DOMINANT AUSLANDER-GORENSTEIN ALGEBRAS AND MIXED CLUSTER TILTING AARON CHAN OSAMU IYAMA AND REN E MARCZINZIK
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DOMINANT AUSLANDER-GORENSTEIN ALGEBRAS AND
MIXED CLUSTER TILTING
AARON CHAN, OSAMU IYAMA, AND REN´
E MARCZINZIK
Dedicated to the memory of Hiroyuki Tachikawa
Abstract. We introduce the class of dominant Auslander-Gorenstein algebras as a generalisation of
higher Auslander algebras and minimal Auslander-Gorenstein algebras, and give their basic properties.
We also introduce mixed (pre)cluster tilting modules as a generalisation of (pre)cluster tilting mod-
ules, and establish an Auslander type correspondence by showing that dominant Auslander-Gorenstein
(respectively, Auslander-regular) algebras correspond bijectively with mixed precluster (respectively,
cluster) tilting modules. We show that every trivial extension algebra T(A) of a d-representation-finite
algebra Aadmits a mixed cluster tilting module and show that this can be seen as a generalisation of
the well known result that d-representation-finite algebras are fractionally Calabi-Yau. We show that
iterated SGC-extensions of a gendo-symmetric dominant Auslander-Gorenstein algebra admit mixed
precluster tilting modules.
Contents
Introduction 1
1. Preliminaries 4
2. Dominant Auslander-Gorenstein algebras and basic properties 7
2.1. The definition and the first properties 7
2.2. Relation with the Gorenstein condition 8
3. Mixed precluster tilting modules and dominant Auslander-Gorenstein algebras 10
3.1. Dominant Auslander-Solberg correspondence 10
3.2. Mixed cluster tilting modules and Dominant Auslander correspondence 12
4. Mixed cluster tilting for trivial extensions of d-representation finite algebras 14
5. Mixed precluster tilting for gendo-symmetric algebras 19
6. Gluing dominant Auslander-Gorenstein algebras 22
Acknowledgements 24
References 24
Introduction
The notion of Gorenstein rings is central in algebra. They are commutative Noetherian rings Rwhose
localisation at prime ideals have finite injective dimension. One of the characterisations of Gorenstein
rings due to Bass is that, for a minimal injective coresolution
0→R→I0→I1→ · · ·
of the R-module Rand each i≥0, the term Iiis a direct sum of the injective hull of R/p for all prime
ideals pof height i. In this case, Iihas flat dimension i, and this property plays an important role in the
Date: February 20, 2024.
2010 Mathematics Subject Classification. Primary 16G10, 16E10.
Key words and phrases. Auslander-Gorenstein algebra, Auslander-regular algebra, mixed precluster tilting, cluster tilt-
ing, d-representation finite algebra.
The first author is supported by JSPS Grant-in-Aid for Research Activity Start-up program 19K23401. The second
author is supported by JSPS Grant-in-Aid for Scientific Research (B) 22H01113 and (C) 18K3209. The last named author
was supported by the DFG with the project number 428999796.
1
arXiv:2210.06180v3 [math.RT] 17 Feb 2024
2 CHAN, IYAMA, AND MARCZINZIK
study of non-commutative analogues of Gorenstein rings. A noetherian ring A, which is not necessarily
commutative, satisfies the Auslander condition if there exists an injective coresolution
(0.0.1) 0 →A→I0→I1→ · · ·
of the right module Asuch that the flat dimension of Iiis bounded by ifor all i≥0. This condition
is left-right symmetric, and there are many other equivalent conditions. If, additionally, Ahas finite
injective dimension, Ais said to be Auslander-Gorenstein, and if Ahas finite global dimension, Ais
said to be Auslander-regular. These algebras play important roles in various areas, including homological
algebra, non-commutative algebraic geometry, analytic D-modules, Lie theory and combinatorics, see for
example [AR, Bj, Cl, IM, KMM, VO].
