Auslander-Gorenstein Monomial Algebras

• Auslander-Gorenstein monomial algebras are finite-dimensional algebras defined by minimal injective coresolutions satisfying Auslander’s n-Gorenstein inequalities and finite injective dimension.
• Their structure is governed by combinatorial restrictions on quivers and relations that enforce string algebra behavior and enable classification via perfect path techniques.
• They refine Iwanaga-Gorenstein algebras by connecting homological invariants with combinatorial reductions to gentle and Nakayama algebras, impacting studies in Calabi–Yau tilting.

Auslander-Gorenstein monomial algebras are finite-dimensional monomial algebras $A=kQ/I$ for which the minimal injective coresolution of the regular module satisfies Auslander’s $n$-Gorenstein inequalities in all degrees and $A$ has finite injective dimension. In the finite-dimensional setting, they form a homological refinement of Iwanaga-Gorenstein algebras, but in the monomial case the condition is also markedly combinatorial: it constrains the quiver and relations, forces string-algebra behavior, is detected by a bijective Auslander-Reiten map, and in the acyclic Auslander regular case can be read from a Bruhat factorization of the Coxeter matrix (Klász, 9 Aug 2025, Klász et al., 26 Aug 2025, Klász et al., 2 Apr 2026).

1. Homological framework and finite-dimensional criteria

For a finite-dimensional algebra $A$, one fixes a minimal injective coresolution

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

The algebra is $n$-Gorenstein if

$$\operatorname{pdim} I^i \le i \qquad \text{for all } 0\le i<n.$$

It is Auslander-Gorenstein if it is $n$-Gorenstein for all $n\ge 1$ and $\operatorname{idim}A_A<\infty$. It is Auslander regular if it is Auslander-Gorenstein and has finite global dimension. In parallel, $n$0 is Iwanaga-Gorenstein if

$n$1

For finite-dimensional algebras, Auslander’s left-right symmetry implies

$n$2

so Auslander-Gorenstein is left-right symmetric as well (Klász et al., 26 Aug 2025, Klász, 9 Aug 2025).

A standard numerical invariant is the grade

$n$3

For an Iwanaga-Gorenstein algebra $n$4, a central criterion says that $n$5 is Auslander-Gorenstein if and only if for every indecomposable injective $n$6-module $n$7,

$n$8

The survey literature also records several equivalent Ext-theoretic formulations, and emphasizes that the Auslander-Reiten permutation on simples coincides with Iyama’s grade permutation for Auslander-Gorenstein algebras (Klász et al., 26 Aug 2025).

In the monomial setting, the decisive homological characterization is given by the Auslander-Reiten map. If every indecomposable injective $n$9 has a finite minimal projective resolution whose last non-zero term is indecomposable, then

$A$0

is well-defined. For monomial algebras,

$A$1

This confirms a conjecture of Marczinzik for monomial algebras and provides a genuinely homological characterization inside a class usually governed by quiver combinatorics (Klász, 9 Aug 2025, Klász et al., 26 Aug 2025).

2. Gorenstein-projective modules and the perfect-path mechanism

The combinatorial control of monomial homological algebra begins with the bound-quiver presentation

$A$2

where $A$3 is finite and $A$4 is an admissible monomial ideal generated by paths of length at least two. Writing $A$5 for the finite set of minimal paths generating $A$6, a path is nonzero in $A$7 precisely when it contains no subpath from $A$8. These nonzero paths form a $A$9-basis of $A$0. For any nonzero path $A$1, the cyclic module $A$2 is the basic object in the theory (Chen et al., 2015).

The key combinatorial notion is that of a perfect pair $A$3 of nonzero paths. Such a pair is composable, satisfies $A$4 in $A$5, and realizes $A$6 and $A$7 as the unique minimal annihilators of one another. A path $A$8 is perfect if it lies in a cycle of perfect pairs

$A$9

The central classification theorem states that there is a bijection

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

sending a perfect path $0\to A_A \to I^0 \to I^1 \to \cdots .$1 to the module $0\to A_A \to I^0 \to I^1 \to \cdots .$2. Thus every indecomposable non-projective Gorenstein-projective module over a monomial algebra is isomorphic to $0\to A_A \to I^0 \to I^1 \to \cdots .$3 for a unique perfect path $0\to A_A \to I^0 \to I^1 \to \cdots .$4, and every perfect path yields such a module (Chen et al., 2015).

