arXiv:1608.04179v2 [math.RT] 22 Jan 2018

AUSLANDER–GORENSTEIN ALGEBRAS
AND PRECLUSTER TILTING
OSAMU IYAMA AND ØYVIND SOLBERG
To the memory of Maurice Auslander.
Abstract. We generalize the notions of n-cluster tilting subcategories and τ selfinjective algebras into n-precluster tilting subcategories and τn -selfinjective
algebras, where we show that a subcategory naturally associated to n-precluster
tilting subcategories has a higher Auslander–Reiten theory. Furthermore,
we give a bijection between n-precluster tilting subcategories and n-minimal
Auslander–Gorenstein algebras, which is a higher dimensional analog of Auslander–
Solberg correspondence (Auslander–Solberg, 1993) as well as a Gorenstein
analog of n-Auslander correspondence (Iyama, 2007). The Auslander–Reiten
theory associated to an n-precluster tilting subcategory is used to classify
the n-minimal Auslander–Gorenstein algebras into four disjoint classes. Our
method is based on relative homological algebra due to Auslander–Solberg.

Contents
1. Introduction
2. Preliminaries on relative homological algebra
2.1. Relative homological algebra
2.2. The Auslander–Reiten translation revisited
2.3. Relative tilting theory
3. Elementary properties of n-precluster tilting subcategories
4. Auslander–Solberg correspondence
5. n-fold almost split extensions
6. Four classes of n-minimal Auslander–Gorenstein algebras
References

1
5
5
7
9
12
19
22
29
31

1. Introduction
Higher Auslander–Reiten theory was introduced in [21] by looking at n-cluster
tilting subcategories instead of the whole module category. It is known that any
n-cluster tilting subcategory has n-almost split sequences, and that finite n-cluster
Key words and phrases. Auslander-Reiten theory; cluster tilting; Auslander algebra; relative
homological algebra.
Part of the results were presented at the meeting ICRA XIV, Tokyo, Japan, on August 11th,
2010, and at the meeting “Miniconference on selfinjective algebras”, Torun, Poland, on December
15th 2010. Osamu Iyama’s work on this project was partially supported by JSPS Grant-in-Aid for
Scientific Research (B) 16H03923, (C) 23540045 and (S) 15H05738. Øyvind Solberg’s work on this
project was partially supported by FRINAT grant number 23130 from the Norwegian Research
Council.
1

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IYAMA AND SOLBERG

tilting subcategories correspond to n-Auslander algebras [22]. Moreover Artin algebras Λ which have n-cluster tilting modules are called n-representation-finite [13]
and have been studied. (We do not assume gldim Λ ≤ n in this paper in contrast
with several earlier papers [18, 24, 25].) See for example [1, 19, 20, 26, 28, 29, 30,
31, 34, 36] for further results in higher Auslander–Reiten theory.
In this paper we introduce the weaker notions of n-precluster tilting subcategories (Definition 3.2) and τn -selfinjective algebras (Definition 3.4). These notions
generalize and unify two seemingly different concepts, namely (finite) n-cluster tilting subcategories [21] and τ -selfinjective algebras (or D Tr-selfinjective algebras)
[11]. Both of these concepts are generalizations of algebras of finite representation
type and they are linked through the Wedderburn correspondence introduced in
[2]. The first class corresponds to n-Auslander algebras Γ, that is, Artin algebras
Γ satisfying
domdim Γ ≥ n + 1 ≥ gldim Γ.
The latter class corresponds to algebras Γ that are a Gorenstein analog of Auslander
algebras which satisfy
domdim Γ ≥ 2 ≥ idΓ Γ.
Here domdim, gldim and id denote dominant dimension, global dimension and injective dimension, respectively. A central notion in higher dimensional Auslander–
Reiten theory is n-cluster tilting subcategories, which are the whole module categories in the classical case n = 1. As in the process of going from classical
Auslander–Reiten theory to higher Auslander–Reiten theory, we demonstrate that
the number 2 is quite symbolic, as we show that n-precluster-tilting subcategories
correspond to the following class of Artin algebras.
Definition 1.1. We call an Artin algebra Γ an n-minimal Auslander–Gorenstein
algebra if it satisfies
domdim Γ ≥ n + 1 ≥ idΓ Γ.
We will observe in Proposition 4.1(b) that this condition is left-right symmetric. Recall that an Artin algebra Γ is called Gorenstein (or Iwanaga–Gorenstein
more precisely) if idΓ Γ and id ΓΓ are finite [15], and a Gorenstein algebra is called
Auslander–Gorenstein if the minimal injective coresolution
0 → Γ → I0 → I1 → · · ·
of the Γ-module Γ satisfies pdΓ I i ≤ i holds for every i ≥ 0 [16, 5, 6]. Thus our
n-minimal Auslander–Gorenstein algebras can be regarded as the most basic class
among Auslander–Gorenstein algebras since pdΓ I i has the ‘minimal’ value 0 for
0 ≤ i ≤ n.
We have the following diagrams of generalizations of Auslander algebras and
representation-finite algebras:
Auslander✤ algebras ❴ ❴ ❴/ Gorenstein algebras Γ with
✤ domdim Γ ≥ 2 ≥ idΓ Γ
✤
✤
❴
❴
❴
❴
❴
/
n-Auslander algebras
n-minimal Auslander–Gorenstein algebras
representation-finite
algebras ❴ ❴ ❴ ❴/ τ -selfinjective
✤ algebras
✤
✤
✤
❴
❴
❴
/
n-representation-finite algebras
τn -selfinjective algebras

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

3

We introduce the concept of n-precluster tilting subcategories to describe these
algebras, where such a subcategory C is defined to be a functorially finite generatingcogenerating subcategory stable under the n-th Auslander–Reiten translate and
selforthogonal in the interval [1, n−1] (that is, Exti (C, C) = 0 for i = 1, 2, . . . , n−1).
More precisely, we show the following results.
Theorem 4.5. Fix n ≥ 1. There is a bijection between Morita-equivalence classes
of n-minimal Auslander–Gorenstein algebras and equivalence classes of finite nprecluster tilting subcategories C of Artin algebras, where the correspondences are
given in Propositions 4.3 and 4.4.
For a Gorenstein algebra Γ, we denote by
CM Γ = {X ∈ mod Γ | ExtiΓ (X, Γ) = 0 for i > 0}
the category of maximal Cohen–Macaulay Γ-modules. We denote by CM Γ the
stable category of CM Γ, that is, CM Γ modulo the ideal in CM Γ generated by
add Γ. It is basic that CM Γ forms a triangulated category. On the other hand, for
an n-precluster tilting subcategory C of mod Λ, let
Z(C) =

{X ∈ mod Λ | ExtiΛ (C, X) = 0 for i = 1, 2, . . . , n − 1},

U(C) =

Z(C)/[C],

where Z(C)/[C] denotes the category Z(C) modulo the ideal [C] in Z(C) generated by
the subcategory C. Note that Z(C) = mod Λ for the case n = 1. We show that the
category Z(C) is a Frobenius category whose stable category is U(C) (Proposition
3.12). Hence U(C) forms a triangulated category. In fact it is an analog of Calabi–
Yau reduction of triangulated categories [27].
Theorem 4.7. Given an Artin algebra Λ with a finite n-precluster tilting subcategory C = add M , let Γ = EndΛ (M ) be the corresponding n-minimal Auslander–
Gorenstein algebra (see Theorem 4.5). Then Z(C) and CM Γ are dual categories
via the functors HomΛ (−, M ) : Z(C) → CM Γ and HomΓ (−, M ) : CM Γ → Z(C).
Moreover these functors induce triangle equivalences between U(C) and (CM Γ)op .
Furthermore we show that there is a higher Auslander–Reiten theory also for
n-precluster tilting subcategories, though with some differences. The first difference is that one cannot define n-fold almost split sequences as for n-cluster tilting
subcategories, but one is forced to introduce n-fold almost split extensions (see
Definition 5.1). This is because in this more general setting there does not exist
a unique exact sequence representing this extension. Namely, we have the following results, where we denote by P(Λ) (respectively I(Λ)) the category of finitely
generated projective (respectively injective) Λ-modules.
Theorem 5.10. Let C be an n-precluster tilting subcategory of mod Λ, X an indecomposable module in Z(C) \ P(Λ), and Y := τn (X) the corresponding indecomposable module in Z(C) \ I(Λ).
(a) For each 0 ≤ i ≤ n − 1, an n-fold almost split extension in ExtnΛ (X, Y ) can be
represented as
0 → Y → Cn−1 → · · · → Ci+1 → Zi → Ci−1 → · · · → C0 → X → 0
with Zi in Z(C) and Cj in C for each j 6= i.

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IYAMA AND SOLBERG

(b) The following sequences are exact.
0 → HomΛ (C, Y ) → HomΛ (C, Cn−1 ) → · · · → HomΛ (C, Ci+1 ) → HomΛ (C, Zi )
→ HomΛ (C, Ci−1 ) → · · · → HomΛ (C, C0 ) → radΛ (C, X) → 0,
0 → HomΛ (X, C) → HomΛ (C0 , C) → · · · → HomΛ (Ci−1 , C) → HomΛ (Zi , C)
→ HomΛ (Ci+1 , C) → · · · → HomΛ (Cn−1 , C) → radΛ (Y, C) → 0.
(c) If X and Y do not belong to C, then the n-fold almost split extension in (a) can
be given as a Yoneda product of a minimal projective resolution of X in Z(C)
0 → ΩiZ(C) (X) → Ci−1 → · · · → C0 → X → 0,
an almost split sequence in Z(C)
−(n−i−1)

0 → ΩZ(C)

(Y ) → Zi → ΩiZ(C) (X) → 0,

and a minimal injective coresolution of Y in Z(C)
−(n−i−1)

0 → Y → Cn−1 → · · · → Ci+1 → ΩZ(C)

(Y ) → 0.

We use this Auslander–Reiten theory to classify the n-minimal Auslander–Gorenstein algebras into four disjoint classes, as was done for the case n = 1 in [11].
We refer to related results. In [12], the authors studied (n, m)-ortho-symmetric
modules, where our n-precluster tilting modules are precisely (n − 1, 0)-orthosymmetric. In [32], the author proved Theorem 4.5 independently.
The paper is organized as follows. In the second section we recall the relative homological algebra over Artin algebras we need (see [8, 9, 10]) and some unpublished
results of Maurice Auslander and the second author. In Section 3 the notions of
n-precluster tilting subcategories and τn -selfinjective algebras are introduced and
their basic properties are discussed. We show in the next section that there is a
one-to-one correspondence between finite n-precluster subcategories and n-minimal
Auslander–Gorenstein Artin algebras, where the n-Auslander algebras are characterized within this class. In the fifth section we show that there is a meaningful
higher Auslander–Reiten theory in n-precluster tilting subcategories also. This theory is transferred in the next section over to the subcategory of maximal Cohen–
Macaulay modules over the n-minimal Auslander–Gorenstein Artin algebras. In
the final section we use higher Auslander–Reiten theory to classify the n-minimal
Auslander–Gorenstein Artin algebras into four disjoint classes.
Notations. Throughout the paper, all modules are left modules. The composition
of morphisms f : X → Y and g : Y → Z is denoted by gf : X → Z.
Let R be a commutative Artinian ring and Λ an Artin R-algebra. We denote by
mod Λ the category of finitely generated Λ-modules, by P(Λ) (respectively I(Λ))
the category of finitely generated projective (respectively injective) Λ-modules, and
by D : mod Λ ↔ mod Λop the duality HomR (−, E), where E is the injective hull of
the R-module R/ rad R. We denote by τ : mod Λ → mod Λ the Auslander–Reiten
translation.
Let X be a full subcategory of mod Λ. We call X a generator (respectively
cogenerator ) if Λ ∈ X (respectively DΛ ∈ X ). A right X -approximation of A ∈
mod Λ is a morphism f : X → A with X ∈ X such that any morphism g : Y → A
with Y ∈ X factors through f . We call X contravariantly finite if any A ∈ mod Λ
has a right X -approximation. Dually, we define a left X -approximation and a

