Higher Auslander–Reiten Theory

• Higher Auslander–Reiten Theory is the higher-dimensional extension of classical Auslander–Reiten theory, replacing short exact sequences with d-exact or n-exangulated sequences.
• It integrates frameworks like n-abelian and (d+2)-angulated categories, using cluster-tilting subcategories to capture advanced homological structures.
• The theory has broad applications, including innovative duality formulas and geometric models, advancing representation theory and homotopy-theoretic studies.

Higher Auslander–Reiten theory is the higher-dimensional extension of classical Auslander–Reiten theory in which short exact almost split sequences are replaced by $d$-exact sequences, $n$-exangles, or $(d+2)$-angles; maximal rigid or cluster-tilting subcategories replace the whole module category as the natural ambient setting; and the classical Auslander–Reiten translation $\tau$ is replaced by higher translations such as $\tau_d$ and $\tau_n$. In its current form it includes the representation theory of Artin algebras, relative higher homological algebra inside $n$-cluster tilting subcategories, higher Serre duality in $(d+2)$-angulated and $n$-exangulated categories, commutative higher-degree Auslander–Reiten dualities, and geometric or homotopy-theoretic models built from surfaces, repetitive quivers, and stable $\infty$-categories (Jasso et al., 2016, Hafezi et al., 2023, He et al., 2024, Zhou, 2019, Sánchez, 4 Nov 2025).

1. Classical foundations and the higher shift

Classical Auslander–Reiten theory for an Artin algebra $n$0 studies almost split sequences

$n$1

for indecomposable non-projective modules $n$2, together with the Auslander–Reiten translation $n$3, stable categories, and the Auslander correspondence. Iyama’s higher-dimensional extension keeps the same structural agenda but changes the homological scale: short exact sequences are no longer the relevant carriers of higher homological information, and rigid or cluster-tilting subcategories replace the full module category as the basic domain (Jasso et al., 2016).

A subcategory $n$4 is $n$5-rigid when

$n$6

and it is $n$7-cluster-tilting when it is functorially finite and satisfies

$n$8

This is the basic higher replacement for the role played by $n$9 in the classical case. Correspondingly, Auslander algebras with $(d+2)$0 are replaced by $(d+2)$1-Auslander algebras satisfying $(d+2)$2, and representation generators are replaced by $(d+2)$3-cluster-tilting modules (Jasso et al., 2016, Jasso et al., 2024).

The higher analogue of an almost split sequence is a $(d+2)$4-almost split sequence

$(d+2)$5

inside a $(d+2)$6-cluster-tilting subcategory. Its end maps are left and right almost split, while the intermediate maps lie in the Jacobson radical when $(d+2)$7. This formally parallels the classical case but lives in a longer exact complex and is controlled by higher Ext-vanishing rather than by $(d+2)$8 alone (Jasso et al., 2016).

2. Higher homological environments

Higher Auslander–Reiten theory is not tied to a single categorical framework. In the module-theoretic setting of Iyama and Jasso, $(d+2)$9-cluster-tilting subcategories of $\tau$0 are $\tau$1-abelian or, more generally, $\tau$2-abelian: every morphism has an $\tau$3-kernel and an $\tau$4-cokernel, and the relevant exact objects are $\tau$5-exact sequences rather than short exact sequences. A sequence

$\tau$6

is $\tau$7-exact when it is both left and right $\tau$8-exact, meaning that the induced Hom-sequences are exact in the prescribed higher sense (Jasso et al., 2016, He et al., 2024).

The $\tau$9-exangulated formalism of Herschend–Liu–Nakaoka unifies $\tau_d$0-exact categories and $\tau_d$1-angulated categories. An $\tau_d$2-exangle

$\tau_d$3

is defined by exactness conditions on the Hom-functors against an extension bifunctor $\tau_d$4. Within this framework, an Auslander–Reiten $\tau_d$5-exangle is a distinguished $\tau_d$6-exangle whose first map is left almost split, whose last map is right almost split, and whose middle maps are radical morphisms when $\tau_d$7 (He et al., 2024, He et al., 2021).

A further ambient setting is that of $\tau_d$8-angulated categories, where the role of triangles is played by $\tau_d$9-angles

$\tau_n$0

These categories arise naturally from $\tau_n$1-cluster-tilting subcategories in triangulated categories. In this setting, Auslander–Reiten $\tau_n$2-angles provide the higher analogue of Auslander–Reiten triangles, and their existence is tightly linked to Serre duality (Zhou, 2019).