The classical Auslander correspondence [A, ARS] is a milestone in representation theory which gives a
bijective correspondence between representation-finite artin algebras Band Auslander algebras A, which
are by definition the artin algebras Awith gldim A≤2≤domdim A, where domdim Xof an A-module
Xwith minimal injective coresolution
0→X→I0→I1→ · · ·
is the infimum of i≥0 such that Iiis non-projective. This was one of the starting points of Auslander-
Reiten theory developed in the 1970s. It encodes the representation theory of Binto the homological
algebra of its Auslander algebra A. More generally, algebras Asatisfying gldim A≤d≤domdim Aare
called (d−1)-Auslander algebras for each d≥0, and there exists a bijective correspondence (called the
higher Auslander correspondence) between d-cluster tilting modules Mover finite dimensional algebras
B(see (1.4.3)) and d-Auslander algebras for each d≥1 given by M7→ A= EndB(M) [I5]. Finite
dimensional algebras Bwith gldim A≤dadmitting d-cluster tilting modules are called d-representation
finite. These notions are central in higher Auslander-Reiten theory of finite dimensional algebras and
Cohen-Macaulay representations [I4].
A further generalisation of the Auslander correspondence to algebras with infinite global dimension
was established recently in [IS], building upon [AS]. In this setting, Auslander algebras are replaced
by minimal Auslander-Gorenstein algebras, which are algebras Awith idim A≤d≤domdim Afor
some d≥0. Similar to the higher Auslander correspondence, we have A≃EndB(M) for some finite
dimensional algebra Band a precluster tilting B-module M. Several applications and interactions with
other fields are found; these include cluster algebras of Fomin-Zelevinsky [FZ] and non-commutative
crepant desingularisations in algebraic geometry [Van], see also [AT, DI, DJL, DJW, GI, IO1, IO2, JKM,
Han, HS, HI1, HI2, HZ1, HZ2, HZ3, JK, Mi, P, S, ST, Vas, Wi, Wu] and [CIM1, CK, DITW, HKV,
Grev, LMZ, LZ, MS, MMZ, Mc, NRTZ, PS, Rin, Z].
The first aim of this paper is to introduce dominant Auslander-Gorenstein algebras, a generalisation
of higher Auslander algebras as well as minimal Auslander-Gorenstein algebras. A dominant Auslander-
Gorenstein algebra Ais an Iwanaga-Gorenstein algebra Asuch that idim P≤domdim Pholds for every
indecomposable projective A-module P. A dominant Auslander-regular algebra is then defined to be
a dominant Auslander-Gorenstein algebra with finite global dimension. We refer to Theorem 2.7 and
Section 3 for other equivalent characterisations of dominant Auslander-regular algebras.
higher Auslander +3
dominant Auslander-regular +3
Auslander-regular
minimal Auslander-Gorenstein +3dominant Auslander-Gorenstein +3Auslander-Gorenstein
The class of dominant Auslander-Gorenstein algebras is much larger than that of minimal Auslander-
Gorenstein algebras, and still enjoys extremely nice homological properties among all Auslander-
Gorenstein algebras. One of the advantages of dominant Auslander-regular algebras is that they are
closed under gluing of algebras (Section 6) and Koszul duality [CIM2] while higher Auslander algebras
are not so in general.
One of our main results gives the Auslander correspondence for dominant Auslander-Gorenstein al-
gebras. For this, we generalise the notion of cluster tilting modules as follows: For an algebra B, we
consider the higher Auslander-Reiten translates
τn:= τΩn−1and τ−
n:= τ−1Ω−(n−1).
Then a generator-cogenerator Mof Bis called mixed precluster tilting if the following condition is satisfied.
DOMINANT AUSLANDER-GORENSTEIN ALGEBRAS AND MIXED CLUSTER TILTING 3
•For each indecomposable non-projective direct summand Xof M, there exists ℓX≥1 such that
Exti
B(X, M ) = 0 for all 1 ≤i < ℓXand τℓX(X)∈add M.
