Auslander–Reiten Conjecture Overview

• Auslander–Reiten Conjecture is a central problem in homological algebra that asserts the invariance of non-projective simple modules under stable equivalence of Morita type.
• It connects derived and stable equivalence classifications by preserving numerical invariants like Cartan determinants and Külshammer ideals.
• The conjecture underpins classification methods in representation theory, aiding in distinguishing finite-dimensional self-injective and symmetric algebras.

The Auslander–Reiten Conjecture is a central problem in the paper of homological properties of (self-injective) algebras and has deep connections to the structure and classification of finite-dimensional algebras, particularly within the framework of representation theory. It asserts that stable equivalences of Morita type between self-injective algebras preserve the number of isomorphism classes of non-projective simple modules. This statement encapsulates crucial invariants under certain homological equivalences and fundamentally links two classification methods: derived equivalence and stable equivalence of Morita type.

1. Formal Statement and Context

Let $A$ and $B$ be finite-dimensional, self-injective algebras (over an algebraically closed field), or more specifically, symmetric or weakly symmetric algebras of polynomial growth. The Auslander–Reiten Conjecture (ARC) pronounces that if $A$ and $B$ are stably equivalent of Morita type, then the number of isomorphism classes of non-projective simple $A$-modules equals that for $B$: $\#\{\text{non-projective simple %%%%6%%%%-modules}\} = \#\{\text{non-projective simple %%%%7%%%%-modules}\}$ Equivalently, the conjecture asserts that certain "numerical invariants" of the stable category, most notably this count of non-projective simple isoclasses, are preserved under stable equivalence of Morita type.

The conjecture directly relates to the problem of distinguishing finite versus infinite representation type and underpins the interface between "local" invariants of algebras (e.g., Cartan invariants, Külshammer ideals) and "global" equivalence classes in the representation theory landscape.

2. Derived vs. Stable Equivalence Classification

The paper studies weakly symmetric algebras of polynomial growth—this class includes both domestic and non-domestic algebras—and systematically analyzes their structural invariants under two core equivalence mechanisms:

• Derived Equivalence: $A$ and $B$ are derived equivalent if there is a triangle equivalence $\mathsf{D}^b(A) \simeq \mathsf{D}^b(B)$. Derived equivalence preserves a large suite of homological invariants.
• Stable Equivalence of Morita Type: $A$ and $B$ are stably equivalent of Morita type if the stable module categories ${\underline{\mathsf{mod}}}\text{-}A$ and ${\underline{\mathsf{mod}}}\text{-}B$ are equivalent in a manner induced by certain bimodules, specifically those satisfying projectivity on both sides.

The classification results in the paper demonstrate:

• For domestic weakly symmetric algebras, the derived equivalence classification coincides exactly with the classification up to stable equivalence of Morita type (Theorem 2.9):

$A \text{ derived %%%%15%%%% stably equivalent (of Morita type) %%%%16%%%% stably equivalent.}$

• For non-domestic weakly symmetric algebras of polynomial growth, the classifications coincide modulo certain scalar issues within parameterized families (e.g., in $A_3(X)$ and $A_1(X)$).

This means that, up to scalars, derived and stable equivalences describe the same partitioning of algebras in these classes, and invariants preserved under derived equivalence are also invariants of stable equivalence of Morita type.

3. Homological and Numeral Invariants Preserved under Stable Equivalences

The proof strategy is built around showing that a series of structural and numerical invariants remain unchanged under the equivalences considered:

• Cartan Determinant: The absolute value $|\det C_A|$ of the Cartan matrix is invariant under stable equivalence of Morita type ([25, Prop. 5.1]):

$|\det C_A| = |\det C_B|$

This invariant is sensitive to the singularity of the Cartan matrix.

