Auslander–Reiten Theory

• Auslander–Reiten theory is a framework in representation theory that uses almost split sequences and AR quivers to classify module categories and reveal homological invariants.
• It extends to higher dimensions via n-cluster tilting subcategories and d-almost split sequences, enabling refined analysis in modern homological algebra.
• The theory informs practical applications in studying singularity categories, quantum symmetries, and invariance under derived and stable equivalences.

Auslander–Reiten theory is a framework in representation theory and homological algebra that organizes the structure of module categories by means of almost split sequences, Auslander–Reiten quivers, and associated functorial and categorical invariants. The theory encodes deep connections between syzygy phenomena, homological invariants, finiteness conditions, and the combinatorics of module categories. Since its original conception for Artin algebras, Auslander–Reiten theory has been generalized along multiple axes: higher homological dimensions, covering and descent techniques, compatibility with triangulated and extriangulated categories, and categorical invariance under equivalences and mutations. Current research focuses both on extending its homological toolkit (e.g., higher cluster tilting, $d$-angulated categories) and on leveraging these ideas in the paper of singularity categories, quantum symmetries, and algebraic geometry.

1. Core Principles and Classical Constructions

Auslander–Reiten theory originated as a means of decomposing module categories over (Artin or finite-dimensional) algebras using almost split (or Auslander–Reiten) sequences, irreducible morphisms, and the translation quiver structure of the Auslander–Reiten (AR) quiver. An almost split sequence

$$0 \longrightarrow \tau M \longrightarrow E \longrightarrow M \longrightarrow 0$$

captures the minimal nontrivial extensions of $M$ and encodes the non-split self-maps in the category. The AR translate $\tau$ (often the composition of $\operatorname{Tr}$ and $D$ for suitable duals and transposes) relates indecomposable non-projective and non-injective modules. The AR quiver, whose vertices index indecomposable modules and arrows correspond to irreducible morphisms, encodes the combinatorial skeleton of morphism spaces and reflects both tame and wild behavior in the underlying category (Külshammer, 2012, Külshammer, 2012, Krebs, 2015, Chaio et al., 2014, Crawley-Boevey, 2017).

In many classical settings, these AR sequences and quivers provide a classification of finite representation type in terms of Dynkin diagrams or their infinite analogues, and serve as a basis for subtle homological and combinatorial invariants. For self-injective and weakly symmetric algebras of polynomial growth, for example, AR theory provides robust invariants such as Cartan determinants and Külshammer–Reynolds ideals that facilitate the classification of derived and stable equivalence classes (Zhou et al., 2010).

2. Higher Auslander–Reiten Theory

The classical approach generalizes to higher dimensions via $n$-cluster-tilting subcategories and $d$-exact sequences (for $d\geq 1$). In higher Auslander–Reiten theory, pioneered by Iyama, the basic building blocks (“$d$-almost split sequences”) are exact sequences of length $d+2$ in suitable “$d$-cluster-tilting” subcategories—these are subcategories $\mathcal{M}$ of $\operatorname{mod} \Lambda$ that satisfy

$$\mathcal{M} = \{ X \mid \forall\ 1 \leq i < d,\, \operatorname{Ext}^i_\Lambda(X, \mathcal{M}) = 0\}, \quad \mathcal{M} = \{ Y \mid \forall\ 1 \leq i < d,\, \operatorname{Ext}^i_\Lambda(\mathcal{M}, Y) = 0 \}$$

with $d$-almost split sequences

$$0 \longrightarrow \tau_d N \longrightarrow M^1 \longrightarrow \cdots \longrightarrow M^d \longrightarrow N \longrightarrow 0,$$

where the higher AR translation $\tau_d$ is typically $D \circ \operatorname{Tr}_d$ (with $$\operatorname{Tr}_d = \operatorname{Tr} \circ \Omega^{d-1}$$ and $D$ a duality). This theory supports higher analogues of almost split functors, defect formulas, and homological invariants, enabling new approaches to classical theorems and enabling the investigation of $d$-angulated and higher-dimensional categories (Jasso et al., 2016, Zhou, 2019, Fedele, 2018, Asadollahi et al., 19 Jun 2025).

