Auslander-Type Correspondence

• Auslander-type correspondence is a framework that encodes a category’s structure via the endomorphism algebra of a distinguished object, with strong homological bounds.
• In higher settings, d-cluster-tilting modules or subcategories replace additive generators, leading to d-Auslander algebras defined by dominant dimension and bounded global dimension.
• Generalizations extend the framework to exact, triangulated, and DG categories, showcasing its versatility in capturing homological and categorical properties.

Searching arXiv for recent and foundational papers on Auslander-type correspondences to ground the article.

Auslander-type correspondence is a family of classification results in representation theory and higher homological algebra in which a category, algebra, or exact or triangulated structure with strong finiteness or rigidity properties is related to an endomorphism algebra of a distinguished generator, generator-cogenerator, cluster-tilting object, or DG object. The prototype is Auslander’s correspondence for representation-finite Artin algebras: if $M$ is an additive generator of $\mathrm{mod}\,\Lambda$, then $\Gamma=\operatorname{End}_\Lambda(M)$ satisfies $$\operatorname{gl.dim}\Gamma \le 2 \le \operatorname{dom.dim}\Gamma$$, and conversely algebras with these homological bounds arise in this way up to Morita equivalence. Later developments replace $M$ by $d$-cluster-tilting, $n$-precluster tilting, $[1]$-additive, or $d\mathbb Z$-cluster-tilting objects, and replace classical Auslander algebras by higher, Gorenstein, exact, triangulated, or DG counterparts (McMahon, 2020, Jasso et al., 2023).

1. Classical paradigm

For a representation-finite Artin algebra $\Lambda$, with additive generator

$\mathrm{mod}\,\Lambda$0

the Auslander algebra is

$\mathrm{mod}\,\Lambda$1

Auslander showed that $\mathrm{mod}\,\Lambda$2 has $\mathrm{mod}\,\Lambda$3 and $\mathrm{mod}\,\Lambda$4, and that, up to Morita equivalence, these homological inequalities characterize precisely the algebras arising from representation-finite Artin algebras in this manner (McMahon, 2020).

Dominant dimension is defined from a minimal injective resolution

$\mathrm{mod}\,\Lambda$5

as the largest $\mathrm{mod}\,\Lambda$6 such that $\mathrm{mod}\,\Lambda$7 are projective-injective. This invariant is the classical marker of the “Auslander side” of the correspondence, while bounded global dimension records the shortness of projective resolutions on the endomorphism-ring side (McMahon, 2020).

The conceptual content of the classical correspondence is that a finite module category can be encoded by the endomorphism ring of a basic additive generator, and that the image of this encoding is described purely by homological conditions on the endomorphism ring. This pattern persists throughout later generalizations.

2. Higher Auslander correspondence

Iyama’s higher Auslander correspondence replaces additive generators by $\mathrm{mod}\,\Lambda$8-cluster-tilting modules or subcategories. A functorially finite subcategory $\mathrm{mod}\,\Lambda$9 is $\Gamma=\operatorname{End}_\Lambda(M)$0-cluster-tilting if

$\Gamma=\operatorname{End}_\Lambda(M)$1

If $\Gamma=\operatorname{End}_\Lambda(M)$2, then $\Gamma=\operatorname{End}_\Lambda(M)$3 is a $\Gamma=\operatorname{End}_\Lambda(M)$4-cluster-tilting module (McMahon, 2020).

For such an $\Gamma=\operatorname{End}_\Lambda(M)$5, the endomorphism algebra

$\Gamma=\operatorname{End}_\Lambda(M)$6

is a $\Gamma=\operatorname{End}_\Lambda(M)$7-Auslander algebra in the sense that

$\Gamma=\operatorname{End}_\Lambda(M)$8

Conversely, if $\Gamma=\operatorname{End}_\Lambda(M)$9 is a $$\operatorname{gl.dim}\Gamma \le 2 \le \operatorname{dom.dim}\Gamma$$0-Auslander algebra and $$\operatorname{gl.dim}\Gamma \le 2 \le \operatorname{dom.dim}\Gamma$$1 is a minimal projective-injective generator, then $$\operatorname{gl.dim}\Gamma \le 2 \le \operatorname{dom.dim}\Gamma$$2 admits a $$\operatorname{gl.dim}\Gamma \le 2 \le \operatorname{dom.dim}\Gamma$$3-cluster-tilting subcategory, and the correspondence becomes a bijection between equivalence classes of $$\operatorname{gl.dim}\Gamma \le 2 \le \operatorname{dom.dim}\Gamma$$4-cluster-tilting modules and Morita-equivalence classes of $$\operatorname{gl.dim}\Gamma \le 2 \le \operatorname{dom.dim}\Gamma$$5-Auslander algebras (McMahon, 2020).