This is equivalent to its dual condition, see Definition-Proposition 3.1. We refer to Sections 4, 5 and 6
for examples of mixed precluster tilting modules. We prove the following Auslander correspondence for
dominant Auslander-Gorenstein algebras.
Theorem 0.1 (Theorem 3.2).There exists a bijection between the following objects.
(1) The Morita equivalence classes of dominant Auslander-Gorenstein algebras Awith domdim A≥2.
(2) The Morita equivalence classes of pairs (B, M )of finite dimensional algebras Band mixed preclus-
ter tilting modules M.
The correspondence from (2) to (1) is given by (B, M)7→ A:= EndB(M).
To a mixed precluster tilting module M, we associate the subcategory
Z(M) := \
X∈ind(add M)
X⊥ℓX−1
of mod B(Definition 3.6), and we prove that there exists an equivalence
HomB(M, −) : Z(M)≃CM A,
where CM Adenotes the subcategory of maximal Cohen-Macaulay A-modules (Theorem 3.7). It follows
that Ais dominant Auslander-regular if and only if Z(M) = add M. Motivated by this result, we call
a mixed precluster tilting module Mmixed cluster tilting if it satisfies Z(M) = add M. Thus we have
implications
d-cluster tilting +3
mixed cluster tilting
d-precluster tilting +3mixed precluster tilting.
We refer to Sections 4 and 6 for examples of mixed cluster tilting modules. Now we are ready to state
the following restriction of the correspondence in Theorem 0.1.
Theorem 0.2 (Theorem 3.9).The bijection in Theorem 0.1 restricts to a bijection between the following
objects.
(1) The Morita equivalence classes of dominant Auslander-regular algebras Awith domdim A≥2.
(2) The Morita equivalence classes of pairs (B, M )of finite dimensional algebras Band mixed cluster
tilting modules M.
Recall that the complexity of a module M∈mod Ais defined as
inf{i∈Z≥0| ∃c∈R>0∀n∈Z≥0: dimKPn≤cni−1},(0.2.1)
where · · · → P1→P0→M→0 is a minimal projective resolution of M. The complexity of an algebra
Ais the supremum of the complexity of its modules in mod A. We prove the following generalisation of
Erdmann and Holm’s result [EH] on the complexity of self-injective algebras admitting d-cluster tilting
modules.
Theorem 0.3 (Theorem 3.11).If a finite dimensional self-injective algebra Aadmits a mixed cluster
tilting module, then Ahas complexity at most one.
The final two sections contain applications and examples of mixed (pre)cluster tilting modules. Recall
that the trivial extension algebra T(A) = A⊕DA of a finite dimensional algebra Ais a Frobenius
algebra that is of central importance in representation theory. Trivial extension algebras as well as its
covering called repetitive algebras give a systematic construction of self-injective algebras using orbit
categories, and it is known that any representation-finite self-injective algebras are obtained in this way
[BLR, Bo, Rie]. It also plays an important role in a recent characterisation of twisted fractionally Calabi-
Yau algebras of finite global dimension using trivial extension algebras in [CDIM]. We show that, for
each Dynkin quiver Q, the trivial extension algebra B=T(KQ) has a mixed cluster tilting module given
by a direct sum of Band all indecomposable non-projective KQ-modules. In fact, our result is much
more general and can be stated as follows.
4 CHAN, IYAMA, AND MARCZINZIK
Theorem 0.4 (Theorem 4.1).Let Abe a d-representation-finite algebra with a basic d-cluster tilting
module M=A⊕Nand let B=T(A)be the trivial extension of A. Then B⊕Nis a mixed cluster
tilting B-module.
This result is interesting from the point of view of Darp¨o and Kringeland’s classification [DK] of cluster
tilting modules over the trivial extension of Dynkin path algebras. It turns out that, for d≥2, d-cluster
tilting modules over this class of algebras only exist in Dynkin types Anand D4; on the other hand,
Theorem 0.4 tells us that non-trivial mixed cluster tilting modules exist for any Dynkin types.