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

5

covariantly finite subcategory. A contravariantly and covariantly finite subcategory
is called functorially finite.
2. Preliminaries on relative homological algebra
A systematic study of relative homological algebra over Artin algebras was carried out in [8, 9, 10]. We recall the relevant background and results, and in addition
we give some unpublished results of Maurice Auslander and the second author.
2.1. Relative homological algebra. We start with the setup for relative homological algebra, where we assume throughout that Λ is an Artin algebra. Relative
homological algebra for us begins with defining a set of exact sequences, and this
is done through giving an additive sub-bifunctor of Ext1Λ (−, −) (see [8] for further
details). Let
F ⊆ Ext1Λ (−, −) : (mod Λ)op × mod Λ → Ab
be an additive sub-bifunctor. Such an additive sub-bifunctor is nothing else than,
for each pair of Λ-modules C and A, a chosen set of short exact sequences, F (C, A),
starting in A and ending in C, which is closed under pullbacks, pushouts and Baer
sums (or direct sums of short exact sequences).
Definition 2.1. An exact sequence η : 0 → A → B → C → 0 is said to be F -exact
if η is in F (C, A).
In the rest, we fix a subcategory X of mod Λ. We consider the following collection
FX (C, A) of short exact sequences given for a pair of modules A and C in mod Λ:
FX (C, A) = {0 → A → B → C → 0 ∈ Ext1Λ (C, A) |
HomΛ (X, B) → HomΛ (X, C) → 0 exact for all X in X }.
X

Dually one defines F . By [8, Proposition 1.7] these collections induce additive
sub-bifunctors FX and F X of Ext1Λ (−, −), and
FX = F τ X and Fτ − X = F X

(2.1)

by [8, Proposition 1.8]. If X = add X holds for some X in mod Λ, we denote FX
and F X by FX and F X respectively.
We can endow mod Λ with a new exact structure by the following result, where
we call a full subcategory C of mod Λ F -extension closed if, for every F -exact
sequence 0 → A → B → C → 0 such that A and C are in C, B is in C.
Proposition 2.2 ([14]). Let X be a full subcategory of mod Λ and F = F X (respectively F = FX ). Then mod Λ has a structure of an exact category whose short exact sequences are precisely the F -exact sequences. More generally, any F -extension
closed subcategory C of mod Λ has a structure of an exact category whose short exact
sequences are precisely the F -exact sequences contained in C.
Proof. The first assertion follows from [14, Propositions 1.4 and 1.7]. The second
assertion is a general property of extension closed subcategories of an exact category.

We denote by (mod Λ, F ) and (C, F ) the exact categories given in Proposition
2.2.
Recall the following definitions from [8], which coincide with the corresponding
notions in the exact category (mod Λ, F ).

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IYAMA AND SOLBERG

Definition 2.3. (i) A Λ-module P is said to be F -projective if all F -exact sequences 0 → A → B → P → 0 split. The full subcategory of mod Λ consisting
of all F -projective modules is denoted by P(F ).
(ii) A Λ-module I is said to be F -injective if all F -exact sequences 0 → I →
B → C → 0 split. The full subcategory of mod Λ consisting of all F -injective
modules is denoted by I(F ).
(iii) F has enough F -projectives (respectively F -injectives) if for each C (respectively A) in mod Λ there exists an F -exact sequence
0 → C′ → P → C → 0
with P in P(F ) (respectively 0 → A → I → A′ → 0 with I in I(F )).
The FX -projectives and FX -injectives are given as
P(FX ) = add{X , P(Λ)} and I(FX ) = add{τ X , I(Λ)}

(2.2)

P(Ext1Λ (−, −))

by [8, Proposition 1.10]. For example, we have P(FΛ ) =
= P(Λ)
and I(F DΛ ) = I(Ext1Λ (−, −)) = I(Λ). Furthermore an additive sub-bifunctor F
of Ext1Λ (−, −) has enough F -projectives and F -injectives if and only if P(F ) is
functorially finite in mod Λ and F = FP(F ) (see [8, Corollary 1.13]). In this case we
denote by Ω1F (X) the kernel of the F -projective cover of X. The Λ-module Ω−1
F (Y )
is defined dually.
Assume from now on that F is an additive sub-bifunctor of Ext1Λ (−, −) with
enough F -projectives and F -injectives. Recall that an exact sequence
fi+1

fi

· · · → Ci+1 −−−→ Ci −→ Ci−1 → · · ·
is called F -exact if all the short exact sequences 0 → Im fi+1 → Ci → Im fi → 0
are F -exact. Given two modules A and C in mod Λ, there exist F -exact sequences
P : · · · → P2 → P1 → P0 → C → 0
with Pi in P(F ) and
I : 0 → A → I0 → I1 → I2 → · · ·
with I j in I(F ). We call these exact sequences an F -projective resolution and an F injective coresolution of C and A, respectively. For i ≥ 1, the i-th homologies of the
complexes HomΛ (P, A) and HomΛ (C, I) are isomorphic, and denoted by ExtiF (C, A)
and called the i-th F -relative extension group (see [8, Section 2]). Then Ext1F (−, −)
is naturally identified with F . Using ExtiF (−, −), we define F -relative projective
dimension, F -relative injective dimension and F -relative global dimension as in the
absolute setting and the basic properties are the same in the F -relative setting. On
the other hand, we denote by
HomF (A, C) (respectively HomF (A, C))
HomΛ (A, C) modulo all the homomorphisms factoring through an F -projective
(respectively F -injective) module. The F -stable category
modF Λ (respectively modF Λ)
has the same objects as mod Λ, and the morphism sets are given by HomF (A, C) (respectively HomF (A, C)). For F = FX , the Auslander–Reiten translation τ : mod Λ ≃
mod Λ induces the F -relative Auslander–Reiten translation
τ : modF Λ ≃ modF Λ.

(2.3)

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

7

Another central result that has an analog in the F -relative setting is the Auslander–
Reiten formula, which we recall next.
Proposition 2.4 ([8, Proposition 2.3]). Let F be an additive sub-bifunctor of
Ext1Λ (−, −) with enough F -projectives (and F -injectives). Then for all modules
A and C in mod Λ we have an isomorphism
Ext1F (C, τ A) ≃ D HomF (A, C).
In general the higher F -relative extension groups ExtiF (C, A) are not necessarily
related to the higher absolute extension groups ExtiΛ (C, A). However, in some
situations one can compute the absolute extensions by F -relative ones, as described
in the next result.
Proposition 2.5 ([33, Proposition 1.3]). Let X be a functorially finite subcategory
of mod Λ.
(a) A module C in mod Λ satisfies ExtiΛ (C, X ) = 0 for 0 < i < n if and only if
ExtiF X (C, A) = ExtiΛ (C, A) holds for 0 < i < n and for all A in mod Λ.
(b) A module A in mod Λ satisfies ExtiΛ (X , A) = 0 for 0 < i < n if and only if
ExtiFX (C, A) = ExtiΛ (C, A) for 0 < i < n and for all C in mod Λ.
Proof. We only show (a). This is proved in [33, Proposition 1.3] under the assumption that X is a cogenerator. We can drop it by considering X ′ = add{X , I(Λ)}
′
and using F X = F X .

2.2. The Auslander–Reiten translation revisited. Next we recall some unpublished results of Maurice Auslander and the second author that we need later.
We show that the Auslander–Reiten translation τ gives a bijection between Ext1F (C, A)
and Ext1G (τ (C), τ (A)) for certain F and G, and moreover τ commutes with relative
syzygies.
The transpose is a duality Tr : mod Λ → mod Λop [3]. However, from an exact
sequence
0→A→B→C →0
in mod Λ, there is not necessarily a naturally associated exact sequence
0 → Tr C → Tr B ⊕ X → Tr A → 0
in mod Λop . We show that when restricting to appropriate classes of exact sequences
we have such a natural correspondence.
Proposition 2.6. Let X be a functorially finite generator-cogenerator in mod Λ,
and let F = F X , Gop = F D(X ) and G = FX .
(a) If 0 → A → B → C → 0 is F -exact in mod Λ, then there are a Gop -exact
sequence
0 → Tr C → Tr B ⊕ P → Tr A → 0
in mod Λop for some projective Λop -module P and a G-exact sequence
0 → τ (A) → τ (B) ⊕ I → τ (C) → 0
in mod Λ for some injective Λ-module I.
(b) For all A and C in mod Λ, we have functorial isomorphisms
F (C, A) ≃ Gop (Tr A, Tr C) ≃ G(τ (C), τ (A)).

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IYAMA AND SOLBERG
f

Proof. (a) Let η : 0 → A → B −
→ C → 0 be an F -exact sequence. By the Horseshoe
Lemma we have the following commutative diagram
0
0
0

/ P1

/ P0

/A

0

/ P1 ⊕ Q1

/ P0 ⊕ Q0

/B

0

/ Q1

/ Q0

/C

0

/0
/0
/0

with Pi and Qi projective Λ-modules for i = 0, 1. Let (−)∗ = HomΛ (−, Λ). Then
this induces the following commutative diagram

0

0

/ C∗

/ Q∗0

0

/ B∗

/ P0∗ ⊕ Q∗0

0

/ A∗

/ P0∗

/0

0


/ Q∗
1


/ P ∗ ⊕ Q∗
1
1


/ P∗
1

/0


/ Tr C

0


/ Tr B ⊕ P

0

0

η′ : 0

g


/ Tr A

0

/0

/0

where P is a projective Λop -module. The columns are exact by the definition of Tr,
and the top row is exact because η is F -exact and X contains Λ. Since the second
and the third rows are split exact, the Snake Lemma shows that the bottom row
η ′ is also exact. It remains to show that η ′ is Gop -exact. Then the dual of this
sequence is G-exact.
Since η is F -exact, the map f : HomΛ (B, X ) → HomΛ (A, X ) is surjective. Since
Tr is a duality, the map g = Tr f : HomΛop (Tr X , Tr B) → HomΛop (Tr X , Tr A)
is surjective. By a standard argument, the map g : HomΛop (Tr X , Tr B ⊕ P ) →
HomΛop (Tr X , Tr A) is also surjective. Since τ Tr X = D(X ) and hence FTr X = Gop
holds, η ′ is Gop -exact.
(b) By (a), we have a morphism F (C, A) → Gop (Tr A, Tr C) of bifunctors. Since
the same argument gives the inverse morphism Gop (Tr A, Tr C) → F (C, A) of bifunctors, we have the first desired isomorphism. The second isomorphism follows
immediately by applying the duality D.

We end this subsection by showing how this induces isomorphisms on relative
extension groups and relative stable homomorphism sets.
Theorem 2.7. i) Let X ⊆ mod Λ be a functorially finite generator-cogenerator, and
let F = F X and G = FX be the corresponding additive sub-bifunctors of Ext1Λ (−, −).
Then the following is true.
(a) τ gives an equivalence modF Λ ≃ modG Λ.
i)Theorem 2.7 was obtained by Maurice Auslander and the second author, but they never got
to be published in printed form before now. However, the results were presented at a seminar at
the University of Bielefeld, Germany.

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

9

(b) The following diagrams commute up to isomorphisms of functors.
modF Λ

τ

ΩG

ΩF


modF Λ

/ mod Λ
G

τ


/ modG Λ

modO F Λ o

τ−

Ω−
F

modF Λ o

modO G Λ
Ω−
G

τ

−

modG Λ

(c) τ induces a functorial isomorphism in both variables
ϕn = ϕC,A,n : ExtnF (C, A) ≃ ExtnG (τ (C), τ (A))
for all pairs of A and C in mod Λ and n ≥ 1.
Proof. (a) We have an equivalence τ : modF Λ ≃ modF Λ in (2.3). Since X is
a generator-cogenerator, I(F ) = P(G) holds by (2.2). Thus modF Λ = modG Λ
holds, and the assertion follows.
(b) We only prove commutativity of the left diagram. Let
0 → ΩF (A) → P → A → 0
be F -exact with P in P(F ). By Proposition 2.6, we have a G-exact sequence
0 → τ ΩF (A) → τ (P ) ⊕ I → τ (A) → 0
with I in I(Λ). Since τ (P ) ⊕ I is in P(G) by (2.2), we have an isomorphism
τ ΩF ≃ ΩG τ of functors.
(c) We have the following functorial isomorphisms
n−1
ExtnF (C, A) ≃ Ext1F (ΩF
(C), A)
n−1
≃ Ext1G (τ ΩF
(C), τ (A))
n−1
≃ Ext1G (ΩG
τ (C), τ (A))

≃ ExtnG (τ (C), τ (A))
by dimension shift, Proposition 2.6(b) and (b), where all the involved isomorphisms
are functorial. The claim follows.

