Auslander-Gorenstein Algebras

• Auslander-Gorenstein algebras are finite-dimensional noetherian rings that satisfy strengthened Gorenstein conditions by controlling the homological complexity of minimal injective coresolutions.
• They feature a tight synchronization among projective dimension, injective dimension, grade, and dominant dimension, with canonical bijections linking injective, projective, and simple modules.
• These algebras have broad applications in representation theory and tilting theory, supported by explicit combinatorial criteria in monomial and gentle settings.

Auslander-Gorenstein algebras are noetherian rings, and in particular finite-dimensional algebras, for which the Gorenstein finiteness condition is strengthened by systematic control over the homological complexity of the injective terms in a minimal injective coresolution. In the finite-dimensional setting, if

$0\to A_A\to I^0\to I^1\to \cdots$

is a minimal injective resolution of the regular right module, then $A$ is $n$-Gorenstein when

$\operatorname{pdim} I^i\le i \qquad (0\le i<n),$

and $A$ is Auslander-Gorenstein when this holds for all $n\ge 1$ and $\operatorname{idim}A_A<\infty$. The subject is characterized by a tight synchronization among projective dimension, injective dimension, grade, dominant dimension, and canonical bijections on injectives, projectives, and simples, and in several important classes these structures admit explicit combinatorial or matrix-theoretic descriptions (Klász, 9 Aug 2025, Klász et al., 26 Aug 2025).

1. Foundational definitions and symmetry

For two-sided noetherian rings, the starting point is the notion of an $n$-Gorenstein ring: in a minimal injective coresolution

$0\to R\to I^0\to I^1\to \cdots,$

one requires

$\operatorname{flatdim} I^i\le i \qquad (0\le i<n).$

A ring is Auslander-Gorenstein if it is $A$0-Gorenstein for all $A$1 and has finite injective dimension, while an Auslander regular ring is an Auslander-Gorenstein ring of finite global dimension. Over finite-dimensional algebras, flat dimension and projective dimension coincide on finitely generated modules, so the condition can be stated entirely in terms of projective dimensions of the $A$2 (Klász et al., 26 Aug 2025).

In the finite-dimensional case, Auslander and Reiten’s theorem implies that Auslander-Gorenstein algebras are Iwanaga-Gorenstein: $A$3 The Auslander condition is left-right symmetric, and if an algebra satisfies it, then the Gorenstein symmetry conjecture holds for that algebra: $A$4 This left-right symmetry is one reason the condition is central in noncommutative Gorenstein homological algebra (Klász, 9 Aug 2025, Klász et al., 26 Aug 2025).

A distinct but closely related convention, due to Iyama and Solberg and used extensively in finite-dimensional representation theory, calls an algebra Auslander-Gorenstein when

$A$5

and calls it higher Auslander when

$A$6

This convention isolates the range in which dominant dimension reflects the injective-side homological size of the algebra, and it leads naturally to the higher and minimal Auslander-Gorenstein theories discussed below (Marczinzik, 2017).

2. Grade, dominant dimension, and canonical bijections

The grade of a finitely generated module $A$7 is

$A$8

and dominant dimension is defined from the initial projective segment of a minimal injective coresolution. For an Iwanaga-Gorenstein algebra $A$9, the survey literature records several equivalent criteria for being Auslander-Gorenstein; the most practical is that for every indecomposable injective $n$0,

$n$1

Equivalent formulations involve grade bounds on submodules of appropriate Ext-modules, composition factors of $n$2 for simple $n$3, and monomorphism preservation under double Ext (Klász et al., 26 Aug 2025).

A sharper simple-module formulation was established recently: a finite-dimensional algebra $n$4 is Auslander-Gorenstein if and only if for every simple module $n$5,

$n$6

where $n$7 is the injective envelope of $n$8. This characterization packages the Auslander condition into a direct comparison between the first nonvanishing Ext from a simple and the projective dimension of its injective hull (Klász et al., 16 Jan 2025).

