UPPSALA DISSERTATIONS IN MATHEMATICS

118

Department of MathematicsUppsala University

UPPSALA 2020

Constructions of n-cluster tilting subcategories using representation-directed algebras

Laertis Vaso

Dissertation presented at Uppsala University to be publicly examined in Polhemsalen,Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Tuesday, 2 June 2020 at 13:15 for thedegree of Doctor of Philosophy. The examination will be conducted in English. Facultyexaminer: Professor Øyvind Solberg (Norwegian University of Science and Technology,Department of Mathematical Sciences).

AbstractVaso, L. 2020. Constructions of n-cluster tilting subcategories using representation-directedalgebras. Uppsala Dissertations in Mathematics 118. 47 pp. Uppsala: Department ofMathematics. ISBN 978-91-506-2824-1.

One of the most useful tools in representation theory of algebras is Auslander–Reiten theory.A higher dimensional analogue has recently appeared, based on the notion of n-cluster tiltingsubcategories. It turns out that the existence of such subcategories in the module category of analgebra gives important information about the whole module category. However it is unclearin general which algebras have module categories that admit an n-cluster tilting subcategory.This thesis provides many different ways to construct n-cluster tilting subcategories. The maintools are representation-directed algebras, methods from classical Auslander–Reiten theory andhomological algebra.

In Paper I we give a characterization of n-cluster tilting subcategories for representation-directed algebras. Using this characterization, we classify acyclic Nakayama algebras withhomogeneous relations whose module categories admit an n-cluster tilting subcategory. Weprovide a formula for the global dimension of such algebras and then classify all Nakayamaalgebras with homogeneous relations whose module categories admit a d-cluster tiltingsubcategory, where d is the global dimension of the algebra.

In Paper II we construct a generalization of n-cluster tilting subcategories for representation-directed algebras called n-fractured subcategories. By defining a gluing procedure we areable to construct n-cluster tilting subcategories for representation-directed algebras by gluingcompatible n-fractured subcategories. As an application of this method, for any positive integer nand any positive integer d ≥ 2n we explicitly construct an algebra with global dimension dwhose module category admits an n-cluster tilting subcategory. If n is odd, then this result canbe improved to any d ≥ n.

In Paper III we further generalize the gluing procedure defined in Paper II. First we givesense to a notion of infinite gluings which gives an n-cluster tilting subcategory for an abeliancategory which is not the module category of an algebra. Next, we apply an orbit constructionto obtain an algebra which is not necessarily representation-directed but its module categoryadmits an n-cluster tilting subcategory. Finally we classify starlike algebras with radical squarezero relations whose module categories admit n-cluster tilting subcategories. Using the theorydeveloped in this paper, we present many families of algebras whose module categories admitn-cluster tilting subcategories.

Keywords: Representation theory, n-cluster tilting subcategory, Auslander–Reiten theory,representation-directed algebra, Nakayama algebra, global dimension

Laertis Vaso, Department of Mathematics, Algebra and Geometry, Box 480, UppsalaUniversity, SE-751 06 Uppsala, Sweden.

© Laertis Vaso 2020

ISSN 1401-2049ISBN 978-91-506-2824-1urn:nbn:se:uu:diva-408540 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-408540)

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I L. Vaso. n-Cluster tilting subcategories of representation-directedalgebras. Journal of Pure and Applied Algebra, 223 (2019) 2101-2122.

II L. Vaso. Gluing of n-cluster tilting subcategories forrepresentation-directed algebras. arXiv: 1805.12180v3, to appear inAlgebras and Representation Theory.

III L. Vaso. n-cluster tilting from gluing systems ofrepresentation-directed algebras. arXiv: 2004.02269, submitted forpublication.

Reprints were made with permission from the publishers.

Additional papers

The following papers are not included in this thesis.

IV M. Herschend, P. Jørgensen, and L. Vaso. Wide subcategories ofd-cluster tilting subcategories. Transactions of the AmericanMathematical Society 373 (2020), 2281-2309.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Representation theory of finite-dimensional algebras . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Algebras and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2 Category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.3 Quivers bound by relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.4 Global dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.5 Auslander–Reiten theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Higher-dimensional Auslander–Reiten theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.1 n-cluster tilting modules and subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.2 n-cluster tilting and global dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Possible future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1 n-cluster tilting subcategories for algebras which are not

representation-finite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Classification of algebras with radical square zero relations whose

module categories admit n-cluster tilting subcategories . . . . . . . . . . . . . . . . . . . . 39

5 Svensk sammanfattningv (Summary in Swedish) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.1 Bakgrund . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Sammanfattning av artiklar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1. Introduction

One of the main aims of representation theory of algebras is to describe thecategory modΛ of finite-dimensional (right) modules of an algebra Λ. TheseΛ-modules can be decomposed in an essentially unique way into indecompos-able summands, and so the easiest case is when there are only finitely manysuch indecomposable Λ-modules. One of the most important results in this di-rection is the Auslander correspondence between module categories of repre-sentation-finite algebras and Auslander algebras. Not only does the Auslandercorrespondence allow one to translate properties from one side to the other,but it is also a precursor to the very useful tool of Auslander–Reiten theory.