In Section 5, we discuss an inductive way to obtain infinitely many mixed cluster tilting modules
starting from algebras with certain properties, we refer to Theorem 5.3 for full details. We just state
here the main result for the most important special case, namely gendo-symmetric algebras. Recall that
a finite dimensional algebra Ais called gendo-symmetric if A≃EndB(M) for a symmetric algebra B
with generator Mof mod B. Gendo-symmetric algebras were introduced by Fang and Koenig in [FK]
and containing important classes of algebras such as Schur algebras S(n, r) for n≥r, blocks of category
Oand centraliser algebras of nilpotent matrices, see for example [KSX] and [CrM]. On the other hand,
the SGC-extension (smallest generator-cogenerator extension) of an algebra Ais EndA(A⊕DA), see e.g.
[CIM1]. The main result of Section 5 can be stated as follows for gendo-symmetric algebras and their
iterated SGC-extensions.
Theorem 0.5 (Corollary 5.4).Let A=A0be a gendo-symmetric algebra. For each i≥0, let Mi:=
Ai⊕DAiand Ai+1 := EndAi(Mi).
(1) Assume that Ais a dominant Auslander-Gorenstein algebra. Then for each i≥0,Aiis a
dominant Auslander-Gorenstein algebra with a mixed precluster tilting module Mi.
(2) Assume that Ais a minimal Auslander-Gorenstein algebra. Then for each i≥0,Aiis a minimal
Auslander-Gorenstein algebra with a precluster tilting module Mi.
In forthcoming work [CIM2] the notion of mixed cluster tilting modules and dominant Auslander-
regular algebras will be of central importance to answer a question of Green in [Gree] about the ho-
mological characterisation of the Koszul dual algebras of Auslander algebras. We will see that Koszul
duality leads in a very natural way to the notion of dominant Auslander-regular algebras when working
with the classical higher Auslander algebras and this will also allow us to discover large new classes of
d-representation-finite algebras and cluster tilting modules.
1. Preliminaries
We assume that all algebras are finite dimensional over a field Kand modules are finitely generated
right modules unless stated otherwise. We assume that the reader is familiar with the basics of repre-
sentation theory of algebras and refer for example to the textbooks [ARS, ASS, SY]. For background on
homological dimensions such as the dominant dimension and related properties we refer for example to
[T, Ya]. For background on Gorenstein homological algebra and maximal Cohen-Macaulay modules we
refer to [AB, Ch].
Throughout, D= HomK(−, K) denotes the natural duality of the module category of an algebra A,
proj Adenotes the full subcategory of projective A-modules and inj Athe full subcategory of injective
A-modules. We will usually omit the bracket and write DM when applying the duality Dto a module
M, unless there is the danger of confusion – for instance, we will write D(Af) for injective A-module
corresponding to an idempotent f. A module Mis called basic if it has no direct summand of the form
L⊕Lfor some indecomposable module L. An algebra Ais called basic if Ais a basic module. We denote
by LL(M) the Loewy length of a module M.
Denote by pdim Mand idim Mthe projective dimension and injective dimension of a module M
respectively. An algebra Ais said to be Iwanaga-Gorenstein if idim A+ pdim DA < ∞; in such a case,
we have idim A= pdim DA and idim Ais called the self-injective dimension of A. A module Mwith
Exti
A(M, A) = 0 for all i > 0 over an Iwanaga-Gorenstein algebra Ais called maximal Cohen-Macaulay
(also known as Gorenstein projective in the literature). Denote by CM Athe full subcategory of maximal
Cohen-Macaulay modules of Aand by Ωn(mod A) the full subcategory of modules that are isomorphic to
the n-th syzygy of a module. One can show that an algebra Ais Iwanaga-Gorenstein with self-injective
dimension nif and only if Ωn(mod A) = CM A, see for example [Ch, Theorem 2.3.3]. Note that for an
DOMINANT AUSLANDER-GORENSTEIN ALGEBRAS AND MIXED CLUSTER TILTING 5
Iwanaga-Gorenstein algebra A, one has gldim A < ∞if and only if CM A= proj A; in which case, the
global dimension coincides with the self-injective dimension.