An important structural bridge between these worlds is provided by cluster-tilting quotients. If $\tau_n$3 is $\tau_n$4-exangulated and $\tau_n$5 is a cluster-tilting subcategory, then under suitable hypotheses the quotient $\tau_n$6 is $\tau_n$7-abelian; if $\tau_n$8 has Auslander–Reiten $\tau_n$9-exangles, then $n$0 has Auslander–Reiten $n$1-exact sequences. In the Frobenius case, the stable category of an $n$2-exangulated category becomes $n$3-angulated, and Auslander–Reiten $n$4-exangles induce Auslander–Reiten $n$5-angles there (He et al., 2024).

3. Higher translations, duality, and almost split phenomena

The higher Auslander–Reiten translation is built from syzygy and transpose. In the Artin algebra setting one uses the higher transpose

$n$6

and the higher Auslander–Reiten translations

$n$7

More generally, in $n$8-cluster tilting contexts one writes

$n$9

These functors identify the left and right ends of higher almost split sequences and induce equivalences between suitable stable and costable categories (Jasso et al., 2016, Hafezi et al., 2023).

A central technical tool is the higher defect formula. For a $(d+2)$0-exact sequence $(d+2)$1, Jasso–Kvamme adapt Krause’s proof of Auslander's defect formula and obtain

$(d+2)$2

which yields the higher Auslander–Reiten duality

$(d+2)$3

inside a $(d+2)$4-cluster-tilting subcategory. This duality is then used to recover the existence of $(d+2)$5-almost split sequences and to develop higher analogues of morphisms determined by objects (Jasso et al., 2016).

In $(d+2)$6-angulated categories, Serre duality controls the entire higher Auslander–Reiten package. Every Serre functor is $(d+2)$7-angulated, a $(d+2)$8-angulated category has a Serre functor if and only if it has Auslander–Reiten $(d+2)$9-angles, and the $n$0-Auslander–Reiten translation is given by

$n$1

This is the direct higher analogue of the classical triangulated formula $n$2 (Zhou, 2019).

In $n$3-exangulated categories, higher Auslander–Reiten theory can also be organized through representability of extension duals. For an Ext-finite, Krull–Schmidt, $n$4-linear $n$5-exangulated category, the subcategories

$n$6

carry a higher Auslander–Reiten translation

$n$7

with quasi-inverse $n$8. Membership in $n$9 or $\infty$0 is equivalent to the existence of Auslander–Reiten $\infty$1-exangles ending or starting at the object in question, and these exangles admit equivalent characterizations via morphisms determined by objects (He et al., 2021).

4. Relative and generalized dualities

A major recent extension replaces the full bifunctor $\infty$2 on an $\infty$3-cluster tilting subcategory $\infty$4 by an additive sub-bifunctor

$\infty$5

This yields a relative higher homological algebra in which only selected $\infty$6-extensions are retained. The corresponding $\infty$7-exact sequences define $\infty$8-projective and $\infty$9-injective objects, relative stable categories

$n$00

and a relative higher Auslander–Reiten translation

$n$01

characterized by the duality formula

$n$02

This relative theory also has $n$03-almost split sequences, relative Grothendieck groups, and a finite-type criterion in which relations generated by $n$04-almost split sequences detect when the ambient $n$05-cluster tilting subcategory is of finite type (Hafezi et al., 2023).

A different generalization appears over commutative noetherian rings. Sadeghi–Takahashi do not formulate their results in the language of higher Auslander–Reiten theory, but their two duality theorems are explicitly higher-degree generalizations of Auslander–Reiten duality and are presented as fitting naturally into its toolkit and philosophy. The first concerns $n$06-torsionfree modules relative to a module $n$07: if $n$08 is $n$09-torsionfree with respect to $n$10 and $n$11, then for $n$12

$n$13

together with a controlled exact sequence in degree $n$14. The second works over a Cohen–Macaulay local ring with canonical module $n$15, and for suitable $n$16 gives

$n$17

and, when $n$18 is locally totally reflexive on the punctured spectrum,

$n$19

in the stable range. These formulas recover Tate Auslander–Reiten duality in the Gorenstein case and are used to prove freeness criteria and partial results on the Auslander–Reiten conjecture (Sadeghi et al., 2018).

These developments show that “higher” in higher Auslander–Reiten theory is not confined to a single formalism. It includes higher exactness, higher cluster-tilting, relative bifunctorial restriction, and higher-degree Ext-dualities in commutative algebra. The common thread is the replacement of first-degree stable duality by duality patterns extending across a homological range.

5. Geometric, combinatorial, and abstract models

One of the most concrete families of higher Auslander–Reiten structures is provided by higher analogues of linearly oriented type-$n$20 representation theory. The universal $n$21-Nakayama category $n$22 is built from weakly decreasing integer $n$23-tuples, and its interval modules form a $n$24-cluster-tilting subcategory

$n$25

Finite truncations $n$26 are the $n$27-Auslander algebras of type $n$28; they are $n$29-representation-finite and $n$30-hereditary, their unique basic $n$31-cluster-tilting modules are indexed by higher intervals, and cyclic quotients produce higher analogues of tubes. In the infinite setting, higher cluster categories of type $n$32 arise, and their cluster-tilting combinatorics are controlled by triangulations of the cyclic apeirotope (Jasso et al., 2024).

