Auslander-Reiten Bijection

• Auslander-Reiten bijection is a correspondence linking stable morphism spaces with first extension groups through almost split sequences and the translation functor τ.
• It applies across various categorical settings—from module categories to exact, extriangulated, and derived frameworks—ensuring a unified approach to indecomposable object classification.
• The bijection extends to higher homological algebra by incorporating d-exact sequences, connecting with Iyama’s grade bijection, and mirroring Coxeter transformations via matrix factorizations.

Auslander-Reiten bijection denotes a family of closely related correspondences in representation theory and higher homological algebra that are organized by almost split sequences, the Auslander-Reiten translate, and duality between morphisms and extensions. In the classical module-theoretic setting for an Artin algebra $\Lambda$, it is expressed by the bifunctorial isomorphism

$$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$

where $\tau$ is the Auslander-Reiten translation. In a different but compatible usage, especially for Auslander-Gorenstein and Auslander regular algebras, it is the permutation sending an indecomposable injective to the indecomposable projective obtained as a suitable syzygy, a permutation proved to coincide with Iyama’s grade bijection (Jasso et al., 2016, Klász et al., 16 Jan 2025).

1. Classical formulation and homological meaning

In classical Auslander-Reiten theory, the bijection is the natural duality between nontrivial morphisms modulo projectives and first extensions involving the Auslander-Reiten translate. For indecomposable non-projective modules $X$ and $Y$, one has

$$D\,\underline{\operatorname{Hom}_\Lambda(X, Y)} \cong \operatorname{Ext}^1_\Lambda(Y, \tau X),$$

with $\tau = D\operatorname{Tr}$. The isomorphism is bifunctorial in $X$ and $Y$, and it is not merely a vector-space identification: it is the mechanism behind the existence and uniqueness phenomena of almost split sequences and the theory of morphisms determined by objects (Jasso et al., 2016).

This duality links two kinds of data that are usually studied separately. On one side lie stable morphism spaces, where maps factoring through projectives are discarded; on the other lie extension classes, which encode how indecomposables are built from one another. The translate $\tau$ mediates between them. In settings where Auslander-Reiten sequences exist, the indecomposable non-projective object $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$0 appears at the end of a sequence

$$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$1

and dually for non-injective objects. In proper abelian subcategories of triangulated categories with a Serre functor, this classical pattern is recovered by constructing $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$2 from a Nakayama functor defined via covers and envelopes (Nkansah, 2023).

2. Translation quivers and component geometry

The bijection is reflected combinatorially in the stable Auslander-Reiten quiver, where $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$3 acts as a translation on components. For Frobenius-Lusztig kernels, the stable Auslander-Reiten quiver exhibits a sharply constrained geometry: every non-periodic component has tree class $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$4, $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$5, or $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$6, and for symmetric Frobenius-Lusztig kernels, including the small quantum group, one has $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$7. Periodic components are infinite tubes, and in the small quantum group they are homogeneous tubes $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$8 (Külshammer, 2012).

For small half quantum groups, the translate has the explicit form

$$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$9

where $\tau$0 is the Nakayama automorphism. The $\tau$1-period of every periodic module for $\tau$2 divides $\tau$3, the $\tau$4-period divides $\tau$5, and for $\tau$6-modules of complexity $\tau$7, the $\tau$8-period is $\tau$9. Components containing gradable modules are of the form $X$0, so the bijection operates along a bi-infinite linear translation quiver in these cases (Külshammer, 2012).

Component structure also controls finiteness. For an Artin algebra, an Auslander-Reiten component with bounded short cycles is almost acyclic and has only finitely many $X$1-orbits, and an algebra is representation-finite if and only if its module category has bounded short cycles. This places the Auslander-Reiten translation at the center of a criterion for representation-finiteness that is combinatorial but formulated inside the radical of the module category (Liu et al., 2017).

3. The injective-projective permutation and its Coxeter realization

A distinct and highly specific use of the term occurs for Auslander-Gorenstein and Auslander regular algebras. For a simple $X$2-module $X$3,

$X$4

Iyama’s grade bijection sends $X$5 to $X$6, while the Auslander-Reiten bijection is defined on indecomposable injectives $X$7 by sending $X$8 to the indecomposable projective $X$9 for which $Y$0 is the $Y$1-th syzygy of $Y$2, with $Y$3. The decisive theorem is that a finite dimensional algebra is Auslander-Gorenstein if and only if for every simple module $Y$4,

$Y$5

and, moreover, the grade permutation coincides with the Auslander-Reiten permutation (Klász et al., 16 Jan 2025).