This classification is accompanied by explicit syzygy formulas. For a nonzero nontrivial path $0\to A_A \to I^0 \to I^1 \to \cdots .$5, if $0\to A_A \to I^0 \to I^1 \to \cdots .$6 denotes the set of right-minimal paths $0\to A_A \to I^0 \to I^1 \to \cdots .$7 with $0\to A_A \to I^0 \to I^1 \to \cdots .$8, then there is an exact sequence

$0\to A_A \to I^0 \to I^1 \to \cdots .$9

hence

$n$0

If $n$1 is a perfect pair, then $n$2 and

$n$3

In this sense, perfect pairs are exactly the configurations in which syzygy rotates between cyclic path modules (Chen et al., 2015).

For quadratic monomial algebras, the description becomes even sharper. Every perfect path is an arrow, and the relation quiver $n$4 has vertices given by arrows of $n$5, with an arrow $n$6 whenever $n$7. An arrow is perfect if and only if it lies in a connected component of $n$8 that is a basic cycle. If the perfect components are $n$9 with $$\operatorname{pdim} I^i \le i \qquad \text{for all } 0\le i<n.$$0 vertices, then

$$\operatorname{pdim} I^i \le i \qquad \text{for all } 0\le i<n.$$1

so the Gorenstein-projective stable category is semisimple triangulated. In the same quadratic setting,

$$\operatorname{pdim} I^i \le i \qquad \text{for all } 0\le i<n.$$2

$$\operatorname{pdim} I^i \le i \qquad \text{for all } 0\le i<n.$$3 is CM-free exactly when $$\operatorname{pdim} I^i \le i \qquad \text{for all } 0\le i<n.$$4 has no perfect component, and $$\operatorname{pdim} I^i \le i \qquad \text{for all } 0\le i<n.$$5 has finite global dimension exactly when every component is acyclic. These criteria are not yet Auslander-Gorenstein criteria, but they supply the Gorenstein-projective infrastructure on which later Auslander-Gorenstein results rest (Chen et al., 2015).

3. Structural restrictions: string algebras, gentle algebras, and 2-Gorenstein shape

A major rigidity theorem states: $$\operatorname{pdim} I^i \le i \qquad \text{for all } 0\le i<n.$$6 This is one of the strongest structural restrictions presently known for the monomial class. Since string algebras are monomial biserial algebras with the usual local uniqueness conditions on relations, the Auslander-Gorenstein property forces a substantial collapse from general monomial combinatorics to the string-algebra regime (Klász, 9 Aug 2025, Klász et al., 26 Aug 2025).

Within string algebras, the gentle case admits a complete combinatorial criterion. If $$\operatorname{pdim} I^i \le i \qquad \text{for all } 0\le i<n.$$7 is gentle, then

$$\operatorname{pdim} I^i \le i \qquad \text{for all } 0\le i<n.$$8

Thus, in the gentle class, Auslander-Gorensteinness is purely local: a vertex has in-degree $$\operatorname{pdim} I^i \le i \qquad \text{for all } 0\le i<n.$$9 exactly when it has out-degree $n$0. The same work also shows that for gentle algebras,

$n$1

This is a particularly clean identification of Auslander-Gorenstein and low Gorenstein dimension inside a monomial subclass (Klász, 9 Aug 2025).

The paper further gives an explicit description of the Auslander-Reiten bijection $n$2 for Auslander-Gorenstein gentle algebras. Vertices of total degree $n$3 or $n$4 are fixed. In the remaining cases, $n$5 is determined by the source of a unique left-maximal path or the target of a unique maximal critical path, depending on the local in/out-degree configuration. This turns the abstract homological bijection into a concrete permutation of vertices in the gentle case (Klász, 9 Aug 2025).

The broader 2-Gorenstein monomial class is also classified by local quiver-relation data. A monomial algebra $n$6 is $n$7-Gorenstein if and only if $n$8 is biserial, every vertex satisfies

$n$9

every degree-$n\ge 1$0 vertex satisfies the specific crossing pattern

$n\ge 1$1

with $n\ge 1$2 not contained in any minimal relations, and any arrow appearing inside a minimal relation is either its start or its end. This theorem is the geometric starting point for the reduction theory of Auslander-Gorenstein monomial algebras (Klász, 9 Aug 2025).