2.3. Relative tilting theory. Tilting theory is an important topic in representation theory of Artin algebras and elsewhere. It also has a relative version, which we
recall from [9, 10]. Here we always assume that our additive sub-bifunctor F has
enough F -projectives and enough F -injectives. For a subcategory C in mod Λ, let
C ⊥F

=

{M ∈ mod Λ | ExtiF (C, M ) = 0 for all i > 0},

⊥F

=

{M ∈ mod Λ | ExtiF (M, C) = 0 for all i > 0}.

C

When F = Ext1Λ (−, −), we simply denote C ⊥F and ⊥F C by C ⊥ and ⊥ C respectively.
Definition 2.8. We call T in mod Λ an F -cotilting module if
(i) idF T < ∞,
(ii) ExtiF (T, T ) = 0 for i > 0,
(iii) for all I in I(F ) there exists an F -exact sequence
0 → Tn → Tn−1 → · · · → T1 → T0 → I → 0
with Ti in add T .

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IYAMA AND SOLBERG

It is shown in [9, Corollary 3.14] that if there exists an F -cotilting module T ,
then P(F ) and hence I(F ) are of finite type. In fact, they contain the same number
of non-isomorphic indecomposable objects as add T .
Next we collect the basic results on relative cotilting modules that we need later.
Theorem 2.9 ([9]). Let X be a generator in mod Λ and F = FX . Let T be an
F -cotilting Λ-module and Γ = EndΛ (T ). Then we have the following.
(a) Λ ≃ EndΓ (T ).
(b) Any C in ⊥F T has an F -exact sequence
f1

f0

f2

0 → C → T0 −→ T1 −→ T2 −→ T3 → · · ·
with Ti in add T and Im fi in ⊥F T for all i ≥ 0.
(c) ExtiF (C, A) ≃ ExtiΓ (HomΛ (A, T ), HomΛ (C, T )) for all modules A and C in ⊥F T
and i ≥ 0.
(d) The module U = HomΛ (X, T ) is a cotilting Γ-module with idF T ≤ idΓ U ≤
idF T + 2 ii).
(e) HomΛ (−, T ) : mod Λ → mod Γ and HomΓ (−, T ) : mod Γ → mod Λ induce quasiinverse dualities HomΛ (−, T ) : ⊥F T → ⊥ U and HomΓ (−, T ) : ⊥ U → ⊥F T .
Proof. (a) is [9, Corollary 3.4], (b) is [9, Theorem 3.2(a)], (c) is [9, Proposition 3.7],
(d) is [9, Theorem 3.13(d)], and (e) is [9, Corollary 3.6(a), Proposition 3.8(b)]. 
Now let F be an additive sub-bifunctor of Ext1Λ (−, −), and X a generator of
mod Λ. Then X -resdimF M is defined to be the infimum of n such that there exists
an F -exact sequence
0 → Xn → Xn−1 → · · · → X1 → X0 → M → 0,
where Xi is in X for all i ≥ 0. We denote by Xb the full subcategory of mod Λ
consisting of all M in mod Λ with X -resdimF M < ∞. Let X -resdimF (mod Λ) :=
sup{X -resdimF M | M ∈ mod Λ}.
The following result is taken from [10] (with the exception of (c)), which connects special direct summands of absolute cotilting modules with relative cotilting
modules. Let Γ be an Artin algebra and X in mod Γ. Recall that a direct summand
Y of X is said to be a dualizing summand of X if there exists an exact sequence
f

0→X −
→Y0 → Y1
with f a left (add Y )-approximation and Y i in add Y . This is shown to be equivalent
to the natural homomorphism
X → HomΛ (HomΓ (X, Y ), Y )
being an isomorphism, where Λ = EndΓ (Y ) [10, Proposition 2.1].
Theorem 2.10. Let Γ be an Artin algebra and U a cotilting Γ-module. For a
dualizing summand T of U , let Λ = EndΓ (T ), X = HomΓ (U, T ) and F = FX an
additive sub-bifunctor of Ext1Λ (−, −). Then the following assertions hold.
(a) Γ ≃ EndΛ (T ).
(b) T is an F -cotilting Λ-module with idF T ≤ max{id Γ U , 2}.
(c) If T is injective as a Γ-module, then idF T ≤ max{id Γ U − 2, 0}.
ii)This inequality is a corrected version of [9, Theorem 3.13(d)] due to an error in the statement
of [9, Proposition 3.11], where the bound should be idF T + 2, instead of max{idF T, 2}.

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

11

Proof. (a) is [10, Proposition 2.4(b)], and (b) is [10, Proposition 2.7(c)].
(c) Let r = id Γ U and t = max{r−2, 0}. We prove that (⊥F T )-resdimF (mod Λ) ≤
t, as this implies that idF T ≤ t. This is done by showing that ΩtF (C) is in ⊥F T for
all C in mod Λ.
Assume that Γ T is injective. Then by [11, Lemma 2.4] the module Λ T is a
cogenerator in mod Λ, and therefore
C ≃ HomΓ (HomΛ (C, T ), T )
for all modules C in mod Λ. In particular, C ≃ HomΓ (B, T ) for some Γ-module B.
Let
f0

f1

f2

f3

P : B −→ U 0 −→ U 1 −→ U 2 −→ · · · ,
be a sequence of minimal left (add U )-approximations of B, and let B 0 = B and
B j = Coker f j−1 for j ≥ 1. Then an F -projective resolution of C is given by
HomΓ (P, T ) : · · · → HomΓ (U 2 , T ) → HomΓ (U 1 , T ) → HomΓ (U 0 , T ) → C → 0,
and therefore we have, for every j ≥ 0,
ΩjF (C) = HomΓ (B j , T ).
Let Σ = EndΓ (U ). Then U is a cotilting Σ-module with id Σ U = id Γ U . The
above complex P gives rise to a projective resolution of the Σ-module HomΓ (B, U ),
· · · → HomΓ (U 2 , U ) → HomΓ (U 1 , U ) → HomΓ (U 0 , U ) → HomΓ (B, U ) → 0.
Given a projective presentation F1 → F0 → B → 0 of the Γ-module B, it induces
an exact sequence
0 → HomΓ (B, U ) → HomΓ (F0 , U ) → HomΓ (F1 , U ) → B ′ → 0
′
of Σ-modules with HomΓ (F0 , U ) and HomΓ (F1 , U ) in add Σ U . Then Ωj+2
Σ (B ) =
j
HomΓ (B , U ) holds for every j ≥ 0.
t
Now we show HomΓ (B t , U ) is in ⊥
Σ U . If r ≤ 2, then HomΓ (B , U ) = HomΓ (B, U ) =
2
′
⊥
t
ΩΣ (B ) is in Σ U since id Γ U = r ≤ 2. If r > 2, then HomΓ (B , U ) = HomΓ (B r−2 , U ) =
ΩrΣ (B ′ ) is in ⊥
Σ U since id Γ U = r.
By (a), (b) and Theorem 2.9(e), we have dualities
⊥
ΣU

We take B ′′ in

⊥
ΓU

HomΓ (−,U)

HomΓ (−,T )

⊥F
T.
←−−−−−−− ⊥
Γ U −−−−−−−→

such that HomΓ (B ′′ , U ) ≃ HomΓ (B t , U ) as Σ-modules. Then

ΩtF (C) = HomΓ (B t , T ) ≃ HomΓ (B ′′ , T ) ∈ ⊥F T
as Λ-modules. Thus the claim holds.



Let F be an additive sub-bifunctor of Ext1Λ (−, −). A full subcategory X of
mod Λ is F -resolving (respectively F -coresolving) if
(i) X is F -extension closed,
(ii) P(F ) (respectively I(F )) is contained in X ,
(iii) if 0 → A → B → C → 0 is F -exact and B and C are in X (respectively A
and B are in X ), then A (respectively C) is in X .
We need the following preparation from Auslander–Buchweitz theory.

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IYAMA AND SOLBERG

Proposition 2.11 ([9, Theorems 2.4, 2.5, Proposition 2.2]). Let F be an additive
sub-bifunctor of Ext1Λ (−, −), and X an F -resolving subcategory of mod Λ. Assume
that the exact category (X , F ) given in Proposition 2.2 has enough F -injectives and
Xb = mod Λ. Then the following assertions hold.

(a) X is a contravariantly finite subcategory of mod Λ and Y := X ⊥F is a covariantly finite subcategory of mod Λ.
(b) X -resdimF (mod Λ) = idF (X ⊥F ) holds.
3. Elementary properties of n-precluster tilting subcategories

Recall that for n ≥ 1 a subcategory C of mod Λ is called n-cluster tilting if C is
functorially finite and
C = ⊥n−1 C = C ⊥n−1 ,
where

⊥n−1

C is the full subcategory of mod Λ given by the modules
⊥n−1

C = {X ∈ mod Λ | ExtiΛ (X, C) = 0 for 0 < i < n}.

The full subcategory C ⊥n−1 is defined dually. In particular, it follows immediately
from the definition that C is a generator-cogenerator for mod Λ and ExtiΛ (C, C) = 0
for 0 < i < n. In [21] the functors
−(n−1)

n−1
: mod Λ → mod Λ and τn− = τ − ΩΛ
τn = τ ΩΛ

: mod Λ → mod Λ

are defined as the n-Auslander–Reiten translations. By [22, Theorem 2.3.1], the pair
(τn− , τn ) forms an adjunction. By [21, Theorem 1.4.1], they induce equivalences
τn :

⊥n−1

Λ → DΛ⊥n−1 and τn− : DΛ⊥n−1 → ⊥n−1 Λ.

(3.1)

In particular τn and τn− give bijections between indecomposable non-projective
modules in ⊥n−1 Λ and indecomposable non-injective modules in DΛ⊥n−1 , while
they do not preserve indecomposability for arbitrary modules. Moreover, for any
n-cluster tilting subcategory C of mod Λ, they restrict to equivalences
τn : C → C and τn− : C → C.
The next result gives a higher analog of Auslander–Reiten duality.
Lemma 3.1 ([21, Theorem 1.5]). We have the following.
n−i
(A, τn (C)) for
(a) HomΛ (C, A) ≃ D ExtnΛ (A, τn (C)) and ExtiΛ (C, A) ≃ D ExtΛ
⊥n−1
Λ and all modules A in mod Λ.
0 < i < n, for all modules C in
n−i −
(b) HomΛ (C, A) ≃ D ExtnΛ (τn− (A), C) and ExtiΛ (C, A) ≃ D ExtΛ
(τn (A), C) for
0 < i < n, for all modules C in mod Λ and all modules A in DΛ⊥n−1 .

Proof. We give a proof of the second isomorphisms in our language of relative
homological algebra.
(a) Let F = F Λ be an additive sub-bifunctor of Ext1Λ (−, −). Let C be in ⊥n−1 Λ,
and let
n−1
η : 0 → ΩΛ
(C) → Pn−2 → · · · → P0 → C → 0

be a minimal projective resolution of C. Since C is in ⊥n−1 Λ, we have that the
exact sequence η is F -exact. Using this we have for 0 < i < n and an arbitrary

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

13

Λ-module A that
D ExtiF (C, A) ≃ D Ext1F (Ωi−1
Λ (C), A)
≃ D HomF (ΩiΛ (C), A)
≃ Ext1F (τ − (A), ΩiΛ (C))
n−i −
n−1
≃ ExtF
(τ (A), ΩΛ
(C))
n−i
≃ ExtΛ
(A, τn (C)),

where the second isomorphism uses that Pi is in I(F ), the third is the Auslander–
Reiten formula, the first and the fourth use dimension shift and the last one is given
by Theorem 2.7 and G = FΛ = Ext1Λ (−, −).
Since C is in ⊥n−1 Λ, we have that ExtiΛ (C, A) ≃ ExtiF (C, A) for 0 < i < n and
for all modules A in mod Λ by Proposition 2.5(a). Thus the assertion follows.
(b) Similar proof as in (a).

