# Nakayama-type phenomena in higher Auslander–Reiten theory

Gustavo Jasso  Mathematisches Institut
Universität Bonn
Endenicher Allee 60
D-53115 Bonn
GERMANY  gustavo.jasso@@math.lu.se https://gustavo.jasso.infoLund University,
Centre for Mathematical Sciences, Sölvegatan 18A, 22100 Lund, Sweden and
Julian Külshammer  Institut für Algebra und Zahlentheorie
Universität Stuttgart
Pfaffenwaldring 57
70569 Stuttgart
GERMANY<EMAIL_ADDRESS>https://www.katalog.uu.se/profile/?id=N18-1115Department of Mathematics
University of Uppsala, Box 480, Uppsala, Sweden

###### Abstract.

This paper surveys recent contructions in higher Auslander–Reiten theory. We
focus on those which, due to their combinatorial properties, can be regarded
as higher dimensional analogues of path algebras of linearly oriented type
$\mathbb{A}$ quivers. These include higher dimensional analogues of Nakayama
algebras, of the mesh category of type $\mathbb{Z}\mathbb{A}_{\infty}$ and the
tubes, and of the triangulated category generated by an $m$-spherical object.
For $m=2$, the latter category can be regarded as the higher cluster category
of type $\mathbb{A}_{\infty}$ whose cluster-tilting combinatorics are
controlled by the triangulations of the cylic apeirotope.

Part of this research was carried out during a visit of the second author at
MPIM Bonn. The authors would like to thank the anonymous referee for helpful
comments.

##  1\. Introduction

The category of finite dimensional representations of the quiver

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is well understood. The indecomposable representations are the interval
modules and its Auslander–Reiten quiver is of type
$\mathbb{Z}\mathbb{A}_{\infty}$. The category of nilpotent finite dimensional
representations of the cylic quiver $\tilde{\mathbb{A}}_{n-1}$ with $n$
vertices, obtained as the quotient of $\mathbb{A}_{\infty}^{\infty}$ by
identifyng to vertices if they are equal modulo $n$, has a similar
description. In this case its Auslander–Reiten quiver is rank $n$ tube. Note
that both the mesh category of type $\mathbb{ZA}_{\infty}$ and the tubes are
frequently occurring shapes of Auslander–Reiten components. For our purposes,
it is important to note that the categories of representations of the Nakayama
algebras embedd into those of the above quivers and hence admit a nice
combinatorial description.

In this survey we introduce higher dimensional analogues of the quivers
$\mathbb{A}_{\infty}^{\infty}$ and $\tilde{\mathbb{A}}_{n-1}$ from the
viewpoint of Iyama’s higher Auslander–Reiten theory [Iya07b, Iya07a]. This
allows us to construct higher dimensional analogues of the Nakayama algebras,
whose higher Auslander–Reiten theory is controlled by higher dimensional
analogues of the mesh category of type $\mathbb{ZA}_{\infty}$ and the tubes
[JK16a]. To complement loc. cit., we give a few examples to illustrate how the
homological behaviour of the higher Nakayama algebras differs from that of
their classical versions.

We also construct certain Calabi–Yau triangulated categories which can be
regarded as higher dimensional analogues of the triangulated categories
generated by spherical objects of dimension greater than or equal to $2$. In
particular, in analogy to the case of a $2$-dimensional spherical object, we
obtain a higher version of the cluster category of type $\mathbb{A}_{\infty}$
introduced by Holm and Jørgensen [HJ12]. In this case, the cluster-tilting
combinatorics of this category is controlled by triangulations of the cyclic
apeirotope (a higher dimensional analog of the $\infty$-gon).

Let us briefly desribe the structure of this article. Section 2 recalls the
Auslander correspondence and Iyama’s generalisation of it obtained by
introducing cluster-tilting subcategories. We also include a short
introduction to higher Auslander–Reiten theory. In Section 3 we recall the
construction of the higher Nakayama algebras in terms of the higher
dimensional analogues of the quivers $\mathbb{A}_{\infty}^{\infty}$ and
$\tilde{\mathbb{A}}_{n-1}$. In Section 4 we give examples of higher Nakayama
algebras which have different homological properties than those of their
classical versions. Finally, in Section 5 we introduce the higher dimensional
analogues of the triangulated categories generated by spherical objects and of
the cluster category of type $\mathbb{A}_{\infty}$.

######  Conventions 1.1.

We fix a positive integer $d$ as well as a field $\mathbbm{k}$. We denote the
duality of the category of finite dimensional $\mathbbm{k}$-vector spaces by
$D:=\operatorname{Hom}_{\mathbbm{k}}(-,\mathbbm{k})$. By algebra we mean
associative unital $\mathbbm{k}$-algebra. All modules we consider are finite
dimensional left modules. Two algebras $\Lambda$ and $\Lambda^{\prime}$ are
_Morita equivalent_ if their categories of modules are equivalent. Let
$\Lambda$ be a finite dimensional algebra. We denote by
$\operatorname{ind}\Lambda$ a complete set of representatives of the
isomorphisms classes of finite dimensional indecomposable $\Lambda$-modules.
Let $M$ be a $\Lambda$-module (or, more generally, an object in some additive
category). We denote by $\operatorname{add}M$ the full subcategory of
$\operatorname{mod}\Lambda$ consisting of all $\Lambda$-modules which are
direct summands of $M^{n}$ for some positive integer $n$.

##  2\. Preliminaries

In this section we recall the basics of higher Auslander–Reiten theory, namely
Auslander–Iyama correspondence and the basic theory of $d$-representation-
finite $d$-hereditary algebras. The reader is referred to [Iya07a, Iya11,
Iya08, IO11, IJ16, JK16c] for further reading.

###  2.1. The Auslander–Iyama correspondence

For the purpose of motivation we first recall classical Auslander
correspondence. This establishes a relationship between representation-
finiteness, a _representation-theoretic_ property, and being an Auslander
algebra, a _homological_ property. Moreover, the minimal projective
resolutions of simple modules of projective dimension $2$ over Auslander
algebras provided the first evidence for the existence of almost-split
sequences. Thus, Auslander correspondence can also be considered as one of the
seminal results in Auslander–Reiten theory.

On the representation-theoretic side let $\Lambda$ be a finite dimensional
algebra. Recall that $\Lambda$ is _representation-finite_ if there are only
finitely many indecomposable $\Lambda$-modules up to isomorphism. An
equivalent way of saying this is that there exists a finite dimensional
$\Lambda$-module $M$ such that every indecomposable $\Lambda$-module is a
direct summand of $M$. Such a $\Lambda$-module $M$ is called a _representation
generator of $\operatorname{mod}\Lambda$_.

On the homological side, let $\Gamma$ be a finite dimensional algebra and

${0\to\Gamma\to I^{0}\to I^{1}\to\cdots\to I^{d}\to\cdots}$

a minimal injective coresolution of $\Gamma$. We remind the reader that the
_dominant dimension of $\Gamma$_ is defined as

$\operatorname{domdim}\Gamma:=1+\sup\left\\{\,i\in\mathbb{Z}\,\middle|\,I^{i}\text{
is projective}\,\right\\}.$

A finite dimensional algebra $\Gamma$ is an _Auslander algebra_ if

$\operatorname{gldim}\Gamma\leq 2\leq\operatorname{domdim}\Gamma.$

Auslander algebras were introduced by the eponymous author in [Aus71] where
the following theorem is also proven.

######  Theorem 2.1 (Auslander correspondence).

There is a one-to-one correspondence between Morita equivalence classes of
representation-finite algebras and Morita equivalence classes of Auslander
algebras. The correspondence associates to a representation-finite algebra
$\Lambda$ the algebra $\operatorname{End}_{\Lambda}(M)^{\mathrm{op}}$ for a
representation generator $M$ of $\operatorname{mod}\Lambda$. In the reverse
direction, it associates to an Auslander algebra $\Gamma$ the algebra
$\operatorname{End}_{\Gamma}(I^{0})^{\mathrm{op}}$ where $I^{0}$ is the
injective hull of the regular representation.

Auslander correspondence is in fact a particular instance of Morita–Tachikawa
correspondence. This correspondence shifts the focus on the ‘representation-
theoretic’ side away from the algebra $\Lambda$ to the category
$\operatorname{add}M$. In Auslander correspondence knowledge of $\Lambda$ is
(up to Morita equivalence) of course equivalent to knowledge of
$\operatorname{add}M$ for $M$ a representation generator.