In recent years a higher-dimensional analogue to the above situation has ap-peared. Instead of focusing on modΛ, one restricts to a suitable subcategoryC of modΛ called an n-cluster tilting subcategory for some positive integer n;if moreover C has an additive generator M, then M is called an n-cluster tiltingmodule and Λ is called n-representation-finite. Iyama showed that there ex-ists an n-Auslander correspondence between n-cluster tilting subcategories ofn-representation-finite algebras and n-Auslander algebras [23]. Furthermore,an n-dimensional Auslander–Reiten theory for n-cluster tilting subcategorieswas developed by Iyama and others. Both the n-Auslander correspondenceand n-dimensional Auslander–Reiten theory reduce to their classical versionsfor n = 1. Apart from representation theory, higher-dimensional Auslander–Reiten theory also has applications in algebraic geometry, especially in non-commutative crepant resolutions [30, 31, 32]. Connections to algebraic topol-ogy have also been found recently, more specifically in the study of Fukayacategories [15].

There always exists a unique 1-cluster tilting subcategory of modΛ, namelymodΛ itself. If n > 1, then an n-cluster tilting subcategory may not exist. Alot of research has been done in trying to find or construct n-cluster tiltingsubcategories. Many special cases are of particular interest. The case n = 2coincides with the notion of cluster tilting which is used to categorify clusteralgebras introduced by Fomin and Zelevinsky [17, 18]. By definition of n-cluster tilting subcategories, if Λ is not semisimple, then n ≤ d where d isthe global dimension of Λ. The extreme case n = d has many interestingproperties and has also been studied before [28, 20, 21, 29, 13]. In particulard-representation-finite algebras can be thought of as generalizations of repre-sentation-finite hereditary algebras.

Finding an n-cluster tilting subcategory is not a trivial task and so thereare still many questions about the properties of n-cluster tilting subcategories.

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For example, it is unknown if there exists an n-cluster tilting subcategory ofthe module category of an algebra that admits no additive generator. Thisthesis primarily deals with the problem of finding and constructing n-clustertilting subcategories. Certain classes of algebras whose module categoriesadmit n-cluster tilting subcategories are classified, while many more familiesof examples with interesting properties are presented.

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2. Preliminaries

The aim of this chapter is to present the background and context needed tounderstand and motivate the contents of this thesis. We present definitionsand results which are used throughout Papers I-III. We do not include manydetails, but give references for the interested reader.

2.1 Representation theory of finite-dimensional algebrasIn this section we give a brief summary of the notions of representation theoryof finite-dimensional algebras which are required to explain the results of thisthesis. We also motivate and present some ideas behind the main concepts.For more details on the representation theory of finite-dimensional algebraswe refer to the books [11, 2].

2.1.1 Algebras and modulesOne of the most basic algebraic structures is that of a group. A group is a setG together with an associative binary operation G×G→ G such that thereexists a (necessarily unique) identity element and every element of the grouphas an inverse. If moreover the operation is commutative, then we say that thegroup is abelian. For an abelian group we denote the operation by + and callit addition and denote the identity element by 0 and call it the zero element.

There are two ways to expand starting from an abelian group G. One mayadd a second associative binary operation on G which admits an identity el-ement and is distributive with respect to the addition in G. In this case onegets a (unital) ring, the new operation is denoted by · or simply by concate-nation of elements and called ring multiplication and the identity element isdenoted by 1. Maps of rings that respect the ring structure are called ringmorphisms. A commutative ring is a ring where ring multiplication is com-mutative. Commutative rings where the invertible elements with respect tothe ring multiplication are precisely the nonzero elements are called fields.Fields where every nonconstant polynomial has a root are called algebraicallyclosed. As an example, both the set of real numbers R and the set of complexnumbers C are fields with the usual notions of addition and multiplication, butonly C is algebraically closed.

Another notion that one may construct starting from the notion of an abeliangroup is that of a k-vector space. Let G be an abelian group and k be a field.

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Then one defines a scalar multiplication of elements of G with elements of kwhich we denote by concatenation. One requires that this scalar multiplicationis compatible with the field multiplication of k, distributive with respect toaddition in G, and that the identity element 1 of k acts trivially; if these axiomsare satisfied then we say that G is a k-vector space. Maps of k-vector spacesthat respect the k-vector space structure are called k-linear maps. If V is ak-vector space, then a subspace of V is a non-empty subset of V that is closedunder addition and scalar multiplication.