4. Reduction to Nakayama algebras and quantitative consequences

The central technical innovation in the current classification program is a cutting procedure at degree-$n\ge 1$3 vertices. Starting from a 2-Gorenstein monomial algebra and a degree-$n\ge 1$4 vertex $n\ge 1$5 with incoming arrows $n\ge 1$6 and outgoing arrows $n\ge 1$7 satisfying

$n\ge 1$8

one replaces $n\ge 1$9 by two degree-$\operatorname{idim}A_A<\infty$0 vertices $\operatorname{idim}A_A<\infty$1 and reconnects

$\operatorname{idim}A_A<\infty$2

leaving all other vertices, arrows, and relations unchanged. Repeated cutting removes branching and eventually transforms the algebra into a Nakayama algebra (Klász, 9 Aug 2025).

This procedure preserves the relevant homological conditions. For the cut pair $\operatorname{idim}A_A<\infty$3, the theorem states that for every $\operatorname{idim}A_A<\infty$4,

$\operatorname{idim}A_A<\infty$5

for every $\operatorname{idim}A_A<\infty$6,

$\operatorname{idim}A_A<\infty$7

and moreover

$\operatorname{idim}A_A<\infty$8

As a consequence,

$\operatorname{idim}A_A<\infty$9

The survey literature emphasizes the significance of this reduction and also notes that a full classification of Auslander-Gorenstein Nakayama algebras remains open, although several important subclasses have been analyzed (Klász, 9 Aug 2025, Klász et al., 26 Aug 2025).

Two further quantitative results sharpen the homological picture. First, monomial algebras satisfy a parity phenomenon: $n$00 This extends an earlier result of Iwanaga and Fuller from the Nakayama case to all monomial algebras (Klász, 9 Aug 2025, Klász et al., 26 Aug 2025).

Second, if $n$01 is a monomial algebra with $n$02 simple modules and $n$03 is $n$04-Gorenstein, then

$n$05

so $n$06 is $n$07-Iwanaga-Gorenstein. This is described as a stronger version of the Auslander-Reiten Conjecture in the monomial case. The survey presents it as a strong partial result toward the general conjectural picture (Klász, 9 Aug 2025, Klász et al., 26 Aug 2025).

The Nakayama endpoint of the reduction also carries independent combinatorics. The survey records that linear Nakayama algebras correspond to Dyck paths, that $n$08-Gorenstein linear Nakayama algebras were characterized by Dyck paths with no double deficiencies, and that these are counted by Motzkin paths. This suggests that the unresolved part of the general monomial classification is concentrated in an already highly structured uniserial environment (Klász et al., 26 Aug 2025).

5. Auslander-Reiten permutations, Coxeter matrices, and the acyclic Auslander regular case

The acyclic case admits a linear-algebraic classification of Auslander regularity. For a finite-dimensional algebra of finite global dimension, the Coxeter matrix is

$n$09

where $n$10 is the Cartan matrix. The relevant matrix factorization is the Bruhat decomposition

$n$11

with $n$12 upper triangular and $n$13 a permutation matrix. When $n$14, the associated permutation is called the Coxeter permutation (Klász et al., 2 Apr 2026).

For acyclic quiver algebras, the theory is formulated under a natural labelling of vertices, meaning $n$15 whenever there is a path from $n$16 to $n$17 with $n$18. Under the hypotheses used in the paper, a naturally labelled acyclic algebra with a well-defined inverse Auslander-Reiten map and property $n$19 satisfies: $n$20 In this case, the Coxeter permutation is the inverse of $n$21. Moreover, if $n$22 is Auslander regular and naturally labelled, then these conditions hold (Klász et al., 2 Apr 2026).

Specialized to monomial algebras, the main acyclic theorem states that if $n$23 is a $n$24-Gorenstein, acyclic monomial algebra with a natural labelling, then the following are equivalent:

1. $n$25 is Auslander regular.
2. There exists a Bruhat decomposition

$n$26

with $n$27.