A concise proof of this correspondence uses the homological theory of idempotent ideals due to Auslander–Platzeck–Todorov. In that approach, a projective-injective generator $$\operatorname{gl.dim}\Gamma \le 2 \le \operatorname{dom.dim}\Gamma$$6 defines an idempotent ideal $$\operatorname{gl.dim}\Gamma \le 2 \le \operatorname{dom.dim}\Gamma$$7, and the functor $$\operatorname{gl.dim}\Gamma \le 2 \le \operatorname{dom.dim}\Gamma$$8 transfers Ext-vanishing from $$\operatorname{gl.dim}\Gamma \le 2 \le \operatorname{dom.dim}\Gamma$$9 to $M$0, thereby converting higher dominant dimension on $M$1 into the higher Ext-orthogonality that defines a $M$2-cluster-tilting subcategory (McMahon, 2020).

3. Exact and dualizing categorical generalizations

The higher correspondence extends beyond module categories over Artin algebras. For a commutative artinian ring $M$3, Iyama and Jasso showed that, for a dualizing $M$4-variety $M$5, the following are equivalent: $M$6 is $M$7-abelian, $M$8 is a $M$9-Auslander dualizing $d$0-variety with $d$1, and $d$2 is equivalent to a $d$3-cluster-tilting subcategory of $d$4 for some dualizing $d$5-variety $d$6 (Iyama et al., 2016).

In Quillen exact categories, Ebrahimi and Nasr-Isfahani introduced $d$7-Auslander exact categories and proved a bijection between equivalence classes of $d$8-cluster-tilting subcategories of exact categories and equivalence classes of $d$9-Auslander exact categories. A further exact-categorical analogue of the higher Auslander–Solberg correspondence identifies $n$0-precluster tilting subcategories with $n$1-minimal Auslander–Gorenstein exact categories via the category of admissibly presented functors $n$2 and localization by $n$3 (Ebrahimi et al., 2021, Grevstad, 2022).

These results preserve the same formal pattern while changing the ambient category. The distinguished object need not be a single module; it may be a functorially finite subcategory, and the algebraic side may be an exact category rather than a ring.

Setting	Distinguished object	Corresponding algebraic side
Representation-finite Artin algebra	additive generator $n$4	Auslander algebra $n$5 with $n$6
Higher module theory	$n$7-cluster-tilting module or subcategory	$n$8-Auslander algebra with $n$9
Dualizing $[1]$0-varieties	$[1]$1-cluster-tilting subcategory	$[1]$2-Auslander dualizing $[1]$3-variety
Exact categories	$[1]$4-cluster tilting or $[1]$5-precluster tilting subcategory	$[1]$6-Auslander exact category or $[1]$7-minimal Auslander–Gorenstein exact category

The passage from abelian to exact or dualizing settings shows that Auslander-type correspondence is not restricted to finite module categories. It also provides intrinsic characterizations of higher exactness, such as $[1]$8-abelianity, in terms of endomorphism-theoretic or exact-categorical data.

4. Triangulated and derived correspondences

For triangulated categories, Amiot, Guo, and Keller’s line of development yields two triangulated analogues. If $[1]$9 is a finite $d\mathbb Z$0-linear, Hom-finite, idempotent-complete triangulated category with additive generator $d\mathbb Z$1, then its Auslander algebra

$d\mathbb Z$2

is characterized by twisted $d\mathbb Z$3-periodicity. More precisely, finite triangulated categories with additive generators correspond to finite-dimensional algebras $d\mathbb Z$4 that are self-injective and satisfy

$d\mathbb Z$5

for some automorphism $d\mathbb Z$6. In the $d\mathbb Z$7-finite setting, with a $d\mathbb Z$8-additive generator $d\mathbb Z$9, the graded $\Lambda$0-Auslander algebra

$\Lambda$1

is characterized as a $\Lambda$2-twisted $\Lambda$3-periodic graded algebra; algebraic $\Lambda$4-finite triangulated categories are classified by Dynkin diagrams of type $\Lambda$5 (Hanihara, 2018).