If one starts with an abelian group and makes it both into a ring and into avector space such that all defined operations are compatible with each other,then one gets an (associative) k-algebra.

Definition 2.1.1. A k-algebra Λ is a ring which is also a k-vector space andsuch that for all λ ,λ ′ ∈ Λ and k ∈ k we have k(λ ·λ ′) = (kλ ) ·λ ′ = λ · (kλ ′).

A morphism of k-algebras f : Λ→ Λ′ is a k-linear ring morphism.

For the rest of this thesis we fix a field k and say algebra instead of k-algebra. Given an algebra Λ, we can always consider the opposite algebraΛop where the k-vector space structure is the same, but ring multiplication isgiven by a ·op b := b · a. A nonempty subset I ⊆ Λ of an algebra Λ is calleda left ideal if λx ∈ I for all λ ∈ Λ and x ∈ I. It is called a right ideal if thedual condition holds and it is called a two-sided ideal if it is both a left andright ideal. We simply say ideal instead of two-sided ideal. The intersectionof all maximal (with respect to inclusion) proper right ideals of Λ is called the(Jacobson) radical of Λ and is denoted by rad(Λ).

An immediate example of an algebra is the field k itself with scalar multi-plication given by its ring multiplication. An important example of an alge-bra is the endomorphism algebra Endk(V ) of a vector space V consisting ofall k-linear maps V → V , where addition and scalar multiplication are givenpointwise and ring multiplication is given by composition.

Given that algebras are quite abstract objects, it is useful to find a way tomake them more concrete. The following notion helps in that direction.

Definition 2.1.2. Let Λ be an algebra. A (right) Λ-module is a k-vector spaceM equipped with a binary operation M×Λ→M, (m,λ ) 7→ mλ satisfying

(i) (m+m′)λ = mλ +m′λ ,(ii) m(λ +λ ′) = mλ +mλ ′,

(iii) m(λλ ′) = (mλ )λ ′,(iv) m1 = m, and(v) (km)λ = k(mλ )

for all m,m′ ∈M, λ ,λ ′ ∈ Λ and k ∈ k.A Λ-module morphism from a Λ-module M to a Λ-module N is a k-linear

map f : M→ N such that f (mλ ) = f (m)λ for all m ∈M and λ ∈ Λ.

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We define a left Λ-module to be a right Λop-module. In the following byΛ-module we always mean right Λ-module. A submodule N of a Λ-moduleM is a subspace of M such that nλ ∈ N for all n ∈ N and λ ∈ Λ. A Λ-moduleis called simple if its only submodule is 0. The intersection of all maximalsubmodules of a Λ-module M is called the (Jacobson) radical of M and isdenoted by rad(M). For a Λ-module M the top of M, denoted top(M), isdefined to be the Λ-module M/ rad(M).

As we have seen the field k is also an algebra. It is easy to see that thenotions of k-module and k-vector space coincide. Moreover, any algebra Λ isitself a Λ-module where the operation Λ×Λ→ Λ is given by the ring multi-plication of Λ.

Notice that if M is a Λ-module, then for every λ ∈ Λ we may consider thek-linear map fλ ∈ Endk(M) given by fλ (m) = mλ . The assignment λ 7→ fλ

gives a morphism of algebra Λop → Endk(M). In this way we can representthe elements of an algebra by k-linear maps and the algebra multiplicationby composition of maps. From this point of view we can see representationtheory as a way to reduce questions about algebras to questions about k-vectorspaces and k-linear maps.

If n is a nonnegative integer, then the set kn of n-tuples with elements in k isa k-vector space with componentwise operations. If there exists an invertiblek-linear map V → kn for some nonnegative integer n, then this n is unique andcalled the dimension of V . In this case we say that V is finite-dimensional. Ifno such invertible k-linear map exists, then we say that the dimension of V is∞ and call V infinite-dimensional. In this thesis we are interested in studyingfinite-dimensional algebras.

If n is a nonnegative integer, then the set kn×n of n× n matrices with el-ements in k is also an algebra, with usual addition, scalar multiplication andmultiplication of matrices as operations. Let V be a k-vector space. If V hasdimension n<∞, then Endk(V ) can be identified with kn×n. Hence in this casewe can further reduce some of the problems of abstract algebras to problemsof matrices and their operations.

Given an algebra Λ, we have motivated the study of Λ-modules and mor-phisms of Λ-modules. The proper framework for this study is described in thenext section.