3. Covering Techniques, Galois Descent, and Equivariance

An essential topic is the preservation and transfer of AR and higher AR structures under category coverings and quotient constructions. In the classical setting, covering functors (e.g., from mesh categories to module categories) allow one to “lift” composition patterns of irreducible morphisms and relate the graded pieces of radical filtrations in module categories to combinatorial data of the universal AR quiver (Chaio et al., 2014).

In the higher setting, recent work on Galois coverings shows that, for a group $G$ acting freely on a locally bounded category $\mathcal{C}$, the push-down functor $P_*$ associated with $\mathcal{C} \rightarrow \mathcal{C}/G$ preserves $n$-precluster tilting subcategories, $n$-minimal Auslander–Gorenstein property, and locally $\tau_n$-tilting finiteness (Asadollahi et al., 19 Jun 2025). The commutation of $P_*$ with higher AR translation,

$P_*(\tau_n M) \cong \tau_n(P_*(M)),$

ensures the transferability of higher AR-theoretic and tilting-theoretic invariants between the covering and orbit categories. This descent is pivotal for comparing representation-finiteness, cluster-tilting structures, and support cocycle properties in the presence of symmetries.

4. Classification Criteria, Homological Invariants, and Stability

A recurring theme is the role of homological and structural invariants preserved under equivalences and coverings. Derived equivalence, stable equivalence of Morita type, and singular equivalence preserve not only the number of non-projective simple modules (a key content of the Auslander–Reiten conjecture) but also finer invariants such as determinants of Cartan matrices, the Külshammer–Reynolds ideal quotients $Z(A)/T_1(A)$, and Hochschild cohomology structures (Zhou et al., 2010, Chen et al., 2020). Such invariants facilitate the classification of self-injective and symmetric algebras up to derived and stable equivalence, reducing many conjectural statements (e.g., the AR conjecture) to checking invariance of these key quantities under equivalence.

Similarly, finite and infinite representation type in functorially finite resolving or $n$-cluster tilting subcategories is characterized by the shape (tree type) of connected components of the AR quiver: Dynkin diagrams indicate finiteness, while infinite or tame components correspond to infinite families of modules with unbounded length, as shown via sectional or helical paths (Krebs, 2015).

5. Extensions to Triangulated, Extriangulated, and Nonclassical Settings

Auslander–Reiten theory has been adapted to encompass triangulated, extriangulated, and even non-abelian or quasi-abelian categories. In triangulated and $d$-angulated categories, AR theory is formulated in terms of AR triangles and $(d+2)$-angles, with duality realized by Serre functors and higher AR translations $\tau_d = \mathbb{S}\Sigma^{-d}$. Notably, the existence of AR triangles (or $(d+2)$-angles) is equivalent to the existence of a Serre functor, and passage to quotient or extension-closed subcategories often retains these structures provided mutation or stability conditions on the AR translation hold (Zhou, 2019, Iyama et al., 2018, Nkansah, 2023).

Extriangulated categories unify exact and triangulated contexts, and AR theory in this setting hinges on the existence of almost split extensions and their equivalence with AR–Serre duality. The associated stable and costable categories inherit mesh and ladder structures, with combinatorial and homological invariants (such as radical layers and Grothendieck group data) controlled via the AR and mesh category formalism (Iyama et al., 2018).

Key generalizations also include ideal mutations, where classical “approximation by object” is replaced by “approximation by morphism in an ideal,” leading to a generalized AR theory over a Hom-finite Krull–Schmidt triangulated category. The Jacobson radical serves as a canonical AR ideal, and testing for the existence of AR triangles reduces to checking functorial finiteness and ghost map conditions on this ideal (Zhang et al., 2022).

6. Invariance, Descent, and Applications in Commutative Algebra and Singularity Theory

Auslander–Reiten conjectures, classically about vanishing of Ext modules implying projectivity, have found compelling tensorial and annihilator-theoretic generalizations. The behavior of Ext modules under taking syzygies, push-down functors, and quotient by group actions is central. It is now known that invariance of the AR conjecture holds under appropriate singular equivalences induced by adjoint pairs (e.g., via recollements or ladders in derived/singularity categories), enabling reduction of the conjecture to simpler or better understood settings—matrix algebras, recollements of derived categories, and change of rings (Chen et al., 2020).