We introduce the notion of an n-precluster tilting subcategory by relaxing the
conditions for an n-cluster tilting subcategory.
Definition 3.2. A subcategory C of mod Λ is called an n-precluster tilting subcategory if it satisfies the following conditions.
(i)
(ii)
(iii)
(iv)

C is a generator-cogenerator for mod Λ,
τn (C) ⊆ C and τn− (C) ⊆ C,
ExtiΛ (C, C) = 0 for 0 < i < n,
C is a functorially finite subcategory of mod Λ.

If moreover C admits an additive generator M , we say that M is an n-precluster
tilting module.
Using τn and τn− , we define the subcategories
i
∞
Pn = add{τn−i (Λ)}∞
i=0 and In = add{τn (D(ΛΛ ))}i=0 .

For any n-precluster tilting subcategory C of mod Λ, we have
Pn ∨ In ⊆ C and C ⊆ DΛ⊥n−1 ∩ ⊥n−1 Λ
by (i), (ii) and (i), (iii) respectively.
Recall from [11] that an Artin algebra Λ is called τ -selfinjective if P1 is of finite
type, which is shown to be equivalent to that P1 is equal to I1 . We show next that
this is equivalent to the existence of a 1-precluster tilting Λ-module.
Example 3.3. An Artin algebra Λ is τ -selfinjective if and only if Λ has a 1precluster tilting module.
Proof. If Λ is τ -selfinjective, then clearly the additive generator of P1 is a 1precluster tilting module since DΛ ∈ I1 = P1 . If Λ has a finite 1-precluster
subcategory C, then it is clear from the definition that P1 is contained in C. Hence
P1 is of finite type and Λ is τ -selfinjective.

This observation leads us to the following definition.
Definition 3.4. An Artin algebra Λ is called τn -selfinjective if Λ admits an nprecluster tilting module.

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IYAMA AND SOLBERG

This is a common generalization of selfinjective algebras and n-representationfinite algebras. We continue by asking and giving one answer to the natural question: When is an Artin algebra τn -selfinjective?
Proposition 3.5. Let Λ be an Artin algebra and n ≥ 1. Then the following
conditions are equivalent.
(i) Λ is τn -selfinjective.
(ii) Pn ∨ In is of finite type and ExtiΛ (Pn ∨ In , Pn ∨ In ) = 0 for 0 < i < n.
(iii) In is of finite type, In ⊂ ⊥n−1 Λ and ExtiΛ (In , In ) = 0 for 0 < i < n.
(iv) Λ ∈ In and ExtiΛ (In , In ) = 0 for 0 < i < n.
(v) Pn is of finite type, Pn ⊂ DΛ⊥n−1 and ExtiΛ (Pn , Pn ) = 0 for 0 < i < n.
(vi) DΛ ∈ Pn and ExtiΛ (Pn , Pn ) = 0 for 0 < i < n.
Moreover, if these conditions are satisfied, then every additive generator of Pn ∨ In
is an n-precluster tilting Λ-module.
Proof. (i) is equivalent to (ii): Assume that there exists an n-precluster tilting
module M in mod Λ. Since Pn ∨ In ⊂ add M , it is immediate that Pn ∨ In is of
finite type and satisfies ExtiΛ (Pn ∨ In , Pn ∨ In ) = 0 for 0 < i < n. Conversely, if
Pn ∨ In satisfies (ii), then an additive generator of Pn ∨ In is an n-precluster tilting
Λ-module.
(ii) implies (iii): This is immediate.
(iii) implies (iv): For each indecomposable non-projective module X in ⊥n−1 Λ,
we know that τn (X) is indecomposable again by the equivalence (3.1). Since In is
of finite type, then τnl (I) 6= 0 is projective for some l ≥ 0 for all indecomposable
injective modules I. Since the number of indecomposable projectives and of indecomposable injectives coincide, all indecomposable projective modules must occur
in this way, hence Λ is in In and (iv) is satisfied.
(iv) implies (ii): For each indecomposable projective modules P there exists an
indecomposable injective module I such that P ≃ τnl (I). Since τn−i (P ) ≃ τnl−i (I)
−(l+1)
for 0 ≤ i ≤ l and τn
(P ) = 0 hold, we have Pn ∨ In = In . Thus (ii) is satisfied.
The equivalences of (ii), (v) and (vi) are shown dually.

Using the above we have the following consequence for n = 2 thanks to a general
property of I2 .
Proposition 3.6 (cf. [23, Proposition 1.7]). Let Λ be an Artin algebra. Then Λ is
τ2 -selfinjective if and only if Λ ∈ I2 .
Proof. By [23, Proposition 2.5] the subcategory I2 satisfies Ext1Λ (I2 , I2 ) = 0. The
claim then follows immediately from Proposition 3.5(iv)⇒(i).

The next results give analog of properties of higher Auslander–Reiten translation
for n-cluster tilting subcategories.
Proposition 3.7. Let Λ be an Artin algebra and C an n-precluster tilting subcategory with n ≥ 1. Then we have the following.
(a) We have mutually quasi-inverse equivalences τn : C → C and τn− : C → C.
(b) τn and τn− give bijections between indecomposable non-projective modules in C
and indecomposable non-injective modules in C.
(c) We have add{τn− (C), Λ} = C = add{τn (C), DΛ}.
(d) There exists a full subcategory D of mod Λ such that C = add{Pn ∨ In , D},
(Pn ∨ In ) ∩ D = {0} and τn (D) = D = τn− (D).

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

15

Proof. (a) We have C ⊆ DΛ⊥n−1 ∩ ⊥n−1 Λ. Since τn (C) ⊆ C and τn− (C) ⊆ C hold,
the claim follows from the equivalences (3.1).
(b)(c)(d) These follow immediately from (a).

An n-cluster tilting subcategory C satisfies by definition the equalities C =
C = C ⊥n−1 . The same is not true for an n-precluster tilting subcategory C
of mod Λ (e.g. Λ is non-semisimple selfinjective and C = add Λ). But the equality
⊥n−1
C = C ⊥n−1 is shown still to be true.
⊥n−1

Proposition 3.8. Let C be a subcategory of mod Λ satisfying the conditions (i),
(iii) and (iv) in Definition 3.2.
⊥n−1
C) ⊆ ⊥n−1 C and ΩFC (C ⊥n−1 ) ⊆ C ⊥n−1 .
(a) We have Ω−
FC(
(b) Assume n > 1. Then C is an n-precluster tilting subcategory of mod Λ if and
only if C ⊥n−1 = ⊥n−1 C.
Proof. (a) We only prove the first inclusion since the other one is dual.
f

→ C 0 → Y → 0 be an exact sequence with a
Let X be in ⊥n−1 C, and η : 0 → X −
minimal left C-approximation f of X. Applying HomΛ (−, C), one easily shows that
Y is in ⊥n−1 C.
(b) It suffices to show that τn (C) ⊆ C and τn− (C) ⊆ C hold if and only if C ⊥n−1 =
⊥n−1
C holds.
Assume τn (C) ⊆ C and τn− (C) ⊆ C. Since C ⊂ ⊥n−1 Λ, we have
n−i ⊥n−1
C, τn (C)) = 0
(
ExtiΛ (C, ⊥n−1 C) ≃ D ExtΛ

for all 0 < i < n by Lemma 3.1. Thus ⊥n−1 C ⊆ C ⊥n−1 holds. Similarly C ⊥n−1 ⊆
C holds. Consequently C ⊥n−1 = ⊥n−1 C.
Assume that C ⊥n−1 = ⊥n−1 C. By Lemma 3.1, we have

⊥n−1

n−i
n−i
(C, C ⊥n−1 ) = 0
(C, ⊥n−1 C) = D ExtΛ
ExtiΛ (⊥n−1 C, τn (C)) ≃ D ExtΛ

for all 0 < i < n. Thus τn (C) ⊆ (⊥n−1 C)⊥n−1 . We show (⊥n−1 C)⊥n−1 ⊆ C. Since
C ⊆ ⊥n−1 C, we have ⊥n−1 C = C ⊥n−1 ⊇ (⊥n−1 C)⊥n−1 . For any X in (⊥n−1 C)⊥n−1 ,
there exists an exact sequence η : 0 → X → C 0 → Y → 0 with C 0 ∈ C and
Y ∈ ⊥n−1 C by (a). This splits by the assumption n > 1, and X is a direct summand
of C 0 . Hence X is in C.

Similarly we prove that τn− (C) ⊆ C.
Now we introduce the following category U(C), which is an analog of Calabi–Yau
reduction of triangulated categories [27].
Definition 3.9. For an n-precluster tilting subcategory C in mod Λ, let
Z(C) = C ⊥n−1 = ⊥n−1 C and U(C) = Z(C)/[C].
Note that when C is a 1-precluster tilting subcategory, then Z(C) = mod Λ, since
the orthogonality condition is void.
The next result gives basic properties of Z(C) which generalize those of n-cluster
tilting subcategories [21, Theorems 2.3, 2.3.1, 2.2.3]. In particular it gives higher
Auslander–Reiten translation for Z(C) extending Proposition 3.7.
Theorem 3.10. Let Λ be an Artin algebra and C an n-precluster tilting subcategory
of mod Λ with n ≥ 1. Then we have the following.
(a) We have equivalences τn : Z(C) → Z(C) and τn− : Z(C) → Z(C).

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IYAMA AND SOLBERG

(b) τn and τn− give bijections between indecomposable non-projective modules in
Z(C) and indecomposable non-injective modules in Z(C).
(c) We have add{τn (Z(C)), DΛ} = Z(C) = add{τn− (Z(C)), Λ}.
(d) For every X and Y in Z(C) and 0 < i < n, we have functorial isomorphisms
HomΛ (X, Y ) ≃ D ExtnΛ (Y, τn (X)),
HomΛ (X, Y ) ≃

D ExtnΛ (τn− (Y

), X),

n−i
ExtiΛ (X, Y ) ≃ D ExtΛ
(Y, τn (X))
n−i −
ExtiΛ (X, Y ) ≃ D ExtΛ
(τn (Y ), X).

(e) For every X in mod Λ, there exists an FC -exact sequence
0 → Zn−1 → Cn−2 → · · · → C0 → X → 0
with Ci in C for every i and Zn−1 in Z(C).
Proof. (a) Thanks to the equivalences (3.1), it suffices to show τn (Z(C)) ⊆ Z(C)
and τn− (Z(C)) ⊆ Z(C). Using Lemma 3.1, we have
n−i
0 = D ExtiΛ (Z(C), C) ≃ ExtΛ
(C, τn (Z(C)))
n−i −
0 = D ExtiΛ (C, Z(C)) ≃ ExtΛ
(τn (Z(C)), C)

for 0 < i < n. Thus the assertion follows.
(b)(c) Immediate from (a).
(d) This follows from Lemma 3.1.
(e) Let X be in mod Λ, and let 0 → ΩFC (X) → C0 → X → 0 be an FC exact sequence given by an FC -projective cover. Then Ext1Λ (C, ΩFC (X)) = 0 holds.
Taking an FC -exact sequence 0 → Ω2FC (X) → C1 → ΩFC (X) → 0 given by an FC projective cover, it follows that ExtiΛ (C, Ω2FC (X)) = 0 for i = 1, 2. Continuing this
n−1
n−1
process we obtain that ExtiΛ (C, ΩF
(X)) = 0 for 0 < i < n, and hence ΩF
(X)
C
C
is in Z(C).

The following easy property below is useful.
Lemma 3.11. Let C be an n-precluster tilting subcategory of mod Λ with n ≥ 1.
(a) We have FC |Z(C)×Z(C) = F C |Z(C)×Z(C) .
(b) For every 0 < i < n, we have ExtiFC (−, −)|Z(C)×Z(C) = ExtiΛ (−, −)|Z(C)×Z(C) =
ExtiF C (−, −)|Z(C)×Z(C) .
(c) The exact categories (Z(C), FC ) and (Z(C), F C ) are the same. It coincides with
(Z(C), Ext1Λ (−, −)) if n ≥ 2.
Proof. (a) (c) For n = 1, we have FC = F τ C = F C by (2.1) since C is 1-precluster tilting, and for n ≥ 2, we have FC |Z(C)×Z(C) = Ext1Λ (−, −)|Z(C)×Z(C) = F C |Z(C)×Z(C) .
(b) This is immediate from Proposition 2.5.