For Auslander regular algebras this permutation acquires a purely linear-algebraic description. If $Y$6 is the Cartan matrix, then for finite global dimension

$Y$7

is the Coxeter matrix. With an admissible ordering of the simple modules, the Bruhat decomposition has the form

$Y$8

with $Y$9, $$D\,\underline{\operatorname{Hom}_\Lambda(X, Y)} \cong \operatorname{Ext}^1_\Lambda(Y, \tau X),$$0 a permutation matrix corresponding to the grade permutation, and $$D\,\underline{\operatorname{Hom}_\Lambda(X, Y)} \cong \operatorname{Ext}^1_\Lambda(Y, \tau X),$$1 upper triangular with diagonal entries $$D\,\underline{\operatorname{Hom}_\Lambda(X, Y)} \cong \operatorname{Ext}^1_\Lambda(Y, \tau X),$$2. This identifies the grade bijection, and hence the Auslander-Reiten permutation, with the Coxeter permutation, reducing its computation to matrix factorization rather than explicit injective or projective resolutions (Klász et al., 16 Jan 2025).

This linear-algebraic realization has several consequences. The permanent of the Coxeter matrix of an Auslander regular algebra is either $$D\,\underline{\operatorname{Hom}_\Lambda(X, Y)} \cong \operatorname{Ext}^1_\Lambda(Y, \tau X),$$3 or $$D\,\underline{\operatorname{Hom}_\Lambda(X, Y)} \cong \operatorname{Ext}^1_\Lambda(Y, \tau X),$$4; a finite lattice is distributive if and only if its Coxeter matrix can be written as $$D\,\underline{\operatorname{Hom}_\Lambda(X, Y)} \cong \operatorname{Ext}^1_\Lambda(Y, \tau X),$$5 with $$D\,\underline{\operatorname{Hom}_\Lambda(X, Y)} \cong \operatorname{Ext}^1_\Lambda(Y, \tau X),$$6 a permutation matrix and $$D\,\underline{\operatorname{Hom}_\Lambda(X, Y)} \cong \operatorname{Ext}^1_\Lambda(Y, \tau X),$$7 upper triangular; and in blocks of category $$D\,\underline{\operatorname{Hom}_\Lambda(X, Y)} \cong \operatorname{Ext}^1_\Lambda(Y, \tau X),$$8, the set of grades coincides with twice the values of Lusztig’s $$D\,\underline{\operatorname{Hom}_\Lambda(X, Y)} \cong \operatorname{Ext}^1_\Lambda(Y, \tau X),$$9-function, while simple-preserving duality forces the grade bijection to be the identity (Klász et al., 16 Jan 2025).

A related linearization problem concerns the action of $\tau = D\operatorname{Tr}$0 on the Grothendieck group. For hereditary finite-dimensional algebras, the Coxeter transformation extends the action of the Auslander-Reiten translation on non-projective indecomposable modules to a linear endomorphism of $\tau = D\operatorname{Tr}$1. All Nakayama algebras admit such a $\tau = D\operatorname{Tr}$2-map, and if the $\tau = D\operatorname{Tr}$3-quiver is connected and non-acyclic, then the existence of a $\tau = D\operatorname{Tr}$4-map is equivalent to being a cyclic Nakayama algebra (Klapproth, 2022).

4. Exact, Grothendieck, extriangulated, and derived settings

The classical duality extends far beyond module categories. In a Grothendieck abelian category $\tau = D\operatorname{Tr}$5 that is locally finitely presented, for a finitely presented object $\tau = D\operatorname{Tr}$6 and $\tau = D\operatorname{Tr}$7, every injective $\tau = D\operatorname{Tr}$8-module $\tau = D\operatorname{Tr}$9 admits an Auslander-Reiten translate $X$0 characterized by the natural isomorphism

$X$1

Thus $X$2 has a partially defined right adjoint on injective $X$3-modules. For suitable schemes, this recovers Serre duality; for a non-singular projective scheme $X$4 of dimension $X$5, one obtains $X$6 (Krause, 2016).

In exact categories, Auslander's defect formula persists. For a conflation $X$7, one has natural isomorphisms

$X$8

and these fit into a commutative triangle of poset bijections relating right $X$9-determined deflations, submodules of $Y$0, and submodules of $Y$1 (Jiao, 2017). In quasi-abelian, Krull-Schmidt categories, Auslander-Reiten sequences admit equivalent characterizations through local endomorphism rings, minimal left or right almost split morphisms, and irreducibility of the kernel and cokernel maps (Shah, 2018).