Further, the generalization of the AR conjecture to statements about annihilators of Ext-modules, e.g., whether the AR-annihilator ideal coincides with the stable annihilator, provides a finer ideal-theoretic lens on projectivity, stability, and the classification of modules with respect to the AR property. These ideal-theoretic versions have ramifications for the paper of syzygies, maximal Cohen–Macaulay modules, isolated singularities, and almost Gorenstein or Arf rings, and connect with conjectures such as Tachikawa’s (Esentepe, 29 Jul 2024).

Homological invariants such as (complete intersection) CI-dimension and Gorenstein dimension serve as sufficient criteria for the AR conjecture to hold: modules whose (self-)dual or endomorphism algebra has finite CI-dimension or G-dimension satisfy the required vanishing conditions. The use of Ext vanishing, CI-dimension, and stabilization of syzygies in these contexts broadens the class of rings and modules where the AR conjecture is verified, particularly in commutative Noetherian and local settings (Ghosh et al., 2 May 2024, Kumashiro, 2019).

7. Interconnections with Higher Category Theory and Simplicial Structures

Recent perspectives recast higher Auslander–Reiten theory in the language of simplicial and higher category theory. Notions such as $2m$-Segal objects, outer $m$-Kan conditions, and slice mutations provide a combinatorial and categorical framework that unifies descent conditions and truncations in both abelian and stable $\infty$-categories. For example, in a stable $\infty$-category, being $2m$-Segal, being outer $m$-Kan, and the $m$-truncation of the associated filtered object are all equivalent for simplicial objects. This categorical upgrade not only illuminates the geometry underlying AR phenomena (such as horn filling and cube decompositions) but also links the theory to broader topics in higher algebra and $K$-theory (Dyckerhoff et al., 2018).

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Summary Table: Selected Fundamental Notions and Correspondences

Construct	Classical AR Theory	Higher/Generalized Setting
Almost split sequence	Short exact sequence ($d=1$)	$d$-exact, $(d+2)$-angle ($d>1$)
AR translate ($\tau$)	$D \operatorname{Tr}$	$\tau_d = D \operatorname{Tr}_d$
AR triangle/quiver	Triangulated, stable components	$(d+2)$-angulated, higher quivers
Stable equivalence of Morita type	Derived/stable invariants coincide	Invariants (Cartan, center, etc.)
Covering functors	Mesh categories, radical layers	Galois coverings, push-down functor
Tilting rigidity	$\tau$-tilting pairs	$\tau_n$-tilting pairs
Ideal-theoretic AR property	$ARann_R(M)$, $_R(M)$	Annihilator-theoretic generalization

Key Formulas and Invariants:

• $Z(A)/T_1(A) \cong Z(B)/T_1(B)$: invariance of centers modulo Reynolds/Külshammer ideals (Zhou et al., 2010)
• $|\det C_A| = |\det C_B|$: Cartan determinant preservation
• $\tau_n := \tau \Omega^{n-1}$: higher AR translation (Asadollahi et al., 19 Jun 2025)
• Complete intersection dimension: $$CI\text{-}\dim_R(M) := \inf\{\operatorname{pd}_S(M \otimes_R R') - \operatorname{pd}_S(R')\}$$ (Ghosh et al., 2 May 2024)
• $$ARann_R(M) = \bigcap_{i>0} \operatorname{ann}_R \operatorname{Ext}_R^i(M, M \oplus R)$$ (Esentepe, 29 Jul 2024)
• Equivalence: AR conjecture invariant under singular equivalence with suitable adjoint pairs (Chen et al., 2020)

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Conclusion:

Auslander–Reiten theory, both in its classical and higher forms, is a homological and categorical framework essential for understanding the structure and invariants of module categories, with extensions to higher categories, covering techniques, and an increasingly sophisticated collection of homological and ideal-theoretic tools. Its current role encompasses the classification of algebraic objects under equivalence, the analysis of singularities, the paper of tilting and mutation phenomena, and the investigation of projectivity criteria in commutative and noncommutative algebra.