The category Z(C) enjoys the following remarkable properties.
Proposition 3.12. Let C be an n-precluster tilting subcategory of mod Λ for some
n ≥ 1.
(a) Z(C) is extension closed.
(b) Z(C) has a structure of a Frobenius category whose short exact sequences are
precisely FC -exact sequences, and projective-injective objects are precisely C.
(c) U(C) has a structure of a triangulated category with the suspension functor
[1] = Ω−1
FC.

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

17

Proof. (a) This is easily checked by using the long exact sequence of Ext’s.
(b) By Proposition 2.2, we have an exact category (Z(C), FC ), which coincides
with (Z(C), F C ) by Lemma 3.11(c). Thus any object in C is projective-injective in
Z(C). By Proposition 3.8(a), Z(C) has enough projectives and enough injectives.
Therefore the projective and the injective objects coincide, and they are equal to
C. Thus the assertion follows.
(c) This is a general property of Frobenius categories [17].

We show that the triangulated category U(C) admits a Serre functor, which is
an analog of [27, Theorem 4.7].
Theorem 3.13. Let C be an n-precluster tilting subcategory of mod Λ with n ≥ 1.
(a) The triangulated category U(C) admits a Serre functor S given by S = [n] ◦ τn .
(b) The triangulated category U(C) has almost split triangles, i.e. any indecomposable object X in U(C) has almost split triangles in U(C):
SX[−1] → E → X → SX and X → E ′ → S −1 X[1] → X[1].
(c) The Frobenius category Z(C) has almost split sequences, i.e. any indecomposable
module X in Z(C) \ C has almost split sequences in Z(C)
−
0 → τZ(C) (X) → E → X → 0 and 0 → X → E ′ → τZ(C)
(X) → 0,
−(n−1)

where τZ(C) := ΩZ(C)

n−1 −
−
τn .
:= ΩZ(C)
τn and τZ(C)

Proof. (a) Let F = F C and G = FC . Then I(F ) = P(G) = C holds. For all X in
Z(C), take an F -injective coresolution
0 → τn (X) → C 0 → · · · → C n−1 → Ω−n
F (τn (X)) → 0,
which gives a G-projective resolution since Z(C) is Frobenius by Proposition 3.12.
Thus, for every Y in mod Λ, we obtain functorial isomorphisms
−n
(τn (X)))
ExtnG (Y, τn (X)) ≃ Coker(HomΛ (Y, C n−1 ) → HomΛ (Y, ΩF

≃ HomG (Y, Ω−n
F (τn (X))).
Moreover we have functorial isomorphisms
n−1
ExtnG (Y, τn (X)) ≃ ExtnF (τ − (Y ), ΩΛ
(X))

≃ Ext1F (τ − (Y ), X)
≃ D HomF (X, Y ),
where the first isomorphism is given by Theorem 2.7, the second follows from
P(Λ) ⊂ I(F ), and the third is the Auslander–Reiten duality (Proposition 2.4).
Combining them, for every Y in Z(C), we have functorial isomorphisms
−n
HomU (C) (Y, Ω−n
F (τn (X))) = HomG (Y, ΩF (τn (X)))

≃ D HomF (X, Y ) = D(HomU (C) (X, Y )).
Thus U(C) has S = Ω−n
F τn = [n] ◦ τn as a Serre functor.
(b) Since U(C) has a Serre functor, it has almost split triangles [37].
(c) Immediate from (b).

(3.2)



18

IYAMA AND SOLBERG

The next bijective correspondence is an analog of a property of Calabi–Yau
reduction of triangulated categories [27, Theorem 4.9]. One of the consequences is
that, if Pn ∨ In is a functorially finite subcategory of mod Λ, then the classification
problem of n-cluster tilting subcategories in mod Λ can be reduced to the same
problem in the triangulated category U(Pn ∨ In ).
Theorem 3.14. Let C be an n-precluster tilting subcategory of mod Λ with n ≥
1. Then there exists a bijection between n-cluster tilting subcategories of mod Λ
containing C and n-cluster tilting subcategories of U(C) given by C ′ 7→ C ′ /[C].
Proof. Any n-cluster tilting subcategory of mod Λ containing C is clearly contained
in Z(C). On the other hand, let C ′ be a subcategory of mod Λ containing C. It
follows from Lemma 3.11(b) that C ′ is an n-cluster tilting subcategory of mod Λ if
and only if C ′ /[C] is an n-cluster tilting subcategory of U(C). Thus the assertion
follows.

Next we give another proof of existence of almost split sequences in Z(C) by
showing that Z(C) is functorially finite.
Theorem 3.15. Let C be an n-precluster tilting subcategory of mod Λ with n ≥ 1.
(a) Z(C) is FC -resolving and F C -coresolving in mod Λ.
(b) Z(C) is functorially finite in mod Λ with Z(C)-resdimFC (mod Λ) ≤ n − 1 and
idFC C ≤ n − 1.
(c) Z(C) has almost split sequences.
Proof. Since the case n = 1 is clear, we assume n > 1 in the rest.
(a) This is easily checked by using the long exact sequence of Ext’s.
(b) By Theorem 3.10(e) we have Z(C)-resdimFC (mod Λ) ≤ n − 1 and hence
[
Z(C) = mod Λ. Moreover Z(C) is FC -resolving by (a), and the exact category
(Z(C), FC ) has enough injectives C by Proposition 3.12(b). Therefore, by Proposition 2.11, Z(C) is contravariantly finite in mod Λ and idFC C ≤ n − 1 holds. Dual
arguments show that Z(C) is covariantly finite, hence it is functorially finite.
(c) Since Z(C) is extension closed and functorially finite, it has almost split
sequences [7].

We end this section with the following observation on Auslander–Buchweitz type
approximations by Z(C).
Corollary 3.16. Let C be an n-precluster tilting subcategory of Λ with n ≥ 1 and
X in mod Λ.
(a) For every 0 ≤ i ≤ n − 1, there exists an FC -exact sequence
fn−1

fi+2

fi+1

fi

fi−1

f0

f1

0 → Cn−1 −−−→ · · · −−−→ Ci+1 −−−→ Zi −→ Ci−1 −−−→ · · · −→ C0 −→ X → 0
(3.3)
with Zi in Z(C) and Cj in C for every j. Moreover Im fj is in C ⊥j for every j.
(b) For every 0 ≤ i ≤ n − 1, there exists an F C -exact sequence
f0

f1

f i−1

fi

f i+1

f i+2

f n−1

0 → X −→ C 0 −→ · · · −−−→ C i−1 −→ Z i −−−→ C i+1 −−−→ · · · −−−→ C n−1 → 0
with Z i in Z(C) and C j in C for every j. Moreover Im f j is in
j.

⊥j

C for every

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

19

Proof. (a) The case i = n − 1 was shown in Theorem 3.10(e).
Assume that the sequence (3.3) exists. Take an exact sequence 0 → Zi → Ci →
′
′
Zi−1
→ 0 with Ci in C and Zi−1
in Z(C), and consider the pushout diagram.
0


0

/ Cn−1

/ ···

/ Ci+1

0

/ Zi

0

/ Cn−1

/ ···

/ Ci+1


/ Ci


/ Zi−1


′
Zi−1


′
Zi−1


0


0

/ Ci−1

/ Ci−2

/ ···

/ C0

/X

/0

/ Ci−2

/ ···

/ C0

/X

/0

′
Since Zi−1 ≃ Zi−1
⊕ Ci−1 belongs to Z(C), the lower sequence gives the desired
sequence for i − 1. Thus the assertion follows inductively.
(b) This is dual to (a).


4. Auslander–Solberg correspondence
In [22], an Artin algebra Γ is called an n-Auslander algebra if domdim Γ ≥ n+1 ≥
gldim Γ. In addition, there exists a bijection between the set of equivalence classes
of n-cluster tilting subcategories with additive generators and Morita-equivalence
classes of n-Auslander algebras. A slightly more general class of Artin algebras were
considered in [11] for n = 1, namely Artin Gorenstein algebras Γ with domdim Γ =
2 = id Γ Γ. These algebras were classified in [11] in terms of τ -selfinjective algebras
Λ. If one considers the weaker condition domdim Γ ≥ 2 ≥ id Γ Γ, then the difference
is precisely the class of selfinjective algebras (Proposition 4.1).
This section is devoted to extending the above result by exhibiting a bijection between the Morita-equivalence classes of Artin Gorenstein algebras Γ with
domdim Γ ≥ n + 1 ≥ id Γ Γ for n ≥ 1 and equivalence classes of finite n-precluster
tilting subcategories. This class of Gorenstein algebras we call n-minimal Auslander–
Gorenstein algebras.
First we give some general properties of the class of n-minimal Auslander–
Gorenstein algebras.
Proposition 4.1. (a) If Γ is an n-minimal Auslander–Gorenstein algebra, then
either domdim Γ = n + 1 = idΓ Γ holds or Γ is selfinjective. In the latter case
Γ is an m-Auslander–Gorenstein algebra for all m ≥ 0.
(b) An algebra Γ is n-minimal Auslander–Gorenstein if and only if Γop is nminimal Auslander–Gorenstein.
Proof. (a) If Γ is not selfinjective, then the last term of the minimal injective
coresolution of Γ is not projective. Thus the assertion follows from the definition
of an n-minimal Auslander–Gorenstein algebra.
(b) Although the assertion follows from well-known results ([38, Theorem 7.7]
and [5, Corollary 5.5]), we give a direct argument here. Assume that Γ is n-minimal
Auslander–Gorenstein. The minimal injective coresolution
0 → Γ → I 0 → · · · → I n → I n+1 → 0

(4.1)
n+1

n+1

of Γ gives a projective resolution of the injective Γ-module I
. Thus Ω
and
Ω−(n+1) give mutually inverse bijections between indecomposable non-projective

20

IYAMA AND SOLBERG

injective Γ-modules and indecomposable non-injective projective Γ-modules since
their numbers are the same. In particular, all indecomposable non-projective injective Γ-modules appear in I n+1 as a direct summand. Applying D to (4.1), we have
an exact sequence
0 → DI n+1 → DI n → · · · → DI 0 → DΓ → 0
of Γop -modules with projective-injective Γop -modules DI i with 0 ≤ i ≤ n. Since all
indecomposable non-injective projective Γop -modules appear in DI n+1 as a direct
summand, Γop is also n-minimal Auslander–Gorenstein.

Let us recall the following simple observation.
Lemma 4.2 ([35, Lemma 3], [22, d = m = 0 in Theorem 4.2.1]). Let Λ be an
Artin algebra, M a generator-cogenerator of Λ, Γ = EndΛ (M ) and n ≥ 1. Then
domdim Γ ≥ n + 1 holds if and only if ExtiΛ (M, M ) = 0 for 0 < i < n.
The following result shows that a finite n-precluster tilting subcategory gives rise
to an n-minimal Auslander–Gorenstein algebra.
Proposition 4.3. Let Λ be an Artin algebra and M an n-precluster tilting Λ-module
with n ≥ 1.
(a) Γ = EndΛ (M ) is an n-minimal Auslander–Gorenstein algebra and the Γ-module
I = Γ M satisfies P(Γ) ∩ I(Γ) = add I and Λ ≃ EndΓ (I).
(b) M is an FM -cotilting Λ-module with idFM M ≤ n − 1.
(c) M is a projective Λ-module if and only if Γ is selfinjective.
Proof. By Lemma 4.2 we know that domdim Γ ≥ n+1. Let G = FM ⊆ Ext1Λ (−, −).
Then clearly ExtiG (M, M ) = 0 for all i > 0. Since τn− (M ) ∈ add M by our assumption, we have
−(n−1)

I(G) = add{τ (M ), DΛ} = add{ΩΛ

(M ), DΛ}

(4.2)

by Proposition 3.7(a). The start of the minimal injective coresolution
−(n−1)

0 → M → I 0 → I 1 → · · · → I n−2 → ΩΛ

(M ) → 0

ExtiΛ (M, M )

= 0 holds for every 0 < i < n. It follows
of M is G-exact since
\
from (4.2) that idG M ≤ n − 1 and I(G) ⊆ add
M . Therefore M is a G-cotilting
module. By Theorem 2.9(d), HomΛ (M, M ) = Γ is a cotilting Γ-module with id Γ Γ ≤
idG M + 2 ≤ n + 1. Thus Γ is an n-minimal Auslander–Gorenstein algebra.
Since I = HomΛ (Λ, M ) belongs to P(Γ) and DI = HomΛ (M, DΛ) belongs to
P(Γop ), we have I ∈ P(Γ) ∩ I(Γ). Taking a projective cover P → M → 0 in mod Λ
and applying HomΛ (−, M ), we have an exact sequence 0 → Γ → HomΛ (P, M ) in
mod Γ with HomΛ (P, M ) ∈ add I. Thus I is an additive generator of P(Γ) ∩ I(Γ).
In addition, we have Λ ≃ EndΓ (I) from Theorem 2.9(a) since I = M is a G-cotilting
module.
The last claim is easy and left to the reader. This completes the proof.