2.1.2 Category theoryWe start with the definition of a category.

Definition 2.1.3. A category C consists of• a class of objects obj(C ),• a set of morphisms C (x,y) between any pair of objects x,y ∈ obj(C ),

and

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• a way of composing morphisms f ∈ C (x,y) and g ∈ C (y,z) to obtain amorphism g f ∈ C (x,z)

such that composition is associative and there exists an identity morphism idxfor each x ∈ obj(C ).

A functor F : C → D between categories C and D is a mapping thatassociates to each object x ∈ C an object F(x) ∈ D and to each morphismf ∈ C (x,y) a morphism F( f ) ∈D(F(x),F(y)), such that F(idx) = idF(x) andF( f g) = F( f )F(g).

We simply write x ∈ C instead of x ∈ obj(C ). If C is a category, thenC op is the category with same objects as C but with C op(x,y) = C (y,x) andcomposition defined in the obvious way.

If Λ is a finite-dimensional algebra, then finite-dimensional Λ-modules andmorphisms between them form the category modΛ of (right) Λ-modules. Rep-resentation theory of finite-dimensional algebras can be thought of as thestudy of this category. If M,N ∈ modΛ then we denote modΛ(M,N) byHomΛ(M,N). An important example of a functor is the standard duality func-tor D := Homk(−,k) between the categories modΛ and mod(Λop). Noticethat in particular, finite-dimensional k-vector spaces and k-linear maps can beidentified with the category modk.

The algebraic structures that we have defined (group, ring, k-vector space,algebra, module) share the same general pattern of being one or two sets withsome operations defined between them that satisfy some properties. Givenan algebraic structure, we may relabel the elements of the sets but otherwisekeep the operations intact. Although this new algebraic structure is formallydifferent from the one we started with, the underlying structure on both is thesame. To compare the structure of two algebraic structures we need the notionof isomorphism.

Definition 2.1.4. (a) Let C be a category. Then a morphism f ∈ C (x,y) iscalled an isomorphism if there exists a morphism g ∈ C (y,x) such thatf g = idy and g f = idx. In that case g is unique and denoted by f−1,we write x∼= y and we say that x and y are isomorphic.

(b) Let C , D be categories and F : C → D be a functor. Then F induces afunction Fx,y : C (x,y)→D(F(x),F(y)). We say that F is• dense if for every y ∈D there exists an x ∈ C such that F(x)∼= y,• full if Fx,y is surjective for every x,y ∈ C ,• faithfull if Fx,y is injective for every x,y ∈ C .

If F is dense, full and faithfull then we say that F is an equivalence ofcategories. If there exists an equivalence of categories F : C →D , thenwe write C 'D and we say that C and D are equivalent.

It follows that a Λ-module isomorphism is an invertible morphism of Λ-modules and an algebra isomorphism is an invertible morphism of algebras.

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Isomorphic Λ-modules are essentially the same; similarly isomorphic algebrasare also essentially the same. For example, given two algebras Λ1 and Λ2,there are canonical componentwise operations defined on the set product Λ1×Λ2 that turn it into an algebra. If Λ∼= Λ1×Λ2, then we may reduce questionsabout Λ to questions about Λ1 and Λ2. Hence, it is enough to study connectedalgebras only, that is we may assume that Λ is not isomorphic to a productΛ1×Λ2 of algebras.

The category modΛ enjoys many nice properties which can be studied in amore abstract categorical setting. We describe some of them.

Definition 2.1.5. We say that a category C is k-linear if for every x,y ∈ C theset of morphisms C (x,y) is a k-vector space and composition of morphismsis k-bilinear. An ideal R of a k-linear category C is a collection R(i, j) ⊆C (i, j) for every i, j ∈ C such that• if f ∈R(i, j) and g ∈ C ( j,k) then g f ∈R(i,k), and• if f ∈R(i, j) and h ∈ C (m, i) then f h ∈R(m, j).

The category modΛ is indeed k-linear. An important ideal of modΛ is theJacobson radical of modΛ defined by

radΛ(M,N) :=

h ∈ HomΛ(M,N)

∣∣∣∣ idM−gh is invertiblefor all g ∈ HomΛ(N,M)

If f ∈ HomΛ(M,N), then we define the kernel of f by

ker( f ) := m ∈M | f (m) = 0,

the image of f by

im( f ) := n ∈ N | there exists an m ∈M with f (m) = n,

and the cokernel of f by

coker( f ) := N/ im( f ).

Moreover, one can define a notion of direct sum M⊕N of two Λ-modulesM and N. If a Λ-module is not isomorphic to the direct sum of two nonzeroΛ-modules then we say that it is indecomposable. Each Λ-module can bedecomposed as a direct sum of indecomposable Λ-modules in an essentiallyunique way. Using such a decomposition, one can compute Λ-module mor-phisms M→ N for M,N ∈ modΛ by computing Λ-module morphisms fromthe indecomposable direct summands of M to the indecomposable direct sum-mands of N. Hence to compute Λ-module morphisms between Λ-modules,one needs only to compute the Λ-module morphisms between their indecom-posable summands.