Recall that a $\Lambda$-module $M$ is a _generator-cogenerator of
$\operatorname{mod}\Lambda$_ if every indecomposable projective and every
indecomposable injective $\Lambda$-module is a direct summand of $M$. In
particular, a representation generator is an example of a generator-
cogenerator. A _Morita–Tachikawa pair_ is a pair $(\Lambda,M)$ consisting of a
finite dimensional algebra $\Lambda$ and generator-cogenerator $M$ of
$\operatorname{mod}\Lambda$. Two Morita–Tachikawa pairs $(\Lambda,M)$ and
$(\Lambda^{\prime},M^{\prime})$ are _equivalent_ if $\operatorname{add}M$ and
$\operatorname{add}M^{\prime}$ are equivalent categories. Note that this
implies that the algebras $\Lambda$ and $\Lambda^{\prime}$ are Morita
equivalent: Indeed, a module $P$ is a projective object in
$\operatorname{add}M$ (in the sense that every epimorphism to it splits) if
and only if it is a projective $\Lambda$-module. It is easy to see that no
other module can be projective as $M$ is a generator. For the converse let $P$
be a projective $\Lambda$-module and $f\colon M\to P$ be an epimorphism. Let
$g$ be the composition of the projection $P\to\operatorname{Coker}f$ and the
inclusion $\operatorname{Coker}f\to I$ into its injective hull. Then, $gf=0$
implies that $g=0$ as $f$ is an epimorphism. In particular
$\operatorname{Coker}f=0$. It follows that
$\operatorname{add}\Lambda\cong\operatorname{add}\Lambda^{\prime}$ as both are
defined by categorical properties in
$\operatorname{add}M\cong\operatorname{add}M^{\prime}$. Thus, $\Lambda$ and
$\Lambda^{\prime}$ are Morita equivalent.

######  Theorem 2.2 (Morita–Tachikawa correspondence).

There is a one-to-one correspondence between equivalence classes of
Morita–Tachikawa pairs and Morita equivalence classes of finite dimensional
algebras $\Gamma$ such that $\operatorname{domdim}\Gamma\geq 2$. The
correspondence associates to a Morita–Tachikawa pair $(\Lambda,M)$ the algebra
$\operatorname{End}_{\Lambda}(M)^{\mathrm{op}}$ and, in the other direction,
associates to an algebra $\Gamma$ the Morita–Tachikawa pair
$(\operatorname{End}_{\Gamma}(I^{0})^{\mathrm{op}},I^{0})$ for $I^{0}$ the
injective hull of $\Gamma$ as a left $\Gamma$-module.

Almost thirty years later, Iyama realised that Auslander correspondence,
together with it the most fundamental aspects of Auslander–Reiten theory,
could be vastly generalised. In some sense, Iyama’s approach was inverse to
that of Auslander. Indeed, the class of Auslander algebras admits an “obvious”
generalisation: a finite dimensional algebra $\Gamma$ is a _$d$ -Auslander
algebra_ if the inequalities

$\operatorname{gldim}\Gamma\leq d+1\leq\operatorname{domdim}\Gamma$

are satisfied.

In view of the Morita–Tachikawa correspondence, the modules which are in
correspondence with $d$-Auslander algebras must be particular kinds of
generator-cogenerators. These are the $d$-cluster-tilting modules, which we
now define.

######  Definition 2.3 ([Iya07a, Iya11, IO11]).

Let $\Lambda$ be a finite dimensional algebra. A $\Lambda$-module $M$ is _$d$
-cluster-tilting_ if

$\displaystyle\operatorname{add}M$
$\displaystyle=\left\\{\,X\in\operatorname{mod}\Lambda\,\middle|\,\forall
i\in\left\\{1,\dots,d-1\right\\}:\operatorname{Ext}^{i}_{\Lambda}(X,M)=0\,\right\\}$
$\displaystyle=\left\\{\,Y\in\operatorname{mod}\Lambda\,\middle|\,\forall
i\in\left\\{1,\dots,d-1\right\\}:\operatorname{Ext}^{i}_{\Lambda}(M,Y)=0\,\right\\}.$

An algebra $\Lambda$ is called _weakly $d$-representation-finite_ if there
exists a $d$-cluster-tilting module.

It is easy to see that a $1$-cluster-tilting $\Lambda$-module is precisely a
representation generator of $\operatorname{mod}\Lambda$, and that every
$d$-cluster-tilting $\Lambda$-module is a generator-cogenerator.

A _$d$ -Auslander–Iyama pair_ is a pair $(\Lambda,M)$ where $\Lambda$ is a
finite dimensional algebra and $M$ is a $d$-cluster-tilting module. Two
$d$-Auslander–Iyama pairs $(\Lambda,M)$ and $(\Lambda^{\prime},M^{\prime})$
are _equivalent_ if $\operatorname{add}M$ and $\operatorname{add}M^{\prime}$
are equivalent. The following theorem appeared originally as [Iya07a, Theorem
0.2].

######  Theorem 2.4 ($d$-dimensional Auslander–Iyama correspondence).

There is a one-to-one correspondence between equivalence classes of
$d$-Auslander–Iyama pairs and Morita equivalence classes of $d$-Auslander
algebras. The correspondence associates to a $d$-Auslander–Iyama pair
$(\Lambda,M)$ the $d$-Auslander algebra
$\operatorname{End}_{\Lambda}(M)^{\mathrm{op}}$, and in the inverse direction
to a $d$-Auslander algebra $\Gamma$ the pair
$(\operatorname{End}_{\Gamma}(I^{0})^{\mathrm{op}},I^{0})$ where $I^{0}$ is
the injective hull of the regular representation of $\Gamma$.

Recently, there have been further generalisations of this correspondence (and
thus instances of Morita–Tachikawa correspondence) by Iyama–Solberg [IS16] and
Marczinzik [Mar17] replacing $\operatorname{gldim}\Gamma$ by
$\operatorname{injdim}_{\Gamma}\Gamma$ and $\operatorname{findim}\Gamma$,
respectively.

###  2.2. Higher Auslander–Reiten theory

Following the steps in the development of Auslander–Reiten theory, the
$d$-dimensional Auslander–Iyama correspondence served as the starting point
for a $(d+1)$-dimensional Auslander–Reiten theory. The minimal projective
resolutions of simple modules of projective dimension $d+1$ over $d$-Auslander
algebras give rise to the prototypical $d$-almost-split sequences. The only
caveat is that, in order to obtain analogous properties to those in classical
Auslander–Reiten theory, one needs to restrict to modules in a $d$-cluster-
tilting subcategory, a generalisation of the category $\operatorname{add}M$
for a $d$-cluster-tilting module $M$.

######  Definition 2.5 ([Iya07a, (2.4)], [Iya07b, Definition 2.2], [Jas16,
Definition 3.14]).

Let $\mathcal{A}$ be an abelian category. A generating-cogenerating
functorially finite (full) subcategory $\mathcal{C}$ of $\mathcal{A}$ is a
_$d$ -cluster-tilting subcategory_111Note that $d$-cluster-tilting
subcategories where originally called “maximal $(d-1)$-orthogonal
subcategories” in [Iya07a]. if

$\displaystyle\mathcal{C}$
$\displaystyle=\left\\{\,X\in\mathcal{A}\,\middle|\,\forall
i\in\left\\{1,\dots,d-1\right\\}:\operatorname{Ext}^{i}_{\mathcal{A}}(X,\mathcal{C})=0\,\right\\}$
$\displaystyle=\left\\{\,Y\in\mathcal{A}\,\middle|\,\forall
i\in\left\\{1,\dots,d-1\right\\}:\operatorname{Ext}^{i}_{\mathcal{A}}(\mathcal{C},Y)=0\,\right\\}.$

An object $C\in\mathcal{A}$ is a _$d$ -cluster-tilting object_ if
$\operatorname{add}C$ is a $d$-cluster-tilting subcategory of $\mathcal{A}$.

The authors are not aware of any examples of abelian categories $\mathcal{A}$
having a subcategory $\mathcal{C}$ which satisfies the assumptions of a
$d$-cluster-tilting subcategory except for being generating-cogenerating.
However, this assumption is needed in the proofs of even the basic statements
of the theory in [Jas16]. If $\mathcal{A}$ has enough projectives and enough
injectives, this condition is automatic as $\mathcal{C}$ always contains all
the projective and all the injective objects of $\mathcal{A}$.