Auslander and Reiten also defined the canonical bijection underlying the theory. For an $n$9-Gorenstein algebra, indecomposable injectives of projective dimension $\operatorname{pdim} I^i\le i \qquad (0\le i<n),$0 correspond to indecomposable projectives of injective dimension $\operatorname{pdim} I^i\le i \qquad (0\le i<n),$1 via

$\operatorname{pdim} I^i\le i \qquad (0\le i<n),$2

For an Auslander-Gorenstein algebra these combine to the Auslander-Reiten bijection

$\operatorname{pdim} I^i\le i \qquad (0\le i<n),$3

with

$\operatorname{pdim} I^i\le i \qquad (0\le i<n),$4

On simple modules, Iyama’s grade bijection is

$\operatorname{pdim} I^i\le i \qquad (0\le i<n),$5

or, in the later formulation,

$\operatorname{pdim} I^i\le i \qquad (0\le i<n),$6

A central theorem is that the Auslander-Reiten permutation and the grade permutation coincide; thus the syzygy-theoretic bijection on injectives/projectives and the grade-theoretic bijection on simples encode the same permutation (Klász et al., 26 Aug 2025, Klász et al., 16 Jan 2025).

For Auslander regular algebras, the same permutation is visible in linear algebra. If $\operatorname{pdim} I^i\le i \qquad (0\le i<n),$7 denotes the Coxeter matrix, then with an admissible ordering of simples its Bruhat decomposition has the form

$\operatorname{pdim} I^i\le i \qquad (0\le i<n),$8

where $\operatorname{pdim} I^i\le i \qquad (0\le i<n),$9 is the permutation matrix of the grade bijection and $A$0 is upper triangular with diagonal entries $A$1 or $A$2. One consequence is that

$A$3

This gives a purely matrix-theoretic access to a homological invariant (Klász et al., 16 Jan 2025).

3. Higher and minimal Auslander-Gorenstein structures

The higher-dimensional variant uses the inequality

$A$4

An Artin algebra satisfying this is called $A$5-minimal Auslander-Gorenstein, or higher Auslander-Gorenstein. Such algebras are either self-injective or satisfy

$A$6

They form the Gorenstein analogue of higher Auslander algebras, where finite global dimension replaces finite injective dimension (Li et al., 2018, Bragg et al., 2024).

The structural correspondence for this class is the bijection between Morita-equivalence classes of $A$7-minimal Auslander-Gorenstein algebras and equivalence classes of finite $A$8-precluster tilting subcategories. If $A$9 is a finite $n\ge 1$0-precluster tilting module over $n\ge 1$1, then

$n\ge 1$2

is $n\ge 1$3-minimal Auslander-Gorenstein; conversely, every such algebra arises in this way. The associated category

$n\ge 1$4

is Frobenius, its stable category carries a higher Auslander-Reiten theory, and it is dual to the maximal Cohen-Macaulay category of the endomorphism algebra (Iyama et al., 2016).

Within this higher framework, the relation between socles and Gorenstein projective dimension becomes unusually rigid. For a minimal $n\ge 1$5-Auslander-Gorenstein algebra $n\ge 1$6,

$n\ge 1$7

Injective modules are projective precisely when the socle has Gorenstein projective dimension at most $n\ge 1$8. These criteria make the socle a detector of homological complexity in the higher Auslander-Gorenstein setting (Li et al., 2018).

A further enlargement is the class of dominant Auslander-Gorenstein algebras. These are Iwanaga-Gorenstein algebras such that for every indecomposable projective $n\ge 1$9,

$\operatorname{idim}A_A<\infty$0

This class contains both higher Auslander algebras and minimal Auslander-Gorenstein algebras. It is characterized by a bijection with mixed precluster tilting modules, and in the finite global-dimension case the corresponding notion is dominant Auslander-regular, characterized by mixed cluster tilting modules (Chan et al., 2022).

A categorical characterization of higher Auslander-Gorenstein algebras is also available. For an Artin algebra $\operatorname{idim}A_A<\infty$1, being $\operatorname{idim}A_A<\infty$2-minimal Auslander-Gorenstein is equivalent to the abelianness of $\operatorname{idim}A_A<\infty$3, together with the additional conditions

$\operatorname{idim}A_A<\infty$4

The corresponding statement for all modules uses $\operatorname{idim}A_A<\infty$5 in place of the finitely generated subcategory (Bragg et al., 2024).

4. Classified families and combinatorial models

Among finite-dimensional examples, monomial algebras exhibit a particularly rigid form of the Auslander-Gorenstein property. Every Auslander-Gorenstein monomial algebra is a string algebra. The $\operatorname{idim}A_A<\infty$6-Gorenstein monomial algebras admit a purely combinatorial classification: the quiver must be biserial, degree-$\operatorname{idim}A_A<\infty$7 in- and out-vertices must coincide, degree-$\operatorname{idim}A_A<\infty$8 vertices must satisfy a specific cross-relation pattern, and every arrow occurring inside a minimal relation must also occur as a first or last arrow of some minimal relation. In the gentle case the criterion simplifies drastically: $\operatorname{idim}A_A<\infty$9 For gentle algebras, Auslander-Gorenstein is moreover equivalent to being $n$0-Gorenstein, and the Auslander-Reiten bijection can be written explicitly as a permutation of vertices determined by local path combinatorics (Klász, 9 Aug 2025).