In this case, the Coxeter permutation coincides with the Auslander-Reiten permutation. For linear Nakayama algebras with canonical ordering $n$28, this criterion is equivalent to bijectivity of the inverse Auslander-Reiten map as well (Klász et al., 2 Apr 2026).

The proof uses the Euler-form formula

$n$29

which identifies the first nonzero position in each row of $n$30 with the inverse Auslander-Reiten permutation. The survey places this in a broader framework: for Auslander regular algebras with an admissible ordering of simples, the Coxeter permutation coincides with the Auslander-Reiten permutation, so the matrix-theoretic permutation is a genuine homological invariant (Klász et al., 2 Apr 2026, Klász et al., 26 Aug 2025).

This line of work also resolves several questions in special families. For linear Nakayama algebras, Ringel’s homological permutation agrees with the Coxeter permutation, and for any Nakayama algebra and any simple module $n$31,

$n$32

At the same time, the limits of the matrix criterion are explicit: the Bruhat criterion with $n$33 does not extend verbatim to arbitrary cyclic Nakayama algebras, and there are acyclic monomial algebras that are not $n$34-Gorenstein for which the matrix criterion can fail even when the Coxeter matrix exhibits favorable Bruhat behavior (Klász et al., 2 Apr 2026).

6. Related Calabi–Yau and weak Gorenstein regimes

A separate but closely related direction studies monomial Gorenstein algebras through their singularity categories. Over an algebraically closed field of characteristic zero, for a monomial algebra $n$35, the following are equivalent:

1. $n$36 is $n$37-Calabi–Yau tilted.
2. $n$38 is $n$39-Iwanaga–Gorenstein and its singularity category $n$40 is $n$41-Calabi–Yau.
3. $n$42 is Jacobian.

This is a converse to the Keller–Reiten theorem in the monomial setting. For $n$43-Iwanaga–Gorenstein monomial algebras, the minimal zero-relations decompose into disjoint families $n$44 arising from cyclic paths $n$45, and the singularity category has the form

$n$46

If $n$47 is $n$48-Calabi–Yau, then

$n$49

and these numerical conditions yield a potential

$n$50

showing that $n$51 is Jacobian (Elsener, 2018).

In the gentle subclass, the same numerical restriction becomes especially explicit. If a gentle algebra is $n$52-CY tilted, then every relation lies on either a saturated $n$53-cycle or a saturated loop. In the $n$54-angulated families, the resulting algebras are gentle, the only possible saturated cycles are $n$55-cycles, there are at most $n$56 consecutive zero-relations outside a saturated cycle, the algebras are $n$57-Iwanaga–Gorenstein, and

$n$58

These results concern monomial Gorenstein and stably Calabi–Yau behavior rather than Auslander-Gorensteinness itself, but they show that monomial homological rigidity often emerges from numerical cycle data in the singularity category (Elsener, 2018).

A distinct weaker condition is weak Gorensteinness. One abstract states that algebras are left weakly Gorenstein in case the subcategory

$n$59

is representation-finite, and that this applies in particular to all monomial algebras (Marczinzik, 2019). This is substantially weaker than being Auslander-Gorenstein. A common misconception is to identify broad Gorenstein-flavored regularity phenomena in monomial algebras with the Auslander-Gorenstein condition. The current literature instead indicates a hierarchy: perfect-path and relation-quiver methods describe Gorenstein-projective structure; weakly Gorenstein conditions impose Ext-syzygy regularity; $n$60-Iwanaga–Gorenstein plus stably $n$61-CY forces Jacobian and $n$62-CY tilted behavior in the monomial case; and Auslander-Gorensteinness is the stronger condition singled out by the bijective Auslander-Reiten map, the string-algebra restriction, and the reduction to Nakayama algebras (Chen et al., 2015, Elsener, 2018, Marczinzik, 2019, Klász, 9 Aug 2025).

The present picture is therefore both sharp and incomplete. The monomial Auslander-Gorenstein condition is now characterized homologically and strongly constrained combinatorially; gentle algebras are classified, the acyclic Auslander regular case is controlled by Coxeter-matrix Bruhat factorization, and the general problem reduces to Nakayama algebras. At the same time, the survey literature makes clear that a full classification of Auslander-Gorenstein Nakayama algebras remains open, so the reduction theorem isolates, rather than eliminates, the remaining core difficulty (Klász et al., 26 Aug 2025).