Let $\mathcal{C}$ be a $d$-cluster-tilting subcategory of an abelian category
$\mathcal{A}$. Then, a _$d$ -almost split sequence_ is an exact sequence

$0\to C_{0}\to C_{1}\to\dots\to C_{d}\to C_{d+1}\to 0$

with $C_{i}\in\mathcal{C}$ for all $i$ and $C_{0}$ indecomposable such that
every non-split epimorphism $C^{\prime}\to C_{d+1}$ for some
$C^{\prime}\in\mathcal{C}$ factors through the map $C_{d}\to C_{d+1}$;
moreover, one requires that for each $i\in\left\\{0,1,\dots,d\right\\}$ the
morphisms $C_{i}\to C_{i+1}$ lies in the Jacobson radical of
$\operatorname{mod}\Lambda$. In this case, $C_{0}\cong\tau_{d}(C_{d+1})$ where

$\tau_{d}(M):=\tau\Omega^{d-1}(M)$

is the $d$-dimensional replacement of the classical Auslander–Reiten
translation $\tau$. Here, $\Omega$ denotes Heller’s syzygy functor.

In ‘classical’ representation theory, representation-finite hereditary
algebras were among the first algebras whose Auslander–Reiten theory was
systematically investigated, leading to fruitful results, see for example
[ASS06, SS07a, SS07b]. Modelled on this, weakly $d$-representation finite
algebras of small global dimension were among the first to be investigated.

######  Definition 2.6.

A finite dimensional algebra $\Lambda$ is _$d$ -representation-finite
$d$-hereditary_ if it is weakly $d$-representation-finite and
$\operatorname{gldim}\Lambda\leq d$.

As an exercise, the reader can verify that $d$ is the smallest possible global
dimension for a non-semisimple weakly $d$-representation-finite algebra. Thus,
a $d$-representation-finite $d$-hereditary algebra has global dimension either
$0$ or $d$. Much of the research in higher Auslander–Reiten theory has focused
on the class of $d$-representation-finite $d$-hereditary algebras, see for
example [IO11, HI11a, HI11b, OT12, IO13]. In the beginning of Section 3 we
recall Iyama’s construction of the higher Auslander algebras of type
$\mathbb{A}$, which are a particularly important class of $d$-representation-
finite $d$-hereditary algebras.

There is an analogous result to the fact that for a representation-finite
hereditary algebra all indecomposable modules can be constructed from the
indecomposable injectives by iterative application of the Auslander–Reiten
translation.

######  Theorem 2.7 ([Iya11, Proposition 1.3(b)]).

Let $\Lambda$ be a $d$-representation-finite $d$-hereditary algebra with $n$
simple modules $S_{1},\dots,S_{n}$ up to isomorphism. Then, the
$\Lambda$-module

$M=\bigoplus_{i=1}^{n}\bigoplus_{j\geq 0}\tau_{d}^{j}I_{i},$

where $I_{i}$ denotes the indecomposable injective $\Lambda$-module with socle
$S_{i}$, is the (up to isomorphism) unique basic $d$-cluster-tilting
$\Lambda$-module.

Another nice feature of $d$-representation-finite $d$-hereditary algebras is
that their derived categories also have a $d$-cluster-tilting subcategory.
Note that $d$-cluster-tilting subcategories of triangulated categories can be
defined in an obvious way (omitting the condition of being generating-
cogenerating) with the obvious generalisation of almost split triangles to
almost split $d$-angles.

For a finite dimensional algebra $\Lambda$, set

$\mathcal{U}(\Lambda):=\operatorname{add}\left\\{\,\nu_{d}^{i}\Lambda\,\middle|\,i\in\mathbb{Z}\,\right\\},$

where $\nu=D\Lambda\otimes_{\Lambda}^{\mathbb{L}}-$ is the derived Nakayama
functor and $\nu_{d}:=\nu[-d]$. Thus, $\mathcal{U}(\Lambda)$ is a full
subcategory of the bounded derived category
$D^{b}(\operatorname{mod}\Lambda)$.

######  Theorem 2.8.

Let $\Lambda$ be a $d$-representation-finite $d$-hereditary algebra. Then,
$\mathcal{U}(\Lambda)$ is a $d$-cluster-tilting subcategory of
$D^{b}(\operatorname{mod}\Lambda)$.

Note that an analogous construction for more general weakly
$d$-representation-finite algebra is not known and a naive extension is even
not possible, see the authors’ note [JK16b].

##  3\. Higher Nakayama algebras

In this section we recall the construction of higher Nakayama algebras from
the authors’ article [JK16a]. For this, we introduce a certain universal
category whose importance in higher Auslander–Reiten theory is already
mentioned by Iyama in [Iya11, Introduction].

Every partially ordered $(P,\leq)$ has an associated $\mathbbm{k}$-category
$A(P)$ called the _incidence $\mathbbm{k}$-category of the poset $P$_, whose
objects are the elements of $P$ and with morphisms

$A(P)(x,y):=\begin{cases}\mathbbm{k}f_{yx}&\text{if }x\leq y,\\\
0&\text{otherwise},\end{cases}$

where $f_{yx}$ denotes a basis vector of a corresponding one-dimensional
space. The composition in $A(P)$ induced by $f_{zy}\circ f_{yx}:=f_{zx}$. If
the set $P$ is finite, this is just the incidence algebra of the poset
$(P,\leq)$.

######  Definition 3.1.

Let $\leq$ be the product order on $\mathbb{Z}^{d}$, i.e. $\alpha\leq\beta$ if
and only if $\alpha_{i}\leq\beta_{i}$ for all
$i\in\left\\{1,\dots,d\right\\}$. Let $A^{(d)}:=A(\mathbb{Z}^{d},\leq)$.

The $\mathbbm{k}$-category $A^{(d)}$ can be explicitly described as follows.
Let $Q^{(d)}$ be the quiver with vertex set $\mathbb{Z}^{d}$ and arrows

$a_{i}=a_{i}(\alpha)\colon\alpha\to\alpha+e_{i}$

where $i\in\left\\{1,\dots,d\right\\}$. Let $I$ be the two-sided ideal of the
path category $\mathbbm{k}Q^{(d)}$ of $Q^{(d)}$ generated by all commutativity
relations

$\left\\{\,a_{i}a_{j}-a_{j}a_{i}\,\middle|\,1\leq i<j\leq d\,\right\\}.$

With these conventions, $A^{(d)}$ is isomorphic to the quotient category
$\mathbbm{k}Q^{(d)}/I$. From both descriptions it is clear that $A^{(1)}$ is
isomorphic to the path category of $\mathbb{A}_{\infty}^{\infty}$ with linear
orientation.

Note that $A^{(d)}$ is naturally a graded $\mathbbm{k}$-category with respect
to path length. It is elementary to verify that $A^{(d)}$ is the universal
covering of the polynomial ring in $d$ variables in the sense of [Gre83] (see
also [MVdlP83]), that it is of global dimension $d$ and a Koszul category in
the sense of [MOS09].

In the sequel we heavily use a certain automorphism $\varphi_{d}$ on $A^{(d)}$
(and idempotent quotients thereof) defined by

$\displaystyle\varphi_{d}(\mu_{1},\dots,\mu_{d})=(\mu_{1}-1,\dots,\mu_{d}-1),$
$\displaystyle\varphi_{d}(a_{i}(\mu_{1},\dots,\mu_{d}))=a_{i}(\mu_{1}-1,\dots,\mu_{d}-1).$

###  3.1. The universal $d$-Nakayama category

In this subsection, we introduce another category, a certain idempotent
quotient of $A^{(d)}$, which serves as the universal higher Nakayama category,
much like the path category of $\mathbb{A}_{\infty}^{\infty}$ serves as the
universal Nakayama category. Set

$\mathbf{os}^{(d)}:=\left\\{\,(\mu_{1},\dots,\mu_{d})\in\mathbb{Z}^{d}\,\middle|\,\mu_{1}\geq\mu_{2}\geq\dots\geq\mu_{d}\,\right\\}.$

######  Definition 3.2.

For $d\geq 2$, the idempotent quotient $A^{(d)}_{\infty}$ of $A^{(d)}$ by the
objects not in $\mathbf{os}^{(d)}$ is called the _universal $d$-Nakayama
category_.

Note that $A_{\infty}^{(d)}$ is different from the incidence category of
$\mathbf{os}^{(d)}$ as there are certain zero relations at the ‘boundary’ of
$A^{(d)}_{\infty}$. Furthermore observe that $A_{\infty}^{(2)}$ is the mesh
category of a $\mathbb{Z}\mathbb{A}_{\infty}$-component, one of the most
typical Auslander–Reiten components in ‘classical’ Auslander–Reiten theory.
This is why we chose to give them the similar name $A_{\infty}^{(d)}$ as we
see them as higher analogues of this category.