The same paper establishes a reduction theorem for monomial algebras. Starting from a $n$1-Gorenstein monomial algebra, one cuts each degree-$n$2 vertex into two vertices in a way that produces a new monomial algebra $n$3, typically closer to Nakayama type. The relevant homological properties are preserved: $n$4

$n$5

and

$n$6

Hence the classification of Auslander-Gorenstein monomial algebras reduces to the classification of Auslander-Gorenstein Nakayama algebras. The same analysis proves, for monomial algebras,

$n$7

and also shows that

$n$8

If a monomial algebra with $n$9 simple modules is $0\to R\to I^0\to I^1\to \cdots,$0-Gorenstein, then it is already $0\to R\to I^0\to I^1\to \cdots,$1-Iwanaga-Gorenstein (Klász, 9 Aug 2025).

For $0\to R\to I^0\to I^1\to \cdots,$2-Gorenstein acyclic monomial algebras, Auslander regularity can be detected from the Coxeter matrix. Under a natural labelling,

$0\to R\to I^0\to I^1\to \cdots,$3

In the linear Nakayama case this is equivalent to bijectivity of the inverse Auslander-Reiten map, and the Coxeter permutation coincides with the Auslander-Reiten permutation. For cyclic and linear Nakayama algebras, resolution-quiver methods give a separate criterion for minimal Auslander-Gorensteinness, in which the parity of the selfinjective dimension determines whether blackness conditions are imposed on cyclic vertices themselves or on their $0\to R\to I^0\to I^1\to \cdots,$4-shifts (Klász et al., 2 Apr 2026, Shen, 18 Nov 2025).

Incidence algebras provide a second highly explicit family. A finite lattice $0\to R\to I^0\to I^1\to \cdots,$5 is distributive if and only if its incidence algebra $0\to R\to I^0\to I^1\to \cdots,$6 is Auslander regular. In that case

$0\to R\to I^0\to I^1\to \cdots,$7

and the Auslander-Reiten permutation coincides with rowmotion on the lattice. The Coxeter permutation, the grade permutation, and rowmotion therefore agree. Equivalently, distributivity of a finite lattice can be characterized by a Bruhat factorization of the Coxeter matrix with trivial left factor, or by the existence of a factorization $0\to R\to I^0\to I^1\to \cdots,$8 with $0\to R\to I^0\to I^1\to \cdots,$9 a permutation matrix and $\operatorname{flatdim} I^i\le i \qquad (0\le i<n).$0 upper triangular (Klász et al., 26 Aug 2025, Klász et al., 16 Jan 2025).

The same synchronization appears in category $\operatorname{flatdim} I^i\le i \qquad (0\le i<n).$1. Blocks of category $\operatorname{flatdim} I^i\le i \qquad (0\le i<n).$2 are Auslander regular; if $\operatorname{flatdim} I^i\le i \qquad (0\le i<n).$3 is the indecomposable injective corresponding to a Weyl group element $\operatorname{flatdim} I^i\le i \qquad (0\le i<n).$4, then

$\operatorname{flatdim} I^i\le i \qquad (0\le i<n).$5

so the set of grades in a regular block is the set of values of $\operatorname{flatdim} I^i\le i \qquad (0\le i<n).$6. Because these blocks have a simple-preserving duality, their Auslander-Reiten permutation is the identity (Klász et al., 26 Aug 2025).

5. Tilting, relative theories, Gabriel topologies, and Auslander-Yoneda algebras

Tilting theory interacts strongly with Auslander-Gorenstein conditions. For an algebra $\operatorname{flatdim} I^i\le i \qquad (0\le i<n).$7 and $\operatorname{flatdim} I^i\le i \qquad (0\le i<n).$8, the poset $\operatorname{flatdim} I^i\le i \qquad (0\le i<n).$9 of tilting modules of projective dimension at most $A$00 has a minimal element exactly when $A$01 is contravariantly finite. Under the hypotheses

$A$02

the module

$A$03

is a tilting module of projective dimension at most $A$04, is the minimum element in $A$05, and satisfies

$A$06

In particular, every quasi $A$07-Gorenstein algebra ունի such an explicit minimum tilting module. For $A$08-Gorenstein algebras, if $A$09 is an additive generator of the projective-injective modules and $A$10, then

$A$11

is a bijection (Iyama et al., 2018).