Extriangulated categories provide a common framework for exact and triangulated theories. In this setting, almost split $Y$2-extensions are equivalent, under suitable finiteness hypotheses, to Auslander-Reiten-Serre duality, i.e. to the existence of a functor $Y$3 with binatural isomorphisms

$Y$4

When enough projectives and source morphisms exist, the stable category becomes a $Y$5-category, and the Auslander-Reiten quiver controls radical layers, dimensions of Hom-spaces, and mesh-category reconstruction (Iyama et al., 2018).

The derived-categorical formulation is the Auslander-Reiten principle. For a smashing localizing subcategory $Y$6 with right adjoint $Y$7, a bounded injective complex $Y$8, and a complex $Y$9 satisfying $\tau$0, one has

$\tau$1

Classical Auslander-Reiten duality and the Iyama-Wemyss generalization emerge as corollaries, and the principle yields cases of the Auslander-Reiten conjecture over Gorenstein rings of dimension $\tau$2 with one-dimensional singular locus (Ono et al., 2018).

5. Higher Auslander-Reiten theory

Iyama’s higher Auslander-Reiten theory replaces short exact sequences by $\tau$3-exact sequences and the classical translate by

$\tau$4

For a $\tau$5-cluster-tilting subcategory $\tau$6, the higher Auslander-Reiten bijection is the bifunctorial isomorphism

$\tau$7

This higher duality governs $\tau$8-almost split sequences, the equivalence between stable and costable categories induced by $\tau$9, and the defect formula

$$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$00

for $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$01-exact sequences (Jasso et al., 2016).

In $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$02-exangulated categories, Auslander-Reiten theory takes the form of Auslander-Reiten $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$03-exangles. For an Ext-finite, Krull-Schmidt, $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$04-linear $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$05-exangulated category, the existence of Auslander-Reiten $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$06-exangles is equivalent to the existence of Auslander-Reiten-Serre duality. Under this duality, the restricted Auslander bijection becomes part of a commutative triangle of poset anti-isomorphisms connecting classes of deflations, submodules of $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$07, and submodules of $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$08 (He et al., 2021).

These higher correspondences are stable under important categorical constructions. If an $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$09-exangulated category $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$10 with enough projectives and injectives has Auslander-Reiten $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$11-exangles and $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$12 is a cluster-tilting subcategory, then the quotient $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$13 is $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$14-abelian and has Auslander-Reiten $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$15-exact sequences. If $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$16 is Frobenius $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$17-exangulated and has Auslander-Reiten $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$18-exangles, then its stable category has Auslander-Reiten $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$19-angles (He et al., 2024).

6. Related bijection theories and terminological distinctions

A closely related but distinct construction is the Auslander bijection of morphisms determined by objects. For an Artin algebra $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$20, a morphism $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$21 is right $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$22-determined if every morphism $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$23 whose compositions $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$24 with all $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$25 factor through $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$26 must itself factor through $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$27. Auslander’s bijection is the lattice isomorphism

$$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$28

from right equivalence classes of right minimal, right $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$29-determined morphisms ending in $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$30 to submodules of $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$31 (Ringel, 2013).

This theory contains a precise determiner formula. Every morphism is right determined by some module, and the minimal right determiner of $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$32 is

$$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$33

where $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$34 ranges over indecomposable direct summands of the intrinsic kernel of $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$35, and $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$36 ranges over indecomposable projectives almost factoring through $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$37. Thus the Auslander-Reiten translate enters the determination of morphisms even when the bijection under study is not itself the classical Hom-Ext duality (Ringel, 2013).

In abelian categories with Auslander-Reiten duality, universal extensions refine this picture. For $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$38 and $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$39, there is an anti-isomorphism between right equivalence classes of right minimal epimorphisms $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$40 with kernel in $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$41 and finitely generated $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$42-submodules of $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$43. Together with the duality-induced bijection to submodules of $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$44, this yields a commutative bijection triangle, and the restricted Auslander bijection at $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$45 relative to $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$46 always holds in this setting (Chen, 2013).

A recurrent source of confusion is terminological. In the works cited here, “Auslander-Reiten bijection” may refer to the classical duality $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$47, to the translation $$D\,\underline{\operatorname{Hom}_\Lambda(X,Y)} \cong \operatorname{Ext}^1_\Lambda(Y,\tau X),$$48 on isomorphism classes of indecomposables, or to the injective-projective permutation identified with Iyama’s grade bijection. By contrast, “Auslander bijection” is usually reserved for the theory of morphisms determined by objects. A plausible implication is that the phrase is best read relative to the ambient category and the invariants under discussion, rather than as a single universally fixed construction.