The essential properties of modΛ can be axiomatized to the notion of anabelian category and a Krull–Schmidt category, which are both special types

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of additive categories. In any additive category there exists a notion of directsum. Moreover, the notions of kernel, image and cokernel make sense in anyabelian category. Finally, in any Krull–Schmidt there exists a decompositionof objects in indecomposable summands in an essentially unique way.

Two algebras Λ and Λ′ may not be isomorphic, but their module categoriesmay be equivalent; in this case we say that Λ and Λ′ are Morita equivalent.From the point of view of representation theory Morita equivalent algebrasare the same. By writing Λ as a direct sum of indecomposable Λ-modules wemay count the multiplicities of the indecomposable summands; if they are allequal to 1 we say that Λ is a basic algebra. Moreover, for an algebra Λ therealways exists a Morita equivalent basic algebra Λb. Since we are interested inthe study of the module category of an algebra, we may restrict our study tobasic algebras.

2.1.3 Quivers bound by relationsQuivers bound by relations give us a combinatorial way to study algebras andtheir modules. They were first explicitly introduced in [19].

A quiver Q is a directed graph. We denote by Q0 the set of vertices, by Q1the set of arrows, and for an arrow α : i→ j we define its source s(α) to be iand its target t(α) to be j. A path p of length l in Q is a sequence of arrows

p = (α1,α2, . . . ,αl)

such that t(αi) = s(αi+1) for i = 1,2, . . . , l− 1; we call s(α1) the source of pand t(αl) the target of p. To each vertex i ∈ Q0 we associate a trivial pathεi : i→ i of length 0. If Q is a finite quiver, then the path algebra kQ of aquiver Q is the algebra freely generated by the paths of Q (including the trivialpaths) as a k-vector space and where multiplication is given by concatenationof paths on the basis elements as in

(α1, . . . ,αl)(β1, . . . ,βk) =

(α1, . . . ,αl,β1, . . . ,βk), if t(αl) = s(β1),

0, otherwise,(2.1)

and extended linearly to the rest of the elements of kQ. The arrow ideal ofkQ, denoted RQ, is the ideal of kQ generated by Q1.

An ideal R ⊆ kQ is called admissible if RlQ ⊆R ⊆ R2

Q for some l ≥ 2. IfR is an admissible ideal, then kQ/R is a finite-dimensional basic algebra. Arelation on Q is a k-linear combination of paths of length at least two in Q,all of them having the same source and target. If R is a set of relations suchthat the ideal 〈R〉 generated by R is admissible, then we say that Q is boundby the relations R. If the field k is algebraically closed field, then it turns outthat for every basic finite-dimensional algebra Λ there exists a quiver Q anda set of relations R such that Λ is Morita equivalent to kQ/〈R〉. Moreover,

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the quiver Q is connected if and only if the algebra Λ is connected. Since weare interested in studying basic connected finite-dimensional algebras, we mayrestrict to connected quivers bound by relations without losing any generality.

Quivers bound by relations not only provide us with a visualization of al-gebras, they also provide us with a combinatorial way to represent modules.A representation M of a quiver Q bound by relations R is an assignment of ak-vector space Mi at each vertex i∈Q0 and of a linear map φα : Ms(α)→Mt(α)

for each arrow α ∈ Q1, such that it is compatible with the relations R. Moreconcretely, if p = (α1, . . . ,αl) is a path in Q, then we set φp = pαl · · · pα1 ,and then if ∑ci pi ∈ R we require that ∑ciφpi = 0. By defining morphismsbetween representations in an appropriate way, we can define the categoryrep(Q,R) of finite-dimensional representations of Q bound by relations R. Itcan be shown that if Q is a finite quiver, then rep(Q,R) is equivalent to thecategory mod(kQ/〈R〉). We thus may describe kQ/〈R〉-modules using rep-resentations of Q bound by R and this identification is used throughout thisthesis.

If a quiver has infinitely many vertices, then the path algebra is not defined.In that case, we have the following useful notion.

Definition 2.1.6. Let Q be a (potentially infinite) quiver. The path categorykQ of Q is the k-linear category where• obj(kQ) = Q0,• kQ(i, j) for i, j ∈ Q0 is equal to the k-vector space spanned by all paths

from i to j in Q, and• if α ∈ C (i, j) and β ∈ C ( j,k), then β α = αβ where αβ is as in (2.1).

A k-linear category C is called locally bounded, if for every x ∈ C we have

∑y∈C

(dimk C (x,y)+dimk C (y,x))< ∞.