######  Definition 3.3.

Let $\lambda\in\mathbf{os}^{(d+1)}$. The _interval module_ $M_{\lambda}$ is
the $A_{\infty}^{(d)}$-module with composition factors

$[M_{\lambda}\colon S_{\kappa}]=\begin{cases}\mathbbm{k}&\text{if
$\kappa\in[(\lambda_{2},\dots,\lambda_{d+1}),(\lambda_{1},\dots,\lambda_{d})]$,}\\\
0&\text{else,}\end{cases}$

where $[(\lambda_{2},\dots,\lambda_{d+1}),(\lambda_{1},\dots,\lambda_{d})]$
denotes the interval in the partial order on $\mathbf{os}^{(d)}$. The maps on
the arrows $a_{i}$ are identity maps between different copies of $\mathbbm{k}$
and $0$ elsewhere.

The category $A_{\infty}^{(d)}$ satisfies analogous properties to being
$d$-representation-finite in the infinite setting:

######  Theorem 3.4 ([JK16a, Theorem 2.3.5]).

The abelian category $\operatorname{mod}A_{\infty}^{(d)}$ of finite
dimensional $A_{\infty}^{(d)}$-modules is of global dimension $d$ and has a
$d$-cluster-tilting subcategory consisting of the additive hull
$\mathcal{M}_{\infty}^{(d)}$ of all interval modules $M_{\lambda}$ for
$\lambda\in\mathbf{os}^{(d+1)}$.

In the derived situation,

$\mathcal{U}_{\infty}^{(d)}:=\operatorname{add}\left\\{\,M_{\lambda}[di]\,\middle|\,\lambda\in\mathbf{os}^{(d+1)},i\in\mathbb{Z}\,\right\\}$

is a $d$-cluster-tilting subcategory of
$D^{b}(\operatorname{mod}A_{\infty}^{(d)})$.

###  3.2. Higher Nakayama algebras

In this section, we survey on the results of the authors’ paper [JK16a] on the
construction of higher analogues of Nakayama algebras.

######  Definition 3.5.

A finite dimensional algebra $\Lambda$ is called a _Nakayama algebra_ (or
_uniserial_) if every indecomposable projective and every indecomposable
injective module is uniserial.

It is well-known that (over an algebraically closed field), the connected
Nakayama algebras are precisely the quotients by admissible ideals of the path
algebras of the linearly oriented Dynkin quiver

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and the linearly oriented extended Dynkin quiver

$\widetilde{\mathbb{A}}_{n-1}\colon$$0$$1$$2$$\cdots$$n-2$$n-1$

Note that the path algebra of $\mathbb{A}_{n}$ is an idempotent truncation of
$A_{\infty}^{(1)}$ while the path algebra of $\tilde{\mathbb{A}}_{n-1}$ is
obtained from $A_{\infty}^{(1)}$ by dividing out by the action of
$\varphi_{1}^{n}$. There are analogues of the path algebras of
$\mathbb{A}_{n}$ and $\tilde{\mathbb{A}}_{n-1}$, respectively. The former was
already introduced in [Iya11] and studied extensively in [OT12]. For $n=1$,
the latter can be seen as a higher preprojective algebra of type
$\mathbb{A}_{\infty}$. We start by introducing the higher analogues of the
path algebra of $\mathbb{A}_{n}$, in [IO11] called higher Auslander algebras
of type $\mathbb{A}$.

######  Definition 3.6.

Let
$\mathbf{os}^{(d)}_{n}:=\left\\{\,(\mu_{1},\dots,\mu_{d})\in\mathbf{os}^{(d)}\,\middle|\,n-1\geq\mu_{1},\mu_{d}\geq
0\,\right\\}$. The _$d$ -Auslander algebra $A_{n}^{(d)}$ of type
$\mathbb{A}_{n}$_ is defined to be the (full) subcategory of
$A_{\infty}^{(d)}$ on the vertices in $\mathbf{os}^{(d)}_{n}$.

######  Theorem 3.7 ([Iya11, Theorem 1.18]).

Let $d\geq 1$. Then, $A_{n}^{(d)}$ is a $d$-representation-finite
$d$-hereditary algebra. Moreover,
$A_{n}^{(d+1)}\cong\operatorname{End}_{A_{n}^{(d)}}(M_{n}^{(d)})$ where
$M_{n}^{(d)}$ is the unique basic $d$-cluster-tilting $A_{n}^{(d)}$-module.
The indecomposable direct summands of $M_{n}^{(d)}$ are precisely the interval
modules $M_{\lambda}$ for $\lambda\in\mathbf{os}^{(d+1)}_{n}$.

The following theorem of Iyama characterises the higher Auslander algebras of
type $\mathbb{A}$ as the only finite dimensional algebras which are
$d$-representation finite $d$-hereditary and $(d-1)$-Auslander algebras at the
same time. Thus, in particular, these are the only $d$-representation-finite
$d$-hereditary algebra which can be constructed inductively as in Theorem 3.7.

######  Theorem 3.8 ([Iya11, Theorem 1.19]).

Let $d\geq 2$ and $\Lambda$ a finite dimensional algebra. Then, $\Lambda$ is
$d$-representation-finite $d$-hereditary as well as a $(d-1)$-Auslander
algebra if and only if it is Morita equivalent to $A_{n}^{(d)}$ where $n$ is
the number of simple $\Lambda$-modules in the $d$-cluster-tilting subcategory
of $\operatorname{mod}\Lambda$.

For constructing the higher analogue of the path algebra of
$\tilde{\mathbb{A}}$ note that $A_{\infty}^{(d)}$ is closed under
$\varphi_{d}$. Recall that $\varphi_{d}$ was defined to be the quiver
automorphism satisfying

$\displaystyle\varphi_{d}(\mu_{1},\dots,\mu_{d})=(\mu_{1}-1,\dots,\mu_{d}-1),$
$\displaystyle\varphi_{d}(a_{i}(\mu_{1},\dots,\mu_{d}))=a_{i}(\mu_{1}-1,\dots,\mu_{d}-1).$

Define the higher analogue of the path algebra of $\tilde{\mathbb{A}}_{n-1}$
by $\tilde{A}_{n-1}^{(d)}:=A_{\infty}^{(d)}/\varphi_{d}^{n}$. It is easy to
see that the images of the $A_{\infty}^{(d)}$-modules $M_{\lambda}$ and
$M_{\mu}$ under the pushdown functor are isomorphic if and only if
$\lambda\equiv\mu\mod n\mathbb{Z}(1,1,\dots,1)$. As the quiver of this
category contains oriented cycles, we consider the category of finite
dimensional nilpotent modules instead of the category of finite dimensional
modules.

######  Theorem 3.9 ([JK16a, Theorem 2.3.4]).

The category $\operatorname{nil}\tilde{A}^{(d)}_{n-1}$ has a $d$-cluster-
tilting subcategory $\mathcal{T}^{(d)}_{n}$ whose objects are given by the
images of the interval modules $M_{\lambda}$ where
$\lambda\in\mathbf{os}^{(d+1)}_{n}$. Moreover,
$\mathcal{T}^{(d)}_{n}\cong\operatorname{add}\tilde{A}_{n-1}^{(d+1)}$.

The name $\mathcal{T}_{n}^{(d)}$ is chosen because for $d=1$,
$\mathcal{T}_{n}^{(d)}$ is the tube of rank $n$ from ‘classical’
Auslander–Reiten theory.

After having defined the analogues of the path algebras of $\mathbb{A}_{n}$
and $\tilde{\mathbb{A}}_{n-1}$, the next step is to find analogues for
admissible relations for Nakayama algebras. In contrast to classical Nakayama
algebras, which arise as quotients by admissible ideals of the algebras
$A_{n}^{(1)}$ and $\tilde{A}_{n-1}^{(1)}$, the higher Nakayama algebras arise
as idempotent quotients from $A_{n}^{(d)}$ and $\tilde{A}_{n-1}^{(d)}$,
respectively.