This triangulated framework also yields a uniqueness statement: algebraic triangle structures on homotopy categories $\Lambda$6 are unique up to equivalence under the hypotheses used in the paper. In the Dynkin case, $\Lambda$7-finite algebraic triangulated categories are triangle equivalent to $\Lambda$8 for Dynkin quivers $\Lambda$9 (Hanihara, 2018).

A derived analogue, formulated as the Derived Auslander–Iyama Correspondence, replaces algebras by DG algebras. For $\mathrm{mod}\,\Lambda$00, quasi-isomorphism classes of DG algebras $\mathrm{mod}\,\Lambda$01 whose perfect derived category carries a $\mathrm{mod}\,\Lambda$02-cluster tilting object correspond bijectively to equivalence classes of pairs $\mathrm{mod}\,\Lambda$03, where $\mathrm{mod}\,\Lambda$04 is a basic finite-dimensional self-injective algebra and $\mathrm{mod}\,\Lambda$05 is an invertible $\mathrm{mod}\,\Lambda$06-bimodule satisfying

$\mathrm{mod}\,\Lambda$07

in the stable category of $\mathrm{mod}\,\Lambda$08-bimodules. In the $\mathrm{mod}\,\Lambda$09 case relevant to isolated cDV singularities and contraction algebras, this correspondence identifies the DG enhancement by the pair $\mathrm{mod}\,\Lambda$10 (Jasso et al., 2023).

5. Gorenstein and relative variants

Iyama and Solberg introduced $\mathrm{mod}\,\Lambda$11-precluster tilting subcategories as a Gorenstein generalization of $\mathrm{mod}\,\Lambda$12-cluster tilting. An $\mathrm{mod}\,\Lambda$13-precluster tilting subcategory $\mathrm{mod}\,\Lambda$14 is a functorially finite generator-cogenerator satisfying

$\mathrm{mod}\,\Lambda$15

together with closure under the higher Auslander–Reiten translations $\mathrm{mod}\,\Lambda$16 and $\mathrm{mod}\,\Lambda$17. If $\mathrm{mod}\,\Lambda$18, then $\mathrm{mod}\,\Lambda$19 is an $\mathrm{mod}\,\Lambda$20-precluster tilting module, and $\mathrm{mod}\,\Lambda$21 is called $\mathrm{mod}\,\Lambda$22-selfinjective when such a module exists (Iyama et al., 2016).

The algebraic counterpart is an $\mathrm{mod}\,\Lambda$23-minimal Auslander–Gorenstein algebra $\mathrm{mod}\,\Lambda$24, defined by

$\mathrm{mod}\,\Lambda$25

Iyama–Solberg proved a bijection between Morita-equivalence classes of $\mathrm{mod}\,\Lambda$26-minimal Auslander–Gorenstein algebras and equivalence classes of finite $\mathrm{mod}\,\Lambda$27-precluster tilting subcategories, via $\mathrm{mod}\,\Lambda$28. They also associated to an $\mathrm{mod}\,\Lambda$29-precluster tilting subcategory $\mathrm{mod}\,\Lambda$30 a Frobenius category $\mathrm{mod}\,\Lambda$31 and a stable category $\mathrm{mod}\,\Lambda$32 carrying higher Auslander–Reiten theory, including $\mathrm{mod}\,\Lambda$33-fold almost split extensions (Iyama et al., 2016).

A further enlargement replaces uniform cluster-tilting bounds by summand-dependent ones. Cao, Iyama, and Marczinzik introduced dominant Auslander–Gorenstein algebras, defined by the condition that every indecomposable projective $\mathrm{mod}\,\Lambda$34-module $\mathrm{mod}\,\Lambda$35 satisfies

$\mathrm{mod}\,\Lambda$36

and dominant Auslander–regular algebras when, in addition, $\mathrm{mod}\,\Lambda$37. Their module-theoretic counterpart is mixed precluster tilting or mixed cluster tilting, and they established bijections

$\mathrm{mod}\,\Lambda$38

$\mathrm{mod}\,\Lambda$39

up to Morita equivalence, again via endomorphism rings. This strictly enlarges the higher Auslander and minimal Auslander–Gorenstein classes (Chan et al., 2022).