Let R be an ideal of kQ such that kQ/R is a locally bounded category. Arepresentation of kQ/R is then a functor F : kQ/R→modk.

The following two quivers play a special role in this thesis:

Am : 1 2 m,α1 α2 αm−1

10

m 2 .Am :

α0

α1αm

αm−1 α2

17

2.1.4 Global dimensionFor a module M ∈modΛ we denote by add(M) the additive closure of M, thatis the smallest full subcategory of modΛ containing all direct summands ofdirect sums of M. Especially we denote add(Λ) by projΛ and we call objectsof projΛ projective Λ-modules. Similarly we denote add(DΛ) by injΛ and wecall objects of injΛ injective Λ-modules. Let us observe here that the top of anindecomposable projective Λ-module is always simple.

We define a complex M• of Λ-modules to be a sequence

· · · −→Mi+1fi+1−→Mi

fi−→Mi−1 −→ ·· ·

of Λ-module morphisms such that the composition of two consecutive mor-phisms is equal to zero. A complex M• is called exact if ker( fi) = im( fi+1)for all i. A short exact sequence is an exact complex of the form

· · · −→ 0−→Mi+1fi+1−→Mi

fi−→Mi−1 −→ 0−→ ·· · ,

which we denote briefly by 0−→Mi+1fi+1−→Mi

fi−→Mi−1 −→ 0. A short exactsequence is called split if there exists a morphism g : Mi → Mi+1 such thatg fi+1 = idMi+1 or equivalently if there exists a morphism h : Mi−1→Mi suchthat fi h = idMi−1 .

A projective resolution P•(M) of a module M ∈modΛ is an exact complex

· · · −→ P1p1−→ P0

p0−→M −→ 0−→ ·· ·

where every Pi is projective. The length of P•(M) is the smallest natural num-ber n such that Pi = 0 for i > n if such an n exists; otherwise the length isinfinite. Dually we define an injective resolution and its length. By applyingthe functor HomΛ(−,N) : modΛ→modΛ to the complex

· · · −→ P1p1−→ P0

p0−→ 0−→ 0−→ ·· · ,

we get a complex

· · · −→ 0−→ HomΛ(P0,N)−p1−→ HomΛ(P1,N)−→ ·· · .

Then we define ExtiΛ(M,N) := ker(− pi+1)/ im(− pi).

There always exist many different projective resolutions for a Λ-module Mof different lengths. A minimal projective resolution can be constructed in thefollowing inductive way. A projective cover of M is a projective Λ-moduleP(M) and an epimorphism p : P(M)→ M such that no direct summand ofP(M) is in the kernel of p. Projective covers always exist and are unique upto isomorphism; for example if P is an indecomposable projective Λ-module,then it is the projective cover of its simple top P/ rad(P). We denote the kernel

18

of a projective cover by Ω(M) and call it the syzygy of M. A minimal projec-tive resolution is then constructed by setting Pi =P(Ωi(M)) for every i≥ 0. Allminimal projective resolutions have the same (possibly infinite) length whichwe call the projective dimension of M and denote it by pd(M). Dually we de-fine minimal injective resolutions, injective hulls and the injective dimensionof a module.

Definition 2.1.7. The global dimension of an algebra Λ is defined to be

gl.dim(Λ) := suppd(M) |M ∈modΛ.

The global dimension of an algebra is an important numerical invariant. Analgebra Λ has gl.dim(Λ)= 0 if and only if rad(Λ)= 0; such algebras are calledsemisimple. An algebra Λ has gl.dim(Λ) ≤ 1 if and only if all submodulesof projective Λ-modules are projective; such algebras are called hereditary.Moreover, if k is an algebraically closed field, then semisimple algebras areMorita equivalent to kQ for some Q with Q1 =∅ and hereditary algebras areMorita equivalent to a path algebra kQ of an acyclic quiver Q (note that kQ isnot a finite-dimensional algebra if Q is not acyclic).

2.1.5 Auslander–Reiten theoryAuslander–Reiten theory was developed in a series of papers ([3, 4, 5, 6, 7,8, 9, 10] and others) and was very successful in providing important insightsabout the category modΛ.

The following notion is central to Auslander–Reiten theory.

Definition 2.1.8. A short exact sequence 0−→ Lf−→M

g−→N −→ 0 is calledan almost split sequence if L and N are indecomposable, the sequence is notsplit, and

(i) any nonisomorphism h : U → N where U is indecomposable factorsthrough g, and

(ii) any nonisomorphism h′ : L→ U ′ where U ′ is indecomposable factorsthrough f .

The name almost split comes from the fact that if (i) or (ii) was true for iso-morphisms as well, then the sequence would be an actual split sequence. Weproceed by presenting a way to construct almost split sequences in a specialcase.