Recall that a convenient combinatorial way to label Nakayama algebras is by
means of the vector

$(\dim P_{0},\dim P_{1},\dots,\dim P_{n-1})$

where $P_{i}$ is the indecomposable projective to the vertex $i$. This vector
is called the _Kupisch series_ corresponding to the Nakayama algebra. Given a
Kupisch series $\underline{\ell}=(\ell_{0},\dots,\ell_{n-1})$, the
corresponding set of relations one needs to divide out by is given by the
paths $\left\\{\,i-\ell_{i}\leadsto
i\,\middle|\,\ell_{i-1}\geq\ell_{i}\,\right\\}$. Note that the interval
modules $M_{(\mu_{1},\mu_{2})}$ for the Nakayama algebra with Kupisch series
$\underline{\ell}$ satisfy $\mu_{1}-\mu_{2}+1\leq\ell_{\mu_{1}}$ as
$\mu_{1}-\mu_{2}+1$ gives the length of the interval module
$M_{(\mu_{1},\mu_{2})}$ and $\mu_{1}$ gives its top and every module is a
quotient of the corresponding indecomposable projective module with the same
top.

Analogously, in the higher setting for $d\geq 2$, the higher Nakayama algebra
$A_{\underline{\ell}}^{(d)}$ (of type $\mathbb{A}$, i.e. for $\ell_{0}=1$)
associated to the Kupisch series $\underline{\ell}$ is defined to be the
idempotent quotient of $A_{n}^{(d)}$ by the idempotents not in

$\mathbf{os}^{(d)}_{\underline{\ell}}:=\left\\{\,(\mu_{1},\dots,\mu_{d})\in\mathbf{os}_{n}^{(d)}\,\middle|\,\mu_{1}-\mu_{d}+1\leq\ell_{\mu_{1}}\,\right\\}.$

For the Nakayama algebras of type $\tilde{\mathbb{A}}_{n-1}$ given the Kupisch
series $\ell$ one defines an infinite series $\tilde{\ell}$ by periodically
extending $\ell$. One then defines

$\mathbf{os}^{(d)}_{\tilde{\underline{\ell}}}:=\left\\{\,(\mu_{1},\dots,\mu_{d})\in\mathbf{os}^{(d)}\,\middle|\,\mu_{1}-\mu_{d}+1\leq\tilde{\ell}_{\mu_{1}}\,\right\\}.$

Note that by construction, $\mathbf{os}^{(d)}_{\tilde{\underline{\ell}}}$ is
stable under $\varphi_{d}^{n}$. Define $\tilde{A}_{\underline{\ell}}^{(d)}$ to
be the idempotent quotient of $\tilde{A}_{n-1}^{(d)}$ by the idempotents not
corresponding to the ones in $\mathbf{os}^{(d)}_{\tilde{\underline{\ell}}}$.

######  Theorem 3.10.

For every Kupisch series $\underline{\ell}$ and every $d\geq 1$, the algebra
$A_{\underline{\ell}}^{(d)}$ (respectively
$\tilde{A}_{\underline{\ell}}^{(d)}$ in type $\tilde{\mathbb{A}}$) is weakly
$d$-representation-finite. The modules in the corresponding $d$-cluster-
tilting module are the interval modules $M_{\lambda}$ with
$\lambda\in\mathbf{os}^{(d+1)}_{\underline{\ell}}$ (respectively
$\lambda\in\mathbf{os}^{(d+1)}_{\tilde{\underline{\ell}}}$ in type
$\tilde{\mathbb{A}}$).

Among all weakly $d$-representation-finite algebras, the higher Nakayama
algebras belong to a more special class of algebras, since their cluster-
tilting modules $M$ satisfy the stronger condtion

$\operatorname{Ext}^{i}(M,M)\neq 0\implies i\in d\mathbb{Z}.$

One of their special features is that exact sequences of length $d$ in the
cluster-tilting subcategory give rise to long exact
$\operatorname{Ext}^{d}$-sequences, see [IJ16]. Such algebras are called
_$d\mathbb{Z}$ -representation-finite_. They can also be characterised by the
fact that the $d$-cluster-tilting module is closed under $\Omega^{d}$.

##  4\. Obstructions to an alternative definition of higher Nakayama algebras

The reader may ask whether there is a characterisation of higher Nakayama
algebras similar to projectives and injectives being uniserial. The main
problem in answering this question is that it is not clear what the analogue
of a filtration is when there are only higher $\operatorname{Ext}$-groups, but
no $\operatorname{Ext}^{1}$. In this section we discuss a few more problems
with such definition and with statements involving the simples in a cluster-
tilting subcategory.

###  4.1. Simple modules and the $\operatorname{Ext}^{d}$-quiver of the higher
Nakayama algebras

We start by characterising simples only in terms of the cluster tilting
subcategory.

######  Definition 4.1.

Let $\mathcal{A}$ be an additive category and $S\in\mathcal{A}$ a non-zero
object. We say that $S$ is _Schur simple_ if every non-zero morphism $X\to S$
is an epimorphism and every non-zero morphism $S\to Y$ is a monomorphism.

It turns out that, with this definition, Schur simples of a cluster tilting
subcategory are precisely the simple modules.

######  Lemma 4.2.

Let $\mathcal{A}$ be an abelian category and $\mathcal{M}\subseteq\mathcal{A}$
a functorially finite generating-cogenerating subcategory. Then, an object
$S\in\mathcal{M}$ is Schur simple if and only if it is a simple object in
$\mathcal{A}$.

###### Proof.

Let $S\in\mathcal{M}$ be simple in $\mathcal{A}$. Then, of course, it is Schur
simple. For the other direction we show that a Schur simple object in
$\mathcal{M}$ is in fact Schur simple in $\mathcal{A}$, and hence simple. For
this, let $S\in\mathcal{M}$ be Schur simple and $X\in\mathcal{A}$ be arbitrary
with a morphism $f\colon S\to X$. Since $\mathcal{M}$ is covariantly finite,
there exists a left $\mathcal{M}$-approximation $g\colon X\to M$ for some
$M\in\mathcal{M}$. Since $S$ is Schur simple in $\mathcal{M}$, the composition
$gf$ is a monomorphism or zero. Since the left approximation $g$ is a
monomorphism since $\mathcal{M}$ is cogenerating, it follows that $f$ is a
monomorphism or zero. Dually, one shows that every non-zero morphism $X\to S$
in $\mathcal{A}$ is an epimorphism. It follows that $S$ is Schur simple in
$\mathcal{A}$ which is equivalent to being simple. ∎

######  Definition 4.3.

þ Let $\mathcal{A}$ be an abelian category and
$\mathcal{M}\subseteq\mathcal{A}$ a $d$-cluster-tilting subcategory. The
_$\operatorname{Ext}^{d}$ -quiver_ of $\mathcal{M}$ is the quiver which has as
vertices the isomorphism classes of Schur simples in $\mathcal{M}$ and the
number of arrows $[S]\to[S^{\prime}]$ is given by
$\dim_{\mathbbm{k}}\operatorname{Ext}^{d}_{\mathcal{A}}(S,S^{\prime})$.

The $\operatorname{Ext}^{d}$-quivers of the higher Nakayama algebras are
precisely what one would expect, namely equal to the
$\operatorname{Ext}^{1}$-quivers of the corresponding (classical) Nakayama
algebras.

######  Proposition 4.4.

Let $\underline{\ell}$ be a Kupisch series. Then, the the
$\operatorname{Ext}^{d}$-quiver of $A_{\underline{\ell}}^{(d)}$ is
$\mathbb{A}_{n}$ in case $\ell_{0}=1$. while the
$\operatorname{Ext}^{d}$-quiver of $\tilde{A}_{\underline{\ell}}^{(d)}$ is
$\widetilde{\mathbb{A}}_{n-1}$ in case $\ell_{0}\neq 1$.

###### Proof.

This follows immediately from the results of [JK16a]. ∎

We now give an example showing that, in general the
$\operatorname{Ext}^{d}$-quiver for $d$-cluster-tilting subcategories of
module categories of finite dimensional algebras is ill-behaved.

######  Example 4.5.

þ Let $A$ be the preprojective algebra of type $\mathbb{A}_{3}$ which is also
the higher Nakayama algebra $\tilde{A}_{(3)}^{(2)}$, that is, $A$ is the
preprojective algebra of $\mathbb{A}_{3}$, i.e. the path algebra of the quiver

bound by the relations $y_{1}x_{1}=0$, $x_{2}y_{2}=0$ and
$x_{1}y_{1}=y_{2}x_{2}$. The reader can verify that

$M=A\oplus{\small\begin{matrix}1\end{matrix}}\oplus{\small\begin{matrix}2\\\
1\end{matrix}}\oplus{\small\begin{matrix}1\\\
2\end{matrix}}\quad\text{and}\quad N=A\oplus{\small\begin{matrix}2\\\
1\end{matrix}}\oplus{\small\begin{matrix}2\\\
3\end{matrix}}\oplus{\small\begin{matrix}2\\\ 13\end{matrix}}$

are $2$-cluster-tilting $A$-modules, $M$ being the one given by interval
modules. For the convenience of the reader, the Auslander–Reiten quiver of
$\operatorname{mod}A$ is given below, where indecomposable $A$-modules are
described by their radical filtration and the indecomposable direct summands
of $M$ and $N$ are indicated with rectangles and circles respectively.