Next we show the converse, namely n-minimal Auslander–Gorenstein algebras Γ
give rise to a finite n-precluster tilting subcategory.
Proposition 4.4. Let Γ be an n-minimal Auslander–Gorenstein algebra for n ≥ 1.
Let I be an additive generator of P(Γ) ∩ I(Γ), Λ = EndΓ (I) and M = Λ I.
(a) M is an n-precluster tilting Λ-module such that EndΛ (M ) ≃ Γ.

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

21

(b) M is an FM -cotilting Λ-module with idFM M ≤ n − 1.
(c) Γ is selfinjective if and only if M is a projective Λ-module.
Proof. We can assume that I is basic. Since domdim Γ ≥ n + 1 ≥ 2, the module I
is a dualizing summand of the cotilting Γ-module Γ. Thus the claim Γ ≃ EndΛ (M )
follows directly from Theorem 2.10(a).
In the rest, we show that M is an n-precluster tilting Λ-module.
(i) We show that M is a generator-cogenerator for Λ.
Since the Γ-module I belongs to add Γ, the Λ-module Λ = EndΓ (I) belongs to
add HomΓ (Γ, I) = add M . Thus M is a generator of Λ. Since the Γ-module I belongs to add D(ΓΓ ), the Λop -module Λ = EndΓ (I) belongs to add HomΓ (I, D(Γ)) =
add D(M ). Thus M is a cogenerator of Λ.
(ii) It follows from Lemma 4.2 that ExtiΛ (M, M ) = 0 for 0 < i < n.
(iii) It remains to show that both τn− (M ) and τn (M ) are in add M .
Let G = FM ⊆ Ext1Λ (−, −). Then by Theorem 2.10 we infer that M is a Gcotilting module with idG M ≤ max{id Γ Γ − 2, 0} ≤ n − 1. By (ii), the injective
coresolution
−(n−1)

0 → M → I 0 → I 1 → · · · → I n−2 → ΩΛ

(M ) → 0

of the Λ-module M is G-exact, and gives the start of a G-injective coresolution.
−(n−1)
(M ) is in I(G) = add{DΛ, τ (M )}. Hence
Since idG M ≤ n− 1, the module ΩΛ
−(n−1)
(M )) is in add M .
τn− (M ) = τ − (ΩΛ
On the other hand, since Γop is an n-minimal Auslander–Gorenstein algebra such
that P(Γop ) ∩ I(Γop ) = add DI and EndΓop (DI) = Λop , the Λop -module τn− (DM )
is in add DM by the same argument. Thus the Λ-module τn (M ) is in add Λ M .
This completes the proof that M is an n-precluster tilting Λ-module.
The last claim is easy and left to the reader. This completes the proof.

Now we address the bijectivity of the correspondence.
Theorem 4.5. Fix n ≥ 1. There is a bijection between Morita-equivalence classes
of n-minimal Auslander–Gorenstein algebras and equivalence classes of finite nprecluster tilting subcategories C of Artin algebras, where the correspondences are
given in Propositions 4.3 and 4.4.
Proof. The assertions follow from Propositions 4.3 and 4.4.



Remark 4.6. Note that the bijection for n-Auslander algebras given in [22] is dual
to Theorem 4.5.
The category Z(C) associated to a finite n-precluster tilting subcategory C has
the following interpretation in terms of the corresponding n-minimal Auslander–
Gorenstein algebra.
Theorem 4.7. Given an Artin algebra Λ with a finite n-precluster tilting subcategory C = add M , let Γ = EndΛ (M ) be the corresponding n-minimal Auslander–
Gorenstein algebra. Then Z(C) and CM Γ are dual categories via the functors
HomΛ (−, M ) : Z(C) → CM Γ and HomΓ (−, M ) : CM Γ → Z(C). Moreover they
induce triangle equivalences between U(C) and (CM Γ)op .
Proof. Let G = FM . Then M is a G-cotilting module with idG M ≤ n − 1 by the
proof of Proposition 4.3(b), and we have a duality
HomΛ (−, M ) :

⊥G

M → ⊥ Γ = CM Γ

22

IYAMA AND SOLBERG

by Theorem 2.9(e). Let us prove Z(C) = ⊥G M . Since idG M ≤ n − 1, it follows
that ExtiG (−, M ) = 0 for all i ≥ n. Furthermore, since M is in Z(C), we have that
ExtiG (−, M ) = ExtiΛ (−, M ) for 0 < i < n by Proposition 2.5(b). Thus Z(C) =
⊥G
M holds. Since C = add M corresponds to P(Γ) via the duality HomΛ (−, M ),
the last claim follows immediately.

The class of n-Auslander algebras were introduced in [22] as the Artin algebras
Γ with domdim Γ ≥ n + 1 ≥ gldim Γ. Now we characterize this subclass of algebras
within the class of n-minimal Auslander–Gorenstein algebras.
Theorem 4.8. Let Λ be an Artin algebra, C = add M a finite n-precluster tilting
Λ-module with n ≥ 1 and Γ = EndΛ (M ). Then the following are equivalent.
(a) Γ is an n-Auslander algebra.
(b) gldim Γ < ∞.
(c) CM Γ = P(Γ).
(d) Z(C) = C.
Proof. (a) implies (b): This is obvious.
(b) implies (c): If X in CM Γ is non-projective, then Extm
Γ (X, Γ) 6= 0 holds for
m := pdΓ X > 0, a contradiction.
(c) implies (a): For any X in mod Γ, Ωn+1
(X) belongs to CM Γ = P(Γ) by using
Γ
dimension shifting and idΓ Γ ≤ n + 1 in Proposition 4.3(a). Thus gldim Γ ≤ n + 1.
(c) is equivalent to (d): This follows from Theorem 4.7.

For a general Artin Gorenstein algebra Γ, the category of maximal Cohen–
Macaulay modules CM Γ = ⊥ Γ is an extension closed functorially finite subcategory of mod Γ [4]. Therefore the category CM Γ has minimal left (respectively
right) almost split maps and almost split sequences. We denote by τCM Γ the
Auslander–Reiten translation in CM Γ.
We end this section with the following easy observations, which compare almost
split sequences in Z(C) with those in CM Γ.
Proposition 4.9. Let Λ be an Artin algebra, C = add M a finite n-precluster tilting
Λ-module with n ≥ 1 and Γ = EndΛ (M ).
(a) A morphism f : A → B is (minimal) left almost split (respectively (minimal) right almost split) in CM Γ if and only if HomΓ (f, M ) : HomΓ (B, M ) →
HomΓ (A, M ) is (minimal) right almost split (respectively (minimal) left almost
split) in Z(C).
(b) An exact sequence 0 → A → B → C → 0 in CM Γ is almost split in CM Γ if
and only if 0 → HomΓ (C, M ) → HomΓ (B, M ) → HomΓ (A, M ) → 0 is almost
split in Z(C).
−
(c) τZ(C)
(HomΓ (C, M )) ≃ HomΓ (τCM Γ (C), M ) holds for every indecomposable mod−
ule C in CM Γ \ P(Γ), and τCM
Γ (HomΛ (X, M )) ≃ HomΛ (τZ(C) (X), M ) holds
for every indecomposable module X in Z(C) \ C.
Proof. All assertions are immediate from Theorem 4.7.



5. n-fold almost split extensions
Higher Auslander–Reiten theory on n-cluster tilting subcategories was introduced in [21]. An n-almost split sequence in an n-cluster tilting subcategory C is

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

23

an n-fold exact sequence
η : 0 → Y → Cn → · · · → C1 → X → 0
with indecomposable objects X, Y in C and objects {Ci }ni=1 in C, and the sequences
0 → HomΛ (C, Y ) → HomΛ (C, Cn ) → · · · → HomΛ (C, C1 ) → radΛ (C, X) → 0
0 → HomΛ (X, C) → HomΛ (C1 , C) → · · · → HomΛ (Cn , C) → radΛ (Y, C) → 0
are exact. This is equivalent to the statement that η represents an element in
the socle of ExtnΛ (X, Y ) as an EndΛ (X)op - or EndΛ (Y )-module. These properties
play a key role to generalize the notion of n-almost split sequences to more general
categories, and we make the following definition.
Definition 5.1. Let Λ be an Artin algebra, X and Y be modules in mod Λ, and η
a non-zero element in ExtnΛ (X, Y ).
(a) We say that η is a right n-fold almost split extension of X if, for every Z
in mod Λ and a non-zero element ξ in ExtnΛ (X, Z), there exists a morphism
f : Z → Y such that η = f ξ, where f ξ stands for ExtnΛ (X, f )(ξ).
(b) We say that η is a left n-fold almost split extension of Y if, for every Z in mod Λ
and a non-zero element ξ in ExtnΛ (Z, Y ), there exists a morphism g : X → Z
such that η = ξg, where ξg stands for ExtnΛ (g, Y )(ξ).
(c) We say that η is an n-fold almost split extension if it is a right n-fold almost
split extension of X and a left n-fold almost split extension of Y .
We changed the terminology from n-almost split sequence to n-fold almost split
extension, since an element in ExtnΛ (X, Y ) can possibly be represented by several
different long exact sequences.
We need the following isomorphisms to study n-fold almost split extensions.
n−1
(X), −) ≃ D HomΛ (−, τn (X)).
ExtnΛ (X, −) = Ext1Λ (ΩΛ

(5.1)

ExtnΛ (X, τn (X))

In particular, τn (X) 6= 0 implies
≃ DEndΛ (τn (X)) 6= 0. We have
the following characterizations of n-fold almost split extensions.
Lemma 5.2. Let X and Y be modules in mod Λ, and η a non-zero element in
ExtnΛ (X, Y ). Then the following conditions are equivalent.
(i) η is a right n-fold almost split extension of X.
(ii) Soc ExtnΛ (X, −) is a simple functor on mod Λ and generated by η.
(iii) τn (X) is indecomposable, and f η = 0 holds in ExtnΛ (X, Z) for every morphism
f : Y → Z in the radical of mod Λ.
If Y is indecomposable, then the following condition is also equivalent.
(iv) Y ≃ τn (X) and η is a non-zero element in Soc EndΛ (Y ) ExtnΛ (X, Y ).
Proof. (i) is equivalent to (ii): Clearly (i) is equivalent to that η is contained in every
non-zero subfunctor of ExtnΛ (X, −). This is equivalent to (ii) since any subfunctor
of ExtnΛ (X, −) has a non-zero socle by (5.1).
(ii) is equivalent to (iii): Since (5.1) holds, Soc ExtnΛ (X, −) is simple if and only
if τn (X) is indecomposable in mod Λ. In this case, η belongs to the socle if and
only if it is annihilated by the radical of mod Λ. Thus the assertion follows.
(ii) is equivalent to (iv): This is immediate from (5.1).

Now we have the following existence and uniqueness result of right n-fold almost
split extensions.