As we have explained, the category modΛ can be studied through the in-decomposable Λ-modules. If there are only finitely many indecomposable Λ-modules up to isomorphism, then the algebra Λ is called representation-finite.Let Λ be a representation-finite algebra and let M1, . . . ,Mk be a complete

19

and irredundant set of representatives of isomorphism classes of indecompos-able Λ-modules. Let M =

⊕ki=1 Mi. Then add(M) = modΛ by construction.

Moreover, the set EndΛ(M) := HomΛ(M,M) is a finite-dimensional algebracalled the Auslander algebra of Λ.

The Auslander algebra EndΛ(M) has many interesting properties and canbe characterized in a homological way. Of particular interest to us is the factthat if Λ is not semisimple, then gl.dim(EndΛ(M)) = 2. Next notice that ifN ∈modΛ, then we can equip HomΛ(M,N) with the structure of an EndΛ(M)-module using composition of functions. Then the set HomΛ(M,Mi)k

i=1is a complete and irredundant set of representatives of isomorphism classesof indecomposable projective EndΛ(M)-modules. In particular, the top ofHomΛ(M,Mi) is a simple EndΛ(M)-module Si for every i ∈ 1, . . . ,k. Leti ∈ 1, . . . ,k and assume that Mi is not projective. It turns out that in this caseSi has projective dimension 2. Hence the projective resolution of Si has theform

0−→ HomΛ(M,N2)−→ HomΛ(M,N1)−→ HomΛ(M,Mi)−→ Si −→ 0(2.2)

where N1,N2 ∈modΛ. It turns out that such a projective resolution gives riseto an almost split sequence of Λ-modules

0−→ N2 −→ N1 −→Mi −→ 0.

In this way we have constructed an almost split sequence in modΛ endingat Mi. On the other hand, all almost split sequences in modΛ give rise toprojective resolutions of simple EndΛ(M)-modules.

Such a construction of almost split sequences is possible for representation-finite algebras. If the algebra Λ is not representation-finite, this constructioncan be generalized by considering modules over categories instead of algebras.In fact we have the following characterization of almost split sequences.

Proposition 2.1.9. [6] Let 0 −→ Lf−→ M

g−→ N −→ 0 be a short exact se-quence in modΛ with f ∈ rad(L,M) and g ∈ rad(M,N) such that any of thefollowing two equivalent conditions is satisfied.

(i) The sequence 0→ HomΛ(U,L)f−−→ HomΛ(U,M)

g−−→ radΛ(U,N)→ 0is exact for every U ∈modΛ.

(ii) The sequence 0→ HomΛ(N,U)−g−→ HomΛ(M,U)

−g−→ radΛ(L,U)→ 0is exact for every U ∈modΛ.

Then the following are equivalent.(a) L is indecomposable.(b) N is indecomposable.

(c) The sequence 0−→ Lf−→M

g−→ N −→ 0 is an almost split sequence.

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There is another method to construct almost split sequence that is also ofinterest to us. Let us denote by modΛ the stable category of modΛ, that is thecategory with the same objects as modΛ and morphisms spaces HomΛ(M,N)given by taking the quotient of HomΛ(M,N) by all morphisms M → N thatfactor through a projective A-module. Dually we define the costable categorymodΛ.

Next, notice that we have the Λ-dual functor

(−)t := HomΛ(−,Λ) : modΛ←→mod(Λop)

which induces a duality (−)t : projΛ←→ proj(Λop). If P•(M) is a minimalprojective resolution of M, then we define Tr(M) := coker(pt

1). In particular,if P is projective, then one has Tr(P) = 0. One can then check that Tr induces aduality between modΛ and mod(Λop) called the Auslander–Bridger transpose.Next notice that the k-dual functor D := Homk(−,k) : modΛ→ mod(Λop)restricts to a duality modΛ←→ mod(Λop). Hence by composing we get thefunctors

τ := DTr : modΛ→modΛ,

τ− := TrD : modΛ→modΛ,

called the Auslander–Reiten translations. Indecomposable modules and theirAuslander–Reiten translations are connected through almost split sequences.

Theorem 2.1.10. [6](a) For any indecomposable nonprojective Λ-module M there exists an al-

most split sequence 0→ τM→ E →M→ 0 which is unique up to iso-morphism.

(b) For any indecomposable noninjective Λ-module N there exists an almostsplit sequence 0→ N → E → τ−N → 0 which is unique up to isomor-phism.

In particular, the functor τ gives a bijection from the set of isomorphismclasses of indecomposable nonprojective Λ-modules to the set of isomorphismclasses of indecomposable noninjective Λ-modules, with τ− as an inverse.