• Külshammer Ideals: For $A$ symmetric over a field of positive characteristic $p$, the sequence of ideals

$T_n(A) = \{ x \in A \mid x^{p^n} \in [A, A] \}$

and the quotients

$Z(A) / T_n(A)^\perp \cong Z(B) / T_n(B)^\perp$

are preserved under stable equivalences of Morita type.

• Hochschild Cohomology: The dimension $\dim \mathsf{HH}_2(A)$ is also stable invariant.
• Representation Type: Using results by Krause and Zwara, the property of being domestic, tame, etc., remains unchanged.

Collectively, these invariants—especially the number of simple modules—ensure that the partition of the class of weakly symmetric algebras of polynomial growth by "derived equivalence" matches that given by "stable equivalence of Morita type" (modulo certain scalars in the non-domestic case).

4. Theorems and Explicit Formulas

Key formulas and theorem statements underlying the main proof and its consequences include:

• Cartan Determinant Invariance:

$$|\det C_A| = |\det C_B| \quad \text{(under stable equivalence of Morita type)}$$

• Külshammer Ideals and Center Structure:

$$T_n(A) = \{ x \in A \mid x^{p^n} \in [A, A] \}, \quad Z(A)/T_n(A)^\perp \cong Z(B)/T_n(B)^\perp$$

• Theorem 2.9 (Domestic Case): — Two weakly symmetric algebras of domestic type are derived equivalent if and only if they are stably equivalent of Morita type, and thus stably equivalent.
• Theorem 3.8 (Non-Domestic Case): — If $A$ is indecomposable and stably equivalent of Morita type to a non-domestic symmetric algebra $B$ of polynomial growth, then $A$ and $B$ have the same number of simple modules.

These results guarantee that the number of simple modules is a rigid invariant under the considered equivalences, so the Auslander–Reiten Conjecture holds for weakly symmetric algebras of polynomial growth.

5. Proof Outline and Consequences

The argument proceeds by:

1. Case distinction: Domestic vs. non-domestic; further split into standard/non-standard in the domestic case.
2. Classification via explicit representatives: Using explicit quivers with relations and pre-existing classification theorems.
3. Utilization of numerical invariants: Cartan determinant, Külshammer ideals, Hochschild cohomology, and properties of centers.
4. Establishing the coincidence of equivalence classes: Showing that derived and stable equivalence partitions are the same (allowing for scalars in non-domestic families).
5. Deduction of the conjecture: Once equivalence classes coincide, invariance of the number of simple modules under stable equivalence is immediate.

The broader impact is a unified approach to distinguishing representation-theoretic types and classes of self-injective algebras by invariants computable in both homological and combinatorial frameworks.

6. Implications for Homological Algebra and Representation Theory

This unification of derived and stable equivalence invariants has several significant consequences:

• Classification Tools: It makes invariants like Cartan determinant and Külshammer ideals actionable tools for distinguishing algebras up to these equivalences.
• Stable Equivalence Detects Derived Equivalence Modulo Scalars: In practice, for weakly symmetric algebras of polynomial growth, one can use stable invariants to decide derived equivalence classes (with caveats in certain families).
• Strengthens the arsenal for the paper of modular representations and deformation theory: Because the number of non-projective simple modules is crucial in deformation problems, the confirmed invariance is highly relevant.

7. Summary

For weakly symmetric algebras of polynomial growth, the derived equivalence and stable equivalence of Morita type classifications coincide (up to certain scalars in non-domestic cases). This coincidence yields a verification of the Auslander–Reiten Conjecture in this setting: a stable equivalence of Morita type between such algebras necessarily preserves the number of isomorphism classes of non-projective simple modules. The proof is achieved by showing that canonical invariants—Cartan determinant, Külshammer ideals, invariants of the center, and Hochschild homology—are shared under both derived and stable equivalence. These results provide a structural consolidation within self-injective and symmetric algebra representation theory, linking category-theoretic and explicit combinatorial data, and furnish new methodologies for distinguishing algebras within their equivalence classes (1005.1150).