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IYAMA AND SOLBERG

Proposition 5.3. Let Λ be an Artin algebra, and n ≥ 1. A module X in mod Λ
has a right n-fold almost split extension if and only if τn (X) is indecomposable. If
these conditions are satisfied, then the following assertions hold.
(a) Let Y be an indecomposable module in mod Λ and η an element in ExtnΛ (X, Y ).
Then η is a right n-fold almost split extensions of X if and only if Y ≃ τn (X)
and η is a non-zero element in SocEndΛ (Y ) ExtnΛ (X, Y ).
(b) For i = 1, 2, assume that ηi in ExtnΛ (X, Yi ) is a right n-fold almost split extension of X with an indecomposable module Yi . Then there exists an isomorphism
f : Y1 → Y2 such that η2 = f η1 .
Proof. “Only if” part follows from Lemma 5.2(i)⇒(iii), and “if” part follows from
Lemma 5.2(iv)⇒(i).
(a) This was shown in Lemma 5.2(i)⇔(iv).
(b) By our assumption, there exists f : Y1 → Y2 and g : Y2 → Y1 satisfying
η2 = f η1 and η1 = gη2 . Since (1 − gf )η1 = 0 holds, 1 − gf is not an automorphism
of Y1 , and hence gf is an automorphism of Y1 . Similarly f g is an automorphism of
Y2 , and hence f and g are isomorphisms.

We record dual results.
Lemma 5.4. Let X and Y be modules in mod Λ, and η a non-zero element in
ExtnΛ (X, Y ). Then the following conditions are equivalent.
(i) η is a left n-fold almost split extension of Y .
(ii) Soc ExtnΛ (−, Y ) is a simple functor on mod Λ and generated by η.
(iii) τn− (Y ) is indecomposable, and ηf = 0 holds in ExtnΛ (Z, Y ) for every morphism
f : Z → X in the radical of mod Λ.
If X is indecomposable, then the following condition is also equivalent.
(iv) X ≃ τn− (Y ) and η is a non-zero element in Soc ExtnΛ (X, Y )EndΛ (X) .
We also have the following dual result for left n-fold almost split extensions.
Proposition 5.5. Let Λ be an Artin algebra, and n ≥ 1. A module Y in mod Λ
has a left n-fold almost split extension if and only if τn− (Y ) is indecomposable. If
these conditions are satisfied, then the following assertions hold.
(a) Let X be an indecomposable module in mod Λ and η an element in ExtnΛ (X, Y ).
Then η is a left n-fold almost split extensions of Y if and only if X ≃ τn− (Y )
and η is a non-zero element in Soc ExtnΛ (X, Y )EndΛ (X) .
(b) For i = 1, 2, assume that ηi in ExtnΛ (Xi , Y ) is a left n-fold almost split extension
of Y with an indecomposable module Xi . Then there exists an isomorphism
f : X2 → X1 such that η2 = η1 f .
The following easy observation shows that all n-fold almost split extensions can
be obtained from those between indecomposable modules.
Lℓ
Lm
Lemma 5.6. Let X =
j=1 Yj be modules in mod Λ with
i=1 Xi and Y =
indecomposableLsummands Xi and Yj , and η = (ηij )i,j a non-zero element in
ExtnΛ (X, Y ) = i,j ExtnΛ (Xi , Yj ).
(a) η is a right n-fold almost split extension of X if and only if the following
conditions hold.
(i) There exists 1 ≤ i0 ≤ m such that ExtnΛ (Xi , −) = 0 for every i 6= i0 .

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

25

(ii) ηi0 j is either zero or a right n-fold almost split extension of Xi0 for every
j.
(b) η is a left n-fold almost split extension of Y if and only if the following conditions hold.
(i) There exists 1 ≤ j0 ≤ ℓ such that ExtnΛ (−, Yj ) = 0 for every j 6= j0 .
(ii) ηij0 is either zero or a left n-fold almost split extension of Yj0 for every j.
(c) η is an n-fold almost split extension if and only if the following conditions hold.
(i) There exist 1 ≤ i0 ≤ m and 1 ≤ j0 ≤ ℓ such that ExtnΛ (Xi , −) = 0 for
every i 6= i0 and ExtnΛ (−, Yj ) = 0 for every j 6= j0 .
(ii) ηi0 j0 is an n-fold almost split extension.
Proof. (a) The “if” part follows directly from definition. We show the ”only if”
part. It follows from Lemma 5.2(iii) that τn (X) is indecomposable, and hence (i)
holds. One can check (ii) easily from definition.
(b) This is dual of (a).
(c) This follows immediately from (a) and (b).

Now we consider n-fold almost split extensions.
Lemma 5.7. Let X and Y be indecomposable modules in mod Λ satisfying Y ≃
τn (X) and X ≃ τn− (Y ). Then Soc EndΛ (Y ) ExtnΛ (X, Y ) = Soc ExtnΛ (X, Y )EndΛ (X)
holds.
Proof. Let EX := EndΛ (X) and EY := EndΛ (Y ). Since Soc EY ExtnΛ (X, Y ) ≃
D(Top HomΛ (Y, τn (X))EY ) is simple, it is contained in any non-zero submodule of EY ExtnΛ (X, Y ). Since Soc ExtnΛ (X, Y )EX is also a non-zero submodule of
n
n
n
EY ExtΛ (X, Y ), we have Soc EY ExtΛ (X, Y ) ⊂ Soc ExtΛ (X, Y )EX . Since the reverse inclusion holds by the same argument, the desired equality holds.

We have the following characterization of n-fold almost split extensions.
Theorem 5.8. Let Λ be an Artin algebra, and n ≥ 1.
(a) Let X be an indecomposable module in mod Λ. There exists an n-fold almost
split extension in ExtnΛ (X, Y ) for some module Y in mod Λ if and only if τn (X)
is indecomposable and τn− τn (X) ≃ X. If these conditions are satisfied, then any
right n-fold almost split extension η of X in ExtnΛ (X, Y ) with an indecomposable
module Y is an n-fold almost splits extension.
(b) Let Y be an indecomposable module in mod Λ. There exists an n-fold almost
split extension in ExtnΛ (X, Y ) for some module X in mod Λ if and only if τn− (Y )
is indecomposable and τn τn− (Y ) ≃ Y . If these conditions are satisfied, then any
left n-fold almost split extension η of Y in ExtnΛ (X, Y ) with an indecomposable
module X is an n-fold almost splits extension.
Proof. (a) We show the “if” part. Let Y := τn (X). Then Soc EndΛ (Y ) ExtnΛ (X, Y ) =
Soc ExtnΛ (X, Y )EndΛ (X) holds by Lemma 5.7, and any non-zero element is a right
n-fold almost split extension of X by Proposition 5.3(a) and left n-fold almost split
extension of X by Proposition 5.5(a) .
This argument also gives a proof of the second statement.
We show the “only if” part. By Lemma 5.6, we can assume that Y is indecomposable. By Propositions 5.3(a) and 5.5(a), we have Y ≃ τn (X) and X ≃ τn− (Y ).
Thus the assertions hold.
(b) This is dual of (a).


26

IYAMA AND SOLBERG

Applying these general results to n-precluster tilting subcategories, we have the
following result.
Theorem 5.9. Let Λ be an Artin algebra and C an n-precluster tilting subcategory
in mod Λ with n ≥ 1.
(a) Every indecomposable object X in Z(C) \ P(Λ) has an n-fold almost split extension in ExtnΛ (X, τn (X)).
(b) Every indecomposable object Y in Z(C) \ I(Λ) has an n-fold almost split extension in ExtnΛ (τn− (Y ), Y ).
Proof. Since τn (X) and τn− (Y ) are indecomposable by the equivalences (3.1), the
assertions follow immediate from Propositions 5.3 and 5.5.

Now we consider exact sequences representing a right n-fold almost split extension of a module X in mod Λ such that τn (X) is indecomposable. We consider a
projective resolution
n−1
0 → ΩΛ
(X) → Pn−2 → · · · → P0 → X → 0.
n−1
Then ΩΛ
(X) = Z ⊕ P holds for an indecomposable module Z in (mod Λ) \ P(Λ)
and a module P in P(Λ). Let
n−1
(X) → 0
0 → τn (X) → E ⊕ P → ΩΛ

be a direct sum of an almost split sequence 0 → τn (X) → E → Z → 0 and a split
exact sequence 0 → 0 → P → P → 0. Taking the Yoneda product of these exact
sequences, we have an exact sequence
0 → τn (X) → E ⊕ P → Pn−2 → · · · → P0 → X → 0

(5.2)

which represents a right n-fold almost split extensions of a module X. In fact, one
can apply Lemma 5.2(iii)⇒(i).
Dually, a left n-fold almost split extensions of a module Y in mod Λ such that
τn− (Y ) is indecomposable is represented by an exact sequence
0 → Y → I 0 → · · · → I n−2 → E ′ ⊕ I → τn− (Y ) → 0

(5.3)

with modules I 0 , . . . , I n−2 and I in I(Λ).
These constructions are quite general. However, even if X (respectively Y ) in
(5.2) (respectively (5.3)) belongs to an n-precluster tilting subcategory C of mod Λ,
the module E (respectively E ′ ) does not necessarily belong to even Z(C).
Under a certain condition on X (respectively Y ), the next result gives much
nicer representatives, which satisfies similar properties of n-almost split sequences
in n-cluster tilting subcategories. This is an analog of [21, Theorem 3.3.1].
Theorem 5.10. Let C be an n-precluster tilting subcategory of mod Λ, X an indecomposable module in Z(C) \ P(Λ), and Y := τn (X) the corresponding indecomposable module in Z(C) \ I(Λ).
(a) For each 0 ≤ i ≤ n − 1, an n-fold almost split extension in ExtnΛ (X, Y ) can be
represented as
0 → Y → Cn−1 → · · · → Ci+1 → Zi → Ci−1 → · · · → C0 → X → 0
with Zi in Z(C) and Cj in C for each j 6= i.

(5.4)

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

27

(b) The following sequences are exact.
0 → HomΛ (C, Y ) → HomΛ (C, Cn−1 ) → · · · → HomΛ (C, Ci+1 ) → HomΛ (C, Zi )
→ HomΛ (C, Ci−1 ) → · · · → HomΛ (C, C0 ) → radΛ (C, X) → 0,
0 → HomΛ (X, C) → HomΛ (C0 , C) → · · · → HomΛ (Ci−1 , C) → HomΛ (Zi , C)
→ HomΛ (Ci+1 , C) → · · · → HomΛ (Cn−1 , C) → radΛ (Y, C) → 0.
(c) If X and Y do not belong to C, then the n-fold almost split extension in (a) can
be given as a Yoneda product of a minimal projective resolution of X in Z(C)
0 → ΩiZ(C) (X) → Ci−1 → · · · → C0 → X → 0,

(5.5)

an almost split sequence in Z(C)
−(n−i−1)

0 → ΩZ(C)

(Y ) → Zi → ΩiZ(C) (X) → 0,

(5.6)

and a minimal injective coresolution of Y in Z(C)
−(n−i−1)

0 → Y → Cn−1 → · · · → Ci+1 → ΩZ(C)

(Y ) → 0.

(5.7)

Proof. (a) By (5.2), an n-fold almost split extension is given by the Yoneda product
n−1
n−1
of 0 → Y → E → ΩΛ
(X) → 0 (where E⊕P is written as E) and 0 → ΩΛ
(X) →
Pn−2 → · · · → P0 → X → 0. Applying Corollary 3.16 to E, we have an exact
sequence
f0

f1

fi

f i−1

f i+2

f i+1

f n−1

0 → E −→ C 0 −→ · · · −−−→ C i−1 −→ Z i −−−→ C i+1 −−−→ · · · −−−→ C n−1 → 0
where Z i is in Z(C), C j is in C and E j := Im f j is in
simplicity. We have the following pushout diagram.
0

0

0


/Y

0

/Y


/E

/ C0

E1

0

⊥j

C. We write C i := Z i for

0


/ Ωn−1 (X)
Λ

/W

E1

0

/0
/0

Since Ext1Λ (E j , Pn−j ) = 0 holds for any j ≥ 2, a Horseshoe Lemma-type argument
gives the following commutative diagram of exact sequences.