The $\operatorname{Ext}^{2}$-quiver of $\operatorname{add}M$ has one vertex
and one arrow and the $\operatorname{Ext}^{2}$-quiver of $\operatorname{add}N$
is empty since $N$ has no simple summands. Thus, this example shows that in
general

  * •

the $\operatorname{Ext}^{d}$-quiver of $\mathcal{M}$ depends on $\mathcal{M}$
and not only on $A$, and that

  * •

the $\operatorname{Ext}^{d}$-quiver of $\mathcal{M}$ may be empty.

In particular, the $\operatorname{Ext}^{d}$-quiver of $\mathcal{M}$ in general
does not determine $\mathcal{M}$ nor $A$.

###  4.2. Global dimension of the higher Nakayama algebras

A further example shows that even for the higher Nakayama algebras, where the
simple modules in many ways behave like simple modules in the classical sense,
e.g. in the sense that they all lie on a unique $\tau_{d}$-orbit (see
[JK16a]), in some respect they are different. The subsequent example shows
that the global dimension of one of the Nakayama algebras is not attained by a
projective dimension of a simple in the cluster-tilting subcategory as well as
the global dimension of $A^{(d)}_{\underline{\ell}}$ is in general not equal
to $d\cdot\operatorname{gldim}A^{(1)}_{\underline{\ell}}$. Moreover, the
example also shows that, in general,

$\dim\operatorname{Ext}_{A^{(1)}_{\underline{\ell}}}^{2}(S_{i},S_{j})\neq\dim\operatorname{Ext}_{A^{(d)}_{\underline{\ell}}}^{2d}(S_{(i,i,\cdots,i)},S_{(j,j,\cdots,j)}).$

This suggest that there is no straightforward way to generalise relations in
$A_{\underline{\ell}}^{(1)}$, which are given by the
$\operatorname{Ext}_{A_{\underline{\ell}}^{(1)}}^{2}$ between simples, to
higher values of $d$.

######  Example 4.6.

Let $\underline{\ell}=(1,2,2,3,3,4,3)$. The following figure shows the module
category of $A^{(1)}_{\underline{\ell}}$. The numbers on the vertices show the
projective dimensions of the corresponding indecomposable modules.

In particular, the global dimension is $4$ and is attained by the simple
module $S_{6}$. The following figure shows the $2$-cluster-tilting module of
$A_{\underline{\ell}}^{(2)}$ whose indecomposable direct summands are the
interval modules. The numbers on the vertices show the projective dimensions
divided by $d=2$ of the corresponding indecomposable modules.

Note that the global dimension of the algebra is $6$, but is not attained by a
simple module in the cluster tilting subcategory. Also note that

$\operatorname{gldim}A^{(2)}_{\underline{\ell}}=6\neq 2\cdot
4=2\cdot\operatorname{gldim}A^{(1)}_{\underline{\ell}}.$

Finally, note that

$0\neq\dim\operatorname{Ext}_{A^{(1)}_{\underline{\ell}}}^{2}(S_{6},S_{3})\neq\dim\operatorname{Ext}_{A^{(2)}_{\underline{\ell}}}^{4}(S_{(6,6)},S_{(3,3)})=0$

where the rightmost vanishing condition can be shown by noting that the fifth
term in a minimal projective resolution of $S_{(6,6)}$ does not have a summand
$S_{(3,3)}$ in its top.

It is not even true that finiteness of global dimension is preserved when
passing to the higher analogue222The authors want to thank Yiping Chen for
posing this question.. To show this, consider the same example as before but
with the last and the first vertex identified, i.e. the higher Nakayama
algebra of type $\tilde{\mathbb{A}}_{5}$ with Kupisch series $(2,3,3,4,3,2)$.
Then, $\operatorname{projdim}S_{0}=\infty$ for
$\tilde{A}_{{\underline{\ell}}}^{(1)}$, but
$\operatorname{gldim}\tilde{A}_{\underline{\ell}}^{(2)}=6<\infty$.

###  4.3. Dominant dimension of the higher Nakayama algebras

Concerning the dominant dimension, the authors proved in [JK16a] that
$\operatorname{domdim}A\geq d$ for every $d$-Nakayama algebra $A$. It is
however also not true that
$\operatorname{domdim}\tilde{A}_{\underline{\ell}}^{(d)}=d\cdot\operatorname{domdim}\tilde{A}_{\underline{\ell}}^{(1)}$
as the example of the higher Nakayama algebra $\tilde{A}_{(3,4,4)}^{(d)}$
shows. For $d=1$ its dominant dimension is $4$ as $P_{1}=I_{1}$ and
$P_{2}=I_{2}$ are projective-injective and the non-injective module $P_{0}$
has an injective coresolution

$0\to P_{0}\to I_{1}\to I_{2}\to I_{2}\to I_{0}\to 0.$

On the other hand, $\tilde{A}^{(2)}_{(3,4,4)}$ has $11$ indecomposable
projective of which $8$ are projective-injective. We label them by their label
as interval modules for the category $A_{\infty}^{(d)}$. The projective-
injectives are $P_{(1,0)}=I_{(3,1)}$, $P_{(2,1)}=I_{(4,2)}$,
$P_{(2,2)}=I_{(4,3)}$, $P_{(1,1)}=I_{(4,1)}$, $P_{(4,2)}=I_{(2,1)}$,
$P_{(2,0)}=I_{(3,2)}$, $P_{(4,3)}=I_{(2,2)}$, and $P_{(4,1)}=I_{(1,1)}$. The
non-injective modules $P_{(0,0)}$, $P_{(3,1)}$, and $P_{(3,2)}$ have the
following injective coresolutions, respectively.

$\displaystyle 0\to P_{(3,1)}\to I_{(1,1)}\to I_{(4,1)}\to I_{(4,2)}\to
I_{(4,3)}\to I_{(2,2)}\to I_{(3,2)}\to I_{(0,0)}\to 0$ $\displaystyle 0\to
P_{(3,2)}\to I_{(2,1)}\to I_{(4,1)}\to I_{(1,0)}\to I_{(2,0)}\to I_{(2,2)}\to
I_{(3,2)}\to I_{(0,0)}\to 0$ $\displaystyle 0\to P_{(0,0)}\to I_{(3,1)}\to
I_{(4,1)}\to I_{(1,1)}\to I_{(2,1)}\to I_{(2,2)}\to I_{(4,3)}\to I_{(2,0)}\to
0$

Hence, $\operatorname{domdim}\tilde{A}_{(3,4,4)}^{(2)}=6\neq 2\cdot
4=2\cdot\operatorname{domdim}\tilde{A}_{(3,4,4)}^{(1)}$.

##  5\. Cluster categories of type $\mathbb{A}_{n}$ and $\mathbb{A}_{\infty}$

In this section, we survey on the relationship between cluster tilting
subcategories and cluster categories, especially the connection to
triangulations of polygons and generalisations thereof. Already in the first
paper on cluster algebras [FZ02], Fomin and Zelevinsky studied an example of a
cluster algebra of type $\mathbb{A}_{n}$, the coordinate ring of the affine
cone over the Grassmannian of $2$-dimensional subspaces of an
$(n+3)$-dimensional (complex) vector space. The clusters of the cluster
algebra are in bijection with triangulations of the polygon with $n$-vertices.
The flip, that is the operation on triangulations replacing one diagonal by
the unique other one yielding a triangulation, corresponds to mutating a
cluster variable. Cluster algebras were later categorified in [BMR+06], and
the authors of that paper realised that clusters of the cluster algebra of
type $\mathbb{A}_{n}$ correspond to cluster-tilting objects in the
corresponding cluster category of type $\mathbb{A}_{n}$, which is a certain
$2$-Calabi–Yau category. As certain tilting objects in
$D^{b}(\operatorname{mod}A_{n}^{(1)})$ provide cluster-tilting objects in the
cluster category of type $\mathbb{A}_{n}$, the concept of cluster-tilting
objects can be seen as an extension of the concept of tilting objects. There
have been two generalisations of this picture. Staying in the
‘$2$-dimensional’ situation, the situation has been generalised to $n=\infty$
by Holm and Jørgensen in [HJ12] categorifying triangulations of the apeirogon,
in that paper called infinitygon. The corresponding cluster category also
arises as the perfect derived category of the cochain differential graded
algebra associated to the $2$-sphere. The other generalisation was to the
higher setting, replacing the algebra $A_{n}^{(1)}$ by the higher Auslander
algebras $A_{n}^{(d)}$. This generalisation is due to Oppermann and Thomas in
[OT12] and they prove that a corresponding cluster category categorifies the
triangulations of the cyclic polytope in dimension $2d$. In a forthcoming
paper [JK17], the authors will unify the two generalisations. In this section,
we survey on the results of [HJ12, OT12, JK17]. In the first subsection, we
introduce higher analogues of the cluster categories of type $\mathbb{A}_{n}$.
The second subsection gives the correspondence between tilting modules for
$A_{n}^{(d)}$ and triangulations of a cyclic polytope, following [OT12]. The
third subsection is concerned with the results of [OT12] and [JK17] providing
a bijection between the cluster-tilting subcategories of a cluster category
and triangulations of a cyclic polytope and a corresponding infinite version
for the cyclic apeirotope.