A relative version replaces the projective-injective generator by a self-orthogonal module $A$12. A pair $A$13 is a relative $A$14-Auslander-Gorenstein pair when $A$15, $A$16 is $A$17-Iwanaga-Gorenstein, and

$A$18

The basic structural theorem says that, under suitable bounds on $A$19 and $A$20, this is equivalent to the existence of a unique basic module $A$21 that is simultaneously $A$22-tilting and $A$23-cotilting and has prescribed lower bounds on its relative dominant and codominant dimensions with respect to $A$24. This relative theory generalizes classical Auslander pairs and recovers minimal Auslander-Gorenstein algebras as a special case (Cruz et al., 2023).

Higher Auslander-Gorenstein algebras also admit a Gabriel-topological characterization. If $A$25 is $A$26-minimal Auslander-Gorenstein and $A$27 is the maximal injective summand of $A$28, then

$A$29

and the pair

$A$30

is a hereditary torsion pair cogenerated by $A$31. Equivalently,

$A$32

where the right-hand side is the category of $A$33-closed modules for the Gabriel topology induced by $A$34. This yields higher Auslander-Bridger and higher Auslander-Buchsbaum-Serre type characterizations (Keshavarz et al., 2024).

A different enlargement of the subject is obtained from Auslander-Yoneda algebras. If $A$35 is representation-finite of finite global dimension and

$A$36

then $A$37 is Auslander-Gorenstein if and only if $A$38 is bispherical, meaning that every indecomposable left and right $A$39-module is spherical in the sense of Auslander and Bridger. Spherical algebras are characterized by a split torsion pair

$A$40

and in the Nakayama case the bispherical algebras are exactly the truncated linearly oriented $A$41-algebras $A$42. Replicated algebras of hereditary algebras furnish large families of such examples (Bauwens et al., 9 Jun 2026).

6. Conjectures, extensions, and broader context

A major conjectural theme is the Auslander-Reiten Conjecture: if an algebra satisfies the Auslander condition, then it should be Auslander-Gorenstein. The survey literature places this conjecture between the Generalised Nakayama Conjecture and the Nakayama Conjecture: the Generalised Nakayama Conjecture implies the Auslander-Reiten Conjecture, and the Auslander-Reiten Conjecture implies the Nakayama Conjecture. In the monomial setting, the converse direction has now been proved: a monomial algebra is Auslander-Gorenstein exactly when its Auslander-Reiten map is well-defined and bijective (Klász et al., 26 Aug 2025, Klász, 9 Aug 2025).

A related reduction uses weakly Gorenstein conditions. For an Artin algebra satisfying the Auslander condition, the following are equivalent: $A$43 Equivalently,

$A$44

This reformulates the Auslander-Reiten Conjecture as a weak-Gorenstein detection problem and yields new cases in which the conjecture holds (Huang, 2024).

The theory also extends beyond ordinary rings. For directed algebras, especially lower $A$45-algebras, a directed analogue of the Auslander-Gorenstein package has been developed using an $A$46-functor adapted to quotient categories, a double Ext spectral sequence, and a grade-dimension formula

$A$47

This framework yields equidimensionality of characteristic varieties for homogeneous and simple objects and applies to quantizations of symplectic resolutions and to type $A$48 Cherednik algebras (Gordon et al., 2013).

Another systematic source of examples comes from Serre-formal algebras and their replicated algebras. If $A$49 is Serre-formal, then the replicated algebras $A$50 have explicitly computable dominant and self-injective dimensions, are automatically Iwanaga-Gorenstein, and are minimal Auslander-Gorenstein exactly when

$A$51

for some integer $A$52. There are infinitely many minimal Auslander-Gorenstein algebras in the replicated family if and only if the base algebra is twisted fractionally Calabi-Yau. Under an additional Hom-vanishing condition, the same statements transfer to Yamagata’s SGC extensions (Chan et al., 2017).

In this sense, Auslander-Gorenstein algebras form a nexus where homological algebra, higher Auslander-Reiten theory, tilting theory, matrix invariants, and explicit combinatorics converge. The modern literature shows that the condition is simultaneously rigid enough to admit sharp characterizations and broad enough to persist under relative, higher, and geometric generalizations (Klász et al., 26 Aug 2025).