0
0
0

0


0

/ Ωn−1 (X)
Λ


/ Pn−2


/W

/ E1

0


/ C 1 ⊕ Pn−2

/ C1

0

/ ···
/ ···
/ ···

0

0


/ P0


/X

/0

/X

/0


/ C n−1 ⊕ P0

/ C n−1

0


/0

28

IYAMA AND SOLBERG

The Yoneda product 0 → Y → C 0 → C 1 ⊕ Pn−2 → · · · → C n−1 ⊕ P0 → X → 0 of
the middle sequences of the diagrams above represents the same class in ExtnΛ (X, Y )
as the sequence (5.2). Thus it represents an n-fold almost split extension.
(b) This is easily checked (see e.g. [21, Lemma 3.2]).
(c) It suffices to show that the Yoneda product of (5.5), (5.6) and (5.7) is an
n-fold almost split extension. We have functorial isomorphisms
ExtnΛ (X, τn (X)) =

HomU (C) (X, τn (X)[n]) ≃ HomU (C) (X[−i], τn (X)[n − i])
−(n−i−1)

HomU (C) (ΩiZ(C) (X), ΩZ(C)

=

τn (X)[1])

which induces an isomorphism
Soc EndΛ (τn (X)) ExtnΛ (X, τn (X))
≃

−(n−i−1)

Soc End

−(n−i−1)
τn (X))
U(C) (ΩZ(C)

HomU (C) (ΩiZ(C) (X), ΩZ(C)

τn (X)[1]).

Since the almost split sequence belongs to the right hand side, our Yoneda product
belongs to the left hand side. Thus it is an n-fold almost split extension of X by
Theorem 5.8.

Note that the sequences Theorem 5.10(b) are not exact in general if we replace
C by Z(C). Thus the sequences (5.4) is not necessarily n-exact in Z(C) in the sense
of Jasso [29].
In the rest of this section we let C = add M be a finite n-precluster tilting subcategory of mod Λ, and let Γ = EndΛ (M ) be the corresponding n-minimal Auslander–
Gorenstein algebra. We show that the category CM Γ of maximal Cohen–Macaulay
modules over Γ has n-fold almost split extensions. This is used in the next section to
classify the n-minimal Auslander–Gorenstein algebras. We relate the n-fold almost
split extensions to the corresponding n-precluster tilting subcategory and transfer
properties and constructions, as that of the n-fold Auslander–Reiten translate.
The dualities HomΛ (−, M ) : Z(C) → CM Γ and HomΓ (−, M ) : CM Γ → Z(C)
induce dualities
Z(C) ←→ (CM Γ)/[add I] and Z(C) ←→ (CM Γ)/[add Q],
where I := Γ M is an additive generator of P(Γ) ∩ I(Γ) and Q := HomΛ (DΛ, M ) ≃
HomΓop (DI, Γ) ∈ P(Γ).
Definition 5.11. Using the equivalences τn : Z(C) → Z(C) and τn− : Z(C) → Z(C)
given in Theorem 3.10, we define equivalences
σn− : (CM Γ)/[add I] → (CM Γ)/[add Q] and σn : (CM Γ)/[add Q] → (CM Γ)/[add I]
making the following diagrams commutative up to isomorphisms of functors.
Z(C)

HomΛ (−,M)

/ (CM Γ)/[add I]
σ−

τn


Z(C)

HomΛ (−,M)

 n
/ (CM Γ)/[add Q]

Z(C)
O

HomΛ (−,M)

/ (CM Γ)/[add I]
O

−
τn

Z(C)

σn

HomΛ (−,M)

/ (CM Γ)/[add Q].
(5.8)

Let Γ be an n-minimal Auslander–Gorenstein Artin algebra for some n ≥ 1.
Then we use Theorem 5.10 to construct n-fold almost split extensions in CM Γ.

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

29

Corollary 5.12. Let Γ be an n-minimal Auslander–Gorenstein Artin algebra with
n ≥ 1, X an indecomposable module in (CM Γ) \ (add Q) and Y := σn (X) an
indecomposable module in (CM Γ) \ (add I).
(a) If X does not belong to P(Γ), then there exists an n-fold almost split extension
in ExtnΓ (X, Y ). For each 0 ≤ i ≤ n − 1, it is represented as
0 → Y → P 0 → · · · → P i−1 → E i → P i+1 → · · · → P n−1 → X → 0
with Pj for all j in P(Γ) and Ei in CM Γ.
(b) If X belongs to P(Γ), then there exists an exact sequence
0 → Y → P 0 → · · · → P i−1 → E i → P i+1 → · · · → P n−1 → X → X/ rad X → 0,
with Pj for all j in P(Γ) and Ei in CM Γ.
We call the sequence in (b) a fundamental sequence.
Proof. Both assertions follow from Theorem 5.10 and the duality between CM Γ and
Z(C) for the corresponding n-precluster tilting subcategory C (Theorem 4.7). 
6. Four classes of n-minimal Auslander–Gorenstein algebras
1-Auslander–Gorenstein Artin algebras were classified in [11] in the class of selfinjective Artin algebras and three different disjoint classes using the correspondence
with the τ -selfinjective Artin algebras. This section is devoted to extending this
characterization to n-minimal Auslander–Gorenstein algebras for n ≥ 1.
Suppose Γ is an n-minimal Auslander–Gorenstein algebra. By Theorem 4.5,
there exist an Artin algebra Λ and an n-precluster tilting Λ-module M with Γ ≃
EndΛ (M ). By Proposition 3.7(d), add M = add{Pn ∨ In , N } holds for a Λ-module
N satisfying τn (N ) ≃ N . Then four distinct cases can occur:
(A) Λ is selfinjective and N = 0.
(B) Λ is selfinjective and N 6= 0.
(C) Λ is non-selfinjective and N = 0.
(D) Λ is non-selfinjective and N 6= 0.
This also gives rise to a coarser division, namely (A) and (C) together, or in other
words when N = 0.
We characterize all these four cases in terms of properties of Γ, and in particular
we show that an algebra of type (B) or (D) is constructed from an algebra of type
(A) or (C), respectively.
The case (A) is easy, as this occurs if and only if Γ is a selfinjective algebra by
Proposition 4.3. So we move on to describe the correspondence given by type (B).
Proposition 6.1. Let n ≥ 1. The bijection in Theorem 4.5 restricts to a bijection
between
(

{finite n-precluster tilting subcategories of type (B)} and
)
non-selfinjective n-minimal Auslander–Gorenstein algebras Γ with
an additive generator I of P(Γ) ∩ I(Γ) satisfying add Top Γ I = .
add Soc Γ I

Proof. Let Λ be a selfinjective Artin algebra with non-projective module N such
that M = Λ ⊕ N is an n-precluster tilting Λ-module. We want to show that
Γ = EndΛ (M ) is a non-selfinjective n-minimal Auslander–Gorenstein algebra and
add Top Γ I = add Soc Γ I, where I is an additive generator of P(Γ) ∩ I(Γ).

30

IYAMA AND SOLBERG

Let G = FM ⊆ Ext1Λ (−, −). By Proposition 4.3 we have that Γ is a nonselfinjective n-minimal Auslander–Gorenstein algebra, M is a G-cotilting Λ-module,
and I = Γ M is an additive generator of P(Γ) ∩ I(Γ). By [11, Lemma 2.3(b)] we
have that
HomΓ (−, I) : HomΓ (A, C) → HomΛ (HomΓ (C, I), HomΓ (A, I))
is an isomorphism for all C in mod Γ and all A in add I. In particular for A = I
and noting that HomΓ (I, I) ≃ Λ with Λ selfinjective, we infer that HomΓ (I, C) = 0
if and only if HomΓ (C, I) = 0. It follows from this that add Top Γ I = add Soc Γ I.
Conversely, let Γ be a non-selfinjective n-minimal Auslander–Gorenstein algebra,
such that add Top Γ I = add Soc Γ I where I is an additive generator of P(Γ) ∩ I(Γ).
We want to show that Λ = EndΓ (I) is selfinjective, and M = Λ I ≃ Λ ⊕ N with N
non-projective such that M is an n-precluster tilting Λ-module.
Let G = FM ⊆ Ext1Λ (−, −). By Proposition 4.4 the module M is an nprecluster tilting Λ-module and M = Λ I is a G-cotilting Λ-module. The functor
HomΓ (−, I) : mod Γ → mod Λ induces a duality between P(Γ) and add M . Since
M is a cogenerator, there exists P in P(Γ) such that HomΓ (P, I) ≃ D(ΛΛ ). By
[11, Lemma 2.3(b)] we have that
HomΓ (−, I) : HomΓ (P, C) → HomΛ (HomΓ (C, I), HomΓ (P, I))
is an isomorphism for all C in mod Γ. This implies by the choice of P that
HomΓ (P, C) = 0 if and only if HomΓ (C, I) = 0. Hence add Top Γ P = add Soc Γ I =
add Top Γ I, and we obtain that add P = add I. Therefore I(Λ) = add HomΓ (P, I) =
P(Λ) and Λ is selfinjective.

Remark 6.2. By Proposition 6.1, if Γ is an n-minimal Auslander–Gorenstein algebra of type (B), then there exists a selfinjective algebra Λ and a non-projective
n-precluster tilting Λ-module Λ ⊕ N such that τn (N ) ≃ N and Γ ≃ EndΛ (Λ ⊕ N ).
Now we complete the classification.
Theorem 6.3. Let Γ be an n-minimal Auslander–Gorenstein algebra for n ≥ 1.
Denote by I an additive generator of P(Γ) ∩ I(Γ), and let Λ = EndΓ (I). Then four
disjoint cases occur, where σn is the functor given in Definition 5.11:
(A) Γ is selfinjective,
(B) Γ is not selfinjective and add Top Γ I = add Soc Γ I,
(C) add Top Γ I 6= add Soc Γ I, and there exists no indecomposable projective Γmodule P such that (σn )t (P ) ≃ P for some t > 0,
(D) add Top Γ I 6= add Soc Γ I, and there exists an indecomposable projective Γmodule P such that (σn )t (P ) ≃ P for some t > 0.
In the cases (B), (C) and (D) above, the correspondence given in Theorem 4.5
shows that the algebra Γ arises as follows.
(a) In case (B) there exists a selfinjective algebra Λ with an n-precluster tilting
Λ-module M such that Γ ≃ EndΛ (M ).
(b) In case (C) there exists an algebra Λ with an n-precluster tilting Λ-module M
with no non-zero τn -periodic direct summand such that Γ ≃ EndΛ (M ).
(c) In case (D) there exists an algebra Λ with an n-precluster tilting Λ-module M
with a non-zero τn -periodic direct summand such that Γ ≃ EndΛ (M ).
Proof. The cases (A) and (B) are already discussed before in Proposition 6.1. The
cases (A) and (B) occur if and only if add Top Γ I = add Soc Γ I. Hence the cases

AUSLANDER–GORENSTEIN ALGEBRAS AND PRECLUSTER TILTING

31

(C) and (D) occur if and only if add Top Γ I 6= add Soc Γ I. Using this and the
commutative diagram (5.8), the characterizations of the cases (C) and (D) follow.

Remark 6.4. As we saw in Remark 6.2, any algebra of type (B) can be constructed
from an algebra of type (A). Now we argue that type (D) and type (C) are related
in a similar fashion. Let Γ be an n-minimal Auslander–Gorenstein algebra of type
(D). Let I denote an additive generator of P(Γ) ∩ I(Γ), and let Λ = EndΓ (I) and
M = Λ I. Decompose M = M0 ⊕ N with add M0 = Pn ∨ In and τn (N ) ≃ N .
Furthermore, Γ0 = EndΓ (M0 ) is an n-minimal Auslander–Gorenstein algebra of
type (C) since M0 is also an n-precluster tilting Λ-module. Now we claim that Γ
can be constructed from Γ0 .
Since M0 is a cogenerator for mod Λ, the functor
HomΛ (−, M0 ) : mod Λ → mod Γ0
is full and faithful. This implies that
Γ = HomΛ (M, M ) ≃ EndΓ0 (HomΛ (M, M0 )).
Hence Γ is constructed from the algebra Γ0 which is of type (C).
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Osamu Iyama, Graduate School of Mathematics, Nagoya University, Chikusa-ku,
Nagoya, 464-8602, Japan
E-mail address: iyama@math.nagoya-u.ac.jp
Øyvind Solberg, Institutt for matematiske fag, NTNU, N–7491 Trondheim, Norway
E-mail address: oyvind.solberg@ntnu.no