###  5.1. Categorical construction

We start this section by recalling the definitions of a Calabi–Yau
triangulated category and of a spherical object.

######  Definition 5.1.

Let $\mathcal{D}$ be a $\mathbbm{k}$-linear Hom-finite triangulated category.

  1. (i)

A functor $\mathbb{S}\colon\mathcal{D}\to\mathcal{D}$ is called a _Serre
functor_ if there is a bifunctorial isomorphism

$D\operatorname{Hom}_{\mathcal{D}}(X,Y)\cong\operatorname{Hom}_{\mathcal{D}}(Y,\mathbb{S}X).$

  2. (ii)

The category $\mathcal{D}$ is said to be _$\delta$ -Calabi–Yau_ if $[\delta]$,
the $\delta$-th power of the shift functor, is a Serre functor.

  3. (iii)

Let $\mathcal{D}$ be a $\delta$-Calabi–Yau triangulated category. An object
$X\in\mathcal{D}$ is called _$\delta$ -spherical_ if

$\operatorname{Hom}_{\mathcal{D}}(X,X[i])=\begin{cases}\mathbbm{k}&\text{if
$i=0,\delta$},\\\ 0&\text{else.}\end{cases}$

The thick subcategory generated by a $\delta$-spherical object, i.e. the
smallest triangulated category containing the $\delta$-spherical object and
being closed under direct summands is independent of the specific
$\delta$-spherical object (especially of the ambient triangulated category),
see e.g. [KYZ09, Theorem 2.1]. This category arises in topology as follows:
Let $m\geq 2$ be an integer. Let $X=S^{m}$ be the $m$-sphere. Let
$C^{*}(X,\mathbbm{k})$ be the cochain differential graded algebra of $X$ with
coefficients in $\mathbbm{k}$ (with respect to the cup product. Let
$D^{c}(C^{*}(X,\mathbbm{k}))$ be the perfect derived category of
$C^{*}(X,\mathbbm{k})$, i.e. the subcategory spanned by all compact objects in
the derived category of the differential graded algebra
$C^{*}(X,\mathbbm{k})$. The following result is partially due to Jørgensen and
partially folklore.

######  Theorem 5.2 (cf. [Jør04]).

The category $D^{c}(C^{*}(X,\mathbbm{k}))$ is an $m$-Calabi–Yau category. In
particular, it has almost split triangles. Its Auslander–Reiten quiver
consists of $(m-1)$ components of type $\mathbb{Z}\mathbb{A}_{\infty}$.
Furthermore, $D^{c}(C^{*}(X,\mathbbm{k}))$ is classically generated by an
$m$-spherical object $S$, i.e. $D^{c}(C^{*}(X,\mathbbm{k}))$ itself is the
smallest triangulated subcategory of $D^{c}(C^{*}(X,\mathbbm{k}))$ containing
$S$.

There are several other incarnations of this category throughout mathematics.
Almost by definition, $D^{c}(C^{*}(X,\mathbbm{k}))\cong
D^{c}(\mathbbm{k}[\varepsilon]/(\varepsilon^{2}))$ where
$\mathbbm{k}[\varepsilon]/(\varepsilon^{2})$ is regarded as a differential
graded algebra with trivial differential and $\varepsilon$ in degree $m$.
Using Koszul duality, it is also equivalent to the category
$D^{f}(\mathbbm{k}[T])$, the derived category of complexes with finite
dimensional (total) cohomology over the differential graded algebra
$\mathbbm{k}[T]$ with trivial differential and $T$ in degree $-m+1$.

In representation theory, more precisely in the theory of cluster categories,
the category is better known as the $m$-cluster category of type
$\mathbb{A}_{\infty}$ and can be defined as
$D^{b}(\operatorname{mod}A_{\infty}^{(1)})/\mathbb{S}[-m]$ where $\mathbb{S}$
denotes the Serre functor of $D^{b}(\operatorname{mod}A_{\infty}^{(1)})$ which
exists because $D^{b}(\operatorname{mod}A_{\infty}^{(1)})$ has almost-split
triangles.

Our goal in [JK17] was to obtain similar results starting from
$A_{\infty}^{(d)}$ instead of $A_{\infty}^{(1)}$. As explained in the
introduction to this section, the construction of a higher cluster category of
type $\mathbb{A}_{n}$ was achieved by Oppermann–Thomas in [OT12]. Our proof
follows a similar strategy but additional technical complications arise. The
first step, similar to [OT12] is to realise that
$D^{b}(\operatorname{mod}A_{\infty}^{(d)})/\mathbb{S}[-md]$ is not
triangulated. But, thanks to a construction by Keller [Kel05], it is possible
to embed it into its triangulated hull which we denote by
$\mathcal{C}_{\infty,m}^{(d)}$. The technically difficult part is to prove
that this category is in fact $\operatorname{Hom}$-finite. This is achieved by
establishing a derived equivalence
$D^{b}(\operatorname{mod}A_{\infty}^{(d)})\cong
D^{b}(\operatorname{mod}Z_{\infty}^{(d)})$ where $Z_{\infty}^{(d)}$ is a
locally bounded category. Recall that a subcategory
$\mathcal{O}\subseteq\mathcal{D}$ of a triangulated category is called _weakly
$d\mathbb{Z}$-cluster-tilting_ if it satisfies the analogous
$\operatorname{Ext}$-vanishing property to the case of an abelian category,
but is not necessarily functorially finite (nor generating-cogenerating) and
furthermore it is closed under shift by $[d]$.

Then, a straightforward generalisation of the methods in [Ami09, Ami11, Guo11]
shows the following theorem:

######  Theorem 5.3 ([OT12, Theorems 5.14 and 5.25] for finite $n$ and $m=2$,
[JK17]).

Let $n$ be finite or infinity. Let $A_{n}^{(d)}$ be the algebra (or category)
defined in Section 3. Let $\mathcal{D}$ be the category
$D^{b}(\operatorname{mod}A_{n}^{(d)})$. Let $\mathbb{S}$ be the corresponding
Serre functor on $\mathcal{D}$. The triangulated hull
$\mathcal{C}_{n,m}^{(d)}$ of $\mathcal{D}/\mathbb{S}[-md]$ is triangulated
$\operatorname{Hom}$-finite $md$-Calabi–Yau.

Let $\mathcal{U}$ be the subcategory $\mathcal{U}(A_{n}^{(d)})$ of
$D^{b}(\operatorname{mod}A_{n}^{(d)})$. The subcategory
$\mathcal{O}_{n,m}^{(d)}:=\mathcal{U}/\mathbb{S}[-md]$ is weakly
$d\mathbb{Z}$-cluster-tilting in $\mathcal{C}_{\infty,m}^{(d)}$. For finite
$n$ the ‘weakly’ can be omitted. For infinite $n$, the category
$\mathcal{O}_{\infty,m}^{(d)}$ is classically generated by an $md$-spherical
object $S$, i.e. closing the subcategory $S$ under shifts and $(d+2)$-angles
gives the whole category $\mathcal{O}_{\infty,m}^{(d)}$.

An open question is still whether it is possible to get rid of the ‘weakly’ in
the theorem in case $n=\infty$. The problem is that it is not clear whether in
this case $D^{b}(\operatorname{mod}A_{\infty}^{(d)})/\mathbb{S}[-md]$ is
functorially finite in its triangulated hull which would be a sufficient
condition.

Another open question is whether for $n=\infty$ there is a natural
construction associating these categories to certain topological objects which
generalises the case of $d=1$.