Gentle algebras admit a different geometric realization. A coordinated-marked surface $n$33 determines a gentle algebra $n$34, and zigzag $n$35-arcs correspond bijectively to indecomposable string modules. Using the identification of oriented intersections with bases of $n$36-spaces, rigid modules correspond to admissible arc systems, and maximal rigid modules correspond to equivalence classes of admissible $n$37-partial triangulations. If $n$38 is such a triangulation, then the associated maximal rigid module has rank

$n$39

where $n$40 and $n$41 are the numbers of internal $n$42-gons and $n$43-gons. The same surface model is then used to realize $n$44-closures of injective modules as admissible $n$45-partial triangulations and to classify gentle algebras that are $n$46-finite or $n$47-complete (Chang, 10 Mar 2025).

At a more abstract level, Auslander–Reiten quivers and mesh relations can be internalized in stable $n$48-categories. For a finite acyclic quiver $n$49 and a stable $n$50-category $n$51, the repetitive quiver $n$52 with its mesh relations yields a mesh $n$53-category $n$54, and there is a natural equivalence

$n$55

This construction produces abstract reflection functors, Auslander–Reiten translation, Serre-type autoequivalences, and spectral Picard group actions for representations in arbitrary stable homotopy theories (Sánchez, 4 Nov 2025). This suggests that the repetitive-quiver and mesh viewpoint is not specific to derived categories over fields, but is formal at the level of stability itself.

6. Applications, invariants, and structural consequences

A categorical reformulation of higher Auslander–Reiten theory is provided by $n$56-structures on homotopy categories. For an additive subcategory $n$57, Jørgensen–Kato construct a $n$58-structure on $n$59 whose heart $n$60 is an abelian category serving as a natural domain for higher Auslander–Reiten theory. When $n$61 is maximal $n$62-orthogonal, the simple objects of $n$63 are precisely Iyama’s higher Auslander–Reiten sequences, and higher Auslander–Reiten duality is recovered from the Serre functor on $n$64. If $n$65 is functorially finite, then $n$66 is a quotient of the classical heart $n$67, so higher Auslander–Reiten theory appears as a quotient of classical Auslander–Reiten theory at the level of hearts (Uribe et al., 2013).

Higher preprojective algebras provide another bridge between homological algebra and higher Auslander–Reiten translation. For an $n$68-representation-infinite algebra $n$69, Minamoto studies the $n$70-preprojective algebra

$n$71

and shows that graded coherentness of this tensor algebra is tied to the eventual behavior of the higher Auslander–Reiten adjunction maps

$n$72

If $n$73 is graded coherent, these maps are eventually isomorphisms for every finitely generated $n$74-module; for $n$75, the converse also holds (Minamoto, 2012).

Dominant dimension enters higher Auslander–Reiten theory through higher torsion-free Auslander–Reiten sequences. For a finite-dimensional algebra $n$76, the paper on dominant dimension and higher torsion-free AR sequences proves that $n$77 has $n$78-torsion-free Auslander–Reiten sequences if and only if

$n$79

For gendo-symmetric algebras this simplifies to $n$80. If $n$81 is a higher Auslander algebra attached to a $n$82-cluster tilting object $n$83, then $n$84 has $n$85-torsion-free Auslander–Reiten sequences exactly when $n$86 is $n$87-representation-finite. The same work generalizes Reiten’s formula by showing

$n$88

where $n$89 and $n$90 ranges over simple modules (Cruz et al., 2024).

Covering theory also extends to the higher setting. For a locally support-finite category $n$91 with free $n$92-action, the push-down functor along the Galois covering $n$93 sends $n$94-equivariant $n$95-precluster tilting subcategories of $n$96 to $n$97-precluster tilting subcategories of $n$98, and vice versa. Under square-free hypotheses, $n$99 is $(d+2)$00-minimal Auslander–Gorenstein if and only if $(d+2)$01 is so. The same push-down formalism preserves support $(d+2)$02-tilting pairs and local $(d+2)$03-tilting finiteness (Asadollahi et al., 19 Jun 2025).

Taken together, these developments show that higher Auslander–Reiten theory is simultaneously homological, categorical, geometric, and combinatorial. Its core objects are higher translations, higher almost split configurations, and higher duality formulas; its ambient categories range from $(d+2)$04-cluster-tilting subcategories to $(d+2)$05-exangulated, $(d+2)$06-angulated, commutative, and stable $(d+2)$07-categorical settings; and its applications run from finite-type criteria and freeness problems to surface models, graded coherence, dominant dimension, and covering theory.