6. Specialized algebraic refinements

Auslander-type correspondence is often refined to narrow but structurally important classes. For basic left Köthe rings, whose left modules are direct sums of cyclic modules, the Auslander ring $\mathrm{mod}\,\Lambda$40 of a basic left Köthe ring $\mathrm{mod}\,\Lambda$41 is characterized by the additional property of being generalized right QF-2, meaning that every indecomposable projective right $\mathrm{mod}\,\Lambda$42-module has multiplicity-free socle. The paper proves that a basic ring $\mathrm{mod}\,\Lambda$43 is left Köthe iff $\mathrm{mod}\,\Lambda$44 is representation-finite and its Auslander ring is generalized right QF-2, and deduces a bijection between Morita equivalence classes of left Kawada rings and Morita equivalence classes of Auslander generalized right QF-2 rings (Fazelpour et al., 2021).

Another refinement appears in the study of $\mathrm{mod}\,\Lambda$45-locally free modules over the Geiss–Leclerc–Schröer string algebras $\mathrm{mod}\,\Lambda$46 of affine type $\mathrm{mod}\,\Lambda$47. There the Auslander–Reiten quiver is described explicitly using minimal string modules, and the resulting classification shows that positive roots of type $\mathrm{mod}\,\Lambda$48 correspond exactly to rank vectors of indecomposable $\mathrm{mod}\,\Lambda$49-locally free $\mathrm{mod}\,\Lambda$50-modules. Real roots correspond uniquely to preprojective or preinjective modules, while positive imaginary roots correspond to families of regular modules in tubes (Huang et al., 2021).

A different line extends Auslander's original endomorphism-ring map

$\mathrm{mod}\,\Lambda$51

to non-connected graded Calabi–Yau algebras. For the preprojective algebra $\mathrm{mod}\,\Lambda$52 and a finite subgroup $\mathrm{mod}\,\Lambda$53 acting by graded automorphisms, the Auslander map is an isomorphism if and only if $\mathrm{mod}\,\Lambda$54 does not contain all reflections through a vertex. This gives a dihedral-action criterion for when the skew group algebra is recovered as an endomorphism ring over the invariant subring (Kamsvaag et al., 2021).

7. Broader uses of the term

Not every use of “Auslander-type” is an endomorphism-ring correspondence. In homological algebra, Huang used Auslander-type conditions $\mathrm{mod}\,\Lambda$55 and $\mathrm{mod}\,\Lambda$56 for modules to prove equivalences between the Auslander condition on a left and right Noetherian ring $\mathrm{mod}\,\Lambda$57, the same condition for every flat module, linear bounds on the flat dimensions of the terms in minimal injective coresolutions, dual bounds on minimal flat resolutions, and closure properties for injective envelopes and flat covers. In the Artinian case, the paper also proves that an Artinian algebra satisfying the Auslander condition is Gorenstein iff the subcategory of finitely generated modules satisfying the Auslander condition is contravariantly finite (Huang, 2010).

A recent continuation studies weakly Gorenstein Artin algebras under Auslander-type conditions. If $\mathrm{mod}\,\Lambda$58 satisfies appropriate $\mathrm{mod}\,\Lambda$59-conditions on both sides, then $\mathrm{mod}\,\Lambda$60 is Gorenstein iff it is left and right weakly Gorenstein; under the Auslander condition, $\mathrm{mod}\,\Lambda$61 is Gorenstein iff it is left or right weakly Gorenstein. These results are presented as reductions of the Auslander–Reiten conjecture (Huang, 2024).

The term also appears in Auslander–Kleiner’s generalization of Green correspondence. Zimmermann’s triangulated version treats adjoint pairs of triangle functors $\mathrm{mod}\,\Lambda$62 with a splitting

$\mathrm{mod}\,\Lambda$63

and shows that, after Verdier localization by suitable thick subcategories, one obtains induced equivalences between localized thick closures. In the classical group-algebra case this recovers the Carlson–Peng–Wheeler realization of Green correspondence by triangle functors between relative stable categories (Zimmermann, 2020).

Across these variants, the persistent theme is formal rather than uniform in statement: one starts from a distinguished homologically controlled object or subcategory, passes to an endomorphism ring, exact category, or localized quotient, and characterizes the image by dominant dimension, global or injective dimension, periodicity, Gorenstein, QF-2, or Ext-orthogonality conditions. This recurring pattern is what gives the subject its common name.