The dominant dimension domdim Mof a module Mwith a minimal injective coresolution
0→M→I0→I1→ · · ·
is defined as the smallest nsuch that Inis non-projective. The codominant dimension codomdim Mof a
module Mis defined as the dominant dimension of the left module D(M). Domn(A) denotes the full sub-
category of mod Aconsisting of modules having dominant dimension at least n. The dominant dimension
of an algebra is defined as the dominant dimension of Aas an A-module, i.e. domdim A:= domdim(AA).
We note that domdim A= codomdim DA = domdim Aop always holds. We can characterise modules of
large dominant dimension as follows.
Lemma 1.1. [MVi, Proposition 4] Let Abe an algebra of dominant dimension equal to d≥1. Then
Domi(A)=Ωi(mod A)for all 0≤i≤d.
Let Abe a finite dimensional algebra, and let e, f ∈Abe idempotents such that D(Af) and eA are
the basic additive generators of projective-injective modules, i.e.
add D(Af) = proj A∩inj A= add(eA).(1.1.1)
Then we have an isomorphism fAf ≃eAe of algebras. Having domdim A≥1 is equivalent to saying
that there is an injective map 0 →A→Iwith I∈add D(Af), which is also equivalent to (by left-right
symmetry) the existence of surjective map P→DA →0 with P∈add(eA). In classical ring theoretic
language, this amounts to say that eA is a minimal faithful projective-injective A-module and Af is a
minimal faithful projective-injective Aop-module. We call fAf ≃eAe the base algebra of A
The Morita-Tachikawa correspondence [A, Ya] gives a bijection between Morita equivalence classes
of algebras Awith domdim A≥2 and Morita equivalence classes of pairs (B, M) of algebras Band
generator-cogenerators Mof B. The map (B, M)7→ Ais given by the following proposition.
Proposition 1.2. [A] Let Bbe an algebra, Ma generator-cogenerator of B, and A:= EndB(M).
(1) domdim A≥2.
(2) Let e, f be idempotents of Asuch that eA = HomB(M, DB)and fA = HomB(M, B). Then
(1.1.1) is satisfied.
The map A7→ (B, M) is given by the following proposition.
Proposition 1.3. [A] For an algebra Awith domdim A≥2, let e, f ∈Abe idempotents satisfying
(1.1.1), and B:= fAf. Then the following hold.
(1) M:= Af is a generator-cogenerator over B, i.e. B⊕DB ∈add M.
(2) We have an isomorphism A→EndB(M)of algebras sending a∈Ato (a·)∈EndB(M).
(3) For X∈ind(add M), the indecomposable projective A-module HomB(M, X)is injective if, and
only if, X∈inj B. In such a case, we have HomB(M, X)≃DHomB(ν−(X), M ).
Under the setting of Proposition 1.3, the following (B, A)-bimodules are isomorphic, and are additive
generators of proj A∩inj A:
D(Af)≃eA ≃HomB(M, DB)≃HomfAf (Af, D(Af )f)≃DHomB(B, M)≃DHomf Af (eAf, Af).
Proof. For convenience of the reader, we include a proof.
(1) The B-module B=fAf belongs to add Af = add M. Since add D(Af) = add eA as subcategories
of mod A, we have add D(Af)f= add eAf as subcategories of mod B. Thus the B-module DB =
D(fAf)=(D(Af))fbelongs to add eAf ⊂add M.
(2) Take an injective resolution 0 →A→J0→J1with Ji∈add D(Af ). Multiplying ffrom the right,
we have an exact sequence 0 →M→J0f→J1fof B-modules. Applying HomB(M, −) = HomB(Af, −)
to the second sequence and comparing with the first sequence, we have a commutative diagram of exact
sequences
0//EndB(M)//HomB(Af, J0f)//HomB(Af, J1f)
0//A//J0//J1,
摘要：
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