###  5.2. Tilting modules for $A_{n}^{(d)}$

Recall that there is a bijection between the isomorphism classes of
indecomposable basic tilting $A_{n}^{(1)}$-modules and the triangulations of
the regular polygon with $n+2$ vertices; moreover, this bijection is
compatible with tilting mutation on one side and flip of triangulations on the
other side [BK04, OT12].

The higher dimensional analogues of regular polygons are widely considered to
be the cyclic polytopes [DLRS10, Section 6.1]. In order to define them let
$\left\\{\,(t,t^{2},\dots,t^{\delta})\,\middle|\,t\in\mathbb{R}\,\right\\}$ be
the _moment curve_. Choose $p$ points on it. Then, a _cyclic polytope_ is the
convex hull $C(p,\delta)$ of such points. A _triangulation_ of a cyclic
polytope is a subdivision of it into $\delta$-simplices whose vertices are
also vertices of $C(p,\delta)$. By a result of Dey [Dey93] the analogy to
triangulations of regular polygons is particularly strong if $\delta=2d$ is
even. In this case, a triangulation is determined by the $d$-dimensional
internal simplices, more precisely:

######  Theorem 5.4 (cf. [OT12, Theorem 2.4]).

Specifying a triangulation of $C(p,2d)$ is equivalent to giving a collection
of $p-d-1\choose d$ non-intersecting $d$-simplices which do not lie on a lower
boundary facet with vertices in the $p$ points.

For $d=1$, every maximal collection of $1$-simplices has $p-2$ elements. For
$d>2$ this is in general no longer true, see [OT12].

There is an involutive operation on the set of triangulations of $C(p,\delta)$
called “bistellar flip”. On the corresponding collection of non-intersecting
$d$-simplices it can be described by removing a $d$-simplex and replacing it
by the unique other $d$-simplex giving rise to a triangulation. When
$\delta=2$, this reduces to the usual flip of triangulations of planar
polygons, see _loc. cit._ for further details.

The corresponding notion on the categorical side is tilting mutation: Given
two tilting modules $T$ and $T^{\prime}$ contained in
$\operatorname{add}M_{n}^{(d)}$ which only differ in indecomposable direct
summands $X$ and $X^{\prime}$, $T$ is called the mutation of $T^{\prime}$ in
$X^{\prime}$ and vice versa. In this case, there exists either an exact
sequence

$0\to X\to T_{d}\to\dots\to T_{1}\to X^{\prime}\to 0$

with $T_{i}\in\operatorname{add}\overline{T}$ where $\overline{T}$ is a direct
complement to $X$ in $T$ (resp. to $X^{\prime}$ in $T^{\prime}$) or an exact
sequence in the other direction.

######  Theorem 5.5 ([OT12, Thm. 1.1]).

There is a bijection

${\left\\{\text{internal $d$-simplices of
$C(n+2d,2d)$}\right\\}}$${\left\\{\text{indecomposable non-projective-
injective summands of $M_{n}^{(d)}$}\right\\}}$

which in turn induces a bijection

${\left\\{\text{triangulations of
$C(n+2d,2d)$}\right\\}}$${\left\\{\text{basic tilting modules for
$A_{n}^{(d)}$ contained in $M_{n}^{(d)}$}\right\\}.}$

Moreover, this bijection is compatible with tilting mutation on one side and
bistellar flip of triangulations on the other side.

The second bijection follows from the fact that under the first bijection two
$d$-simplices intersect if and only if the corresponding indecomposable
modules have non-trivial $\operatorname{Ext}^{d}$ in some direction. Note that
because of the property of $M_{n}^{(d)}$ being cluster-tilting and the fact
that $\operatorname{gldim}A_{n}^{(d)}=d$, the space $\operatorname{Ext}^{d}$
is the only possible extension not to vanish. This theorem gives a first
categorification of the triangulations of a cyclic polytope. In the
forthcoming subsection, we give a second one using cluster tilting
subcategories of a cluster category instead of tilting modules in a module
category.

###  5.3. Triangulations of cyclic polytopes and apeirotopes

In this subsection, we describe the combinatorial model for the cluster-
tilting subcategories of the category $\mathcal{O}_{n,2}^{(d)}$ for finite or
infinite $n$, introduced in Subsection 5.1, in more detail. This has been done
for finite $n$ by Oppermann–Thomas [OT12] and for $n=\infty$ by the authors in
[JK17] building on earlier work [FZ02, BMR+06, HJ12]. Following [OT12], we
need to replace functorially finiteness in the definition of cluster-tilting
by a stronger condition in the category $\mathcal{O}_{n,2}^{(d)}$:

######  Definition 5.6.

An object $T\in\mathcal{O}_{n,2}^{(d)}$ is called _cluster-tilting_ if

  1. (i)

$\operatorname{Hom}_{\mathcal{O}_{n,2}^{(d)}}(T,T[d])=0$, and

  2. (ii)

Any $X\in\mathcal{O}_{n,2}^{(d)}$ occurs in a $(d+2)$-angle

$X[-d]\to T_{d}\to T_{d-1}\to\dots\to T_{1}\to T_{0}\to X$

with $T_{i}\in\operatorname{add}T$.

With this definition, for finite $n$, Oppermann and Thomas were able to prove
that cluster-tilting subcategories of $\mathcal{O}_{n,2}^{(d)}$ are precisely
the cluster-tilting subcategories of $\mathcal{C}_{n,2}^{(d)}$ contained in
$\mathcal{O}_{n,2}^{(d)}$. For infinite $n$ the problem is again that it is
not clear whether $\mathcal{O}_{\infty,2}^{(d)}$ is functorially finite in
$\mathcal{C}_{\infty,2}^{(d)}$. The only ingridient missing before stating
Oppermann–Thomas’ categorification of triangulations of the cyclic polytope by
cluster-tilting subcategories of the higher cluster category of type
$\mathbb{A}$ is the notion of mutation of cluster-tilting objects.

######  Definition 5.7.

Let $X$ and $Y$ be indecomposable objects of $\mathcal{O}_{n,2}^{(d)}$ and
$\overline{T}$ be such that $X\oplus\overline{T}$ and $Y\oplus\overline{T}$
are cluster-tilting objects. Then $X\oplus\overline{T}$ is called a mutation
of $Y\oplus\overline{T}$ at $Y$ and vice versa.

In this case, there are _exchange $(d+2)$-angles_

$X\to E_{d}\to\dots\to E_{1}\to Y$

and

$Y\to F_{1}\to\dots\to F_{d}\to X$

with $E_{i},F_{j}\in\operatorname{add}\overline{T}$.

######  Theorem 5.8 ([OT12, Thm. 1.2]).

There is a bijection

${\left\\{\text{internal $d$-simplices of
$C(n+2d+1,2d)$}\right\\}}$${\left\\{\text{indecomposable objects in
$\mathcal{O}_{n,2}^{(d)}$}\right\\}}$

which in turn induces a bijection

${\left\\{\text{triangulations of
$C(n+2d+1,2d)$}\right\\}}$${\left\\{\text{basic $2d$-cluster-tilting objects
in $\mathcal{O}_{n,2}^{(d)}$}\right\\}.}$

The bistellar flip on the left hand side corresponds to the mutation of
cluster-tilting objects on the right hand side.

For our generalisation to the infinite situation, we consider the _cyclic
apeirotope_ , i.e. the convex hull $C(\infty,2d)$ of
$\left\\{\,(t,t^{2},\dots,t^{2d})\,\middle|\,t\in\mathbb{Z}\,\right\\}$. An
_Ind-finite triangulation_ is a collection of internal $d$-simplices such that
for all $I\subseteq\mathbb{Z}$ finite, there exists an interval $[a,b]$
containing $I$ such that the arcs with end points in
$\left\\{\,(t,t^{2},\dots,t^{2d})\,\middle|\,t\in\mathbb{Z}\cap[a,b]\,\right\\}$
form the internal $d$-simplices of a triangulation.

######  Theorem 5.9.

There is a bijection

${\left\\{\text{internal $d$-simplices of
$C(\infty,2d)$}\right\\}}$${\left\\{\text{indecomposable objects in
$\mathcal{O}_{\infty,2}^{(d)}$}\right\\}}$

which in turn induces a bijection

${\left\\{\text{Ind-finite triangulations of
$C(\infty,2d)$}\right\\}}$${\left\\{\text{basic $2d$-cluster-tilting
subcategories in $\mathcal{O}_{\infty,2}^{(d)}$}\right\\}.}$

The bistellar flip on the left hand side again corresponds to the mutation of
$d$-cluster-tilting objects on the right hand side.

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