Delta$ and $\epsilon: \Delta \to \Gamma$ a bimodule map, fulfilling

$$\sum_i x_i \epsilon(y_i d) = d = \sum_i \epsilon(d x_i) y_i\quad\text{for all } d \in \Delta,$$

with dual bases $\{x_i\},\{y_i\}$ in $\Delta$.

• Corings: If $\Delta/\Gamma$ is Frobenius, then $\Delta$ admits a $\Gamma$-coring structure, with comultiplication given by

$$\Delta \to \Delta \otimes_\Gamma \Delta,\quad d \mapsto d\cdot e = \sum d e^1 \otimes e^2,$$

and counit $\epsilon: \Delta \to \Gamma$.

Conversely, any Frobenius $\Gamma$-coring structure arises from such a Frobenius extension.

4. Unification of the Stable Equivalence Structure

The upshot is that any stable equivalence of Morita type between $\Lambda$ and $\Gamma$ can be viewed as the stable equivalence arising from a Frobenius extension $\Delta/\Gamma$:

• The pair $$(\mathrm{Ind}_\Gamma^\Delta, \mathrm{Res}_\Gamma^\Delta)$$ forms a Frobenius pair: each is both left and right adjoint to the other.
• The entire stable equivalence data is encoded in the Frobenius system and the associated coring structure, unifying the induction-restriction adjunctions and projective splittings.

This perspective clarifies the structure of stable equivalence of Morita type and facilitates generalization to e.g., derived or graded settings, where Frobenius and separable properties play a key role in lifting equivalences.

5. Implications for Lifting, Structure, and Category Theory

Several algebraic and categorical consequences result from this interpretation:

• The presence of a Frobenius extension structure on $\Delta / \Gamma$ often allows the stable equivalence to be lifted to derived or homotopy categories due to separability and duality properties.
• The explicit realization via a ring inclusion and Frobenius extension enables transfer of classical induction and restriction techniques to the paper of stable categories.
• The unification of the Morita-type data as a single algebraic object—a Frobenius coring—streamlines categorical analysis, especially regarding adjunctions, dual bases, and the reconstruction of equivalences.

6. Centralization of Results and Connections to Further Research

This framework demonstrates that stable equivalence of Morita type is not an abstract or isolated categorical phenomenon but is intimately linked to Frobenius extensions and coring theory. It elucidates how the comparative module-theoretic data of $(M, N)$ is precisely reflected in the extension-theoretic and (co)algebraic structure of $\Delta / \Gamma$. The consequences are pivotal for the extension to derived equivalences, the paper of open conjectures in modular representation theory, and the systematic construction of equivalences in various algebraic settings (Beattie et al., 2012).