A way of encoding the above information is the following. The Auslan-der–Reiten quiver Γ(Λ) of Λ is the quiver with vertices given by isomorphismclasses [M] of indecomposable Λ-modules and arrows [M]→ [N] being in bi-jection to basis elements of the vector space radΛ(M,N)/ rad2

Λ(M,N). Let [M]be a vertex in Γ(Λ) such that M is nonprojective. Let [Ei]k

i=1 be the set ofvertices in Γ(Λ) such that there exists at least one arrow [Ei]→ [M]. Let ei bethe number of arrows [Ei]→ [M]. Then there exist ei arrows [τM]→ [Ei]. Wehave a dual result starting from a noninjective Λ-module and applying τ−.

From the point of view of representation theory, the Auslander–Reiten quiv-er contains all the information about the objects in modΛ and a lot (but pos-sibly not all) information about the morphisms in modΛ. It is a particularly

21

useful tool when studying representation-finite algebras, since in this case theAuslander–Reiten quiver is finite. Furthermore, in this case, it contains allinformation about morphisms in modΛ.

The following class of algebras plays a special role in this thesis.

Definition 2.1.11. [36, 2.4] A representation-directed algebra is a representa-tion-finite algebra with no oriented cycles in its Auslander–Reiten quiver.

In particular, for a representation-directed algebra, one can construct theAuslander–Reiten quiver in an algorithmic way. One can start with the simpleprojective modules and compute their inverse Auslander–Reiten translation.Continuing recursively, one can compute the whole Auslander–Reiten quiverafter a finite number of steps.

2.2 Higher-dimensional Auslander–Reiten theoryA higher-dimensional analogue of Auslander–Reiten theory has been recentlydeveloped by Iyama and collaborators. In this section we give a brief out-line of this theory in comparison to the classical Auslander–Reiten theory aspresented in Section 2.1.5. For more details we refer to the survey articles[25, 34].

2.2.1 n-cluster tilting modules and subcategoriesIyama’s higher-dimensional Auslander–Reiten theory [24, 23] provides a cer-tain generalization of the classical picture. As we have seen, if Λ is represen-tation-finite, then we can construct almost split sequence in modΛ by consid-ering projective resolutions of EndΛ(M)-modules where M satisfies add(M) =modΛ. In this case the algebra EndΛ(M) has global dimension at most 2.

Our aim is then to replace almost split sequences by a higher version calledn-almost split sequences. A necessary condition to develop this theory is thevanishing of certain extensions. This leads to the following definition.

Definition 2.2.1. Let M ∈modΛ be such that

add(M) = X ∈modΛ | ExtiΛ (X ,add(M)) = 0 for all 1≤ i≤ n−1= X ∈modΛ | ExtiΛ (add(M),X) = 0 for all 1≤ i≤ n−1.

Then M is called an n-cluster tilting module and Λ is called an n-representa-tion-finite algebra.

In particular, an algebra is 1-representation-finite if and only if it is rep-resentation-finite. Then if Λ is an n-representation-finite algebra with an n-cluster tilting module M, one may similarly to the classical case consider

22

the n-Auslander algebra EndΛ(M). In this case it turns out that if Λ is notsemisimple, then

gl.dim(EndΛ(M)) = n+1,

and so we recover the classical result for n = 1.In fact one can generalize the above definition. First one may replace

add(M) with a subcategory C ⊆modΛ which satisfies a similar Ext-vanishingproperty to add(M). It turns out that in order to ensure the existence of n-almost split sequences in this general case, one needs a technical conditioncalled functorial finiteness. One may even define n-cluster tilting subcate-gories for general abelian categories.

Definition 2.2.2. Let A be an abelian category. Let C ⊆A be a functoriallyfinite subcategory such that

C = X ∈A | ExtiΛ (X ,C ) = 0 for all 1≤ i≤ n−1= X ∈A | ExtiΛ (C ,X) = 0 for all 1≤ i≤ n−1.

Then C is called an n-cluster tilting subcategory.

In particular, since add(M) is always functorially finite, it follows that M isan n-cluster tilting module if and only if add(M) is an n-cluster tilting subcat-egory.

We can define n-almost split sequences in an n-cluster tilting subcategoryusing the following higher-dimensional version of Proposition 2.1.9.

Proposition 2.2.3. [24] Let C ⊆ modΛ be an n-cluster tilting subcategory.Let

0−→ Yfn+1−→Cn

fn−→ ·· · f2−→C1f1−→ X −→ 0 (2.3)

be an exact sequence where X ,Y,Ci ∈ C and fi ∈ radΛ(Ci,Ci−1), under theidentifications Cn+1 = X and C0 =Y , such that any of the following two equiv-alent conditions is satisfied.

(i) The sequence

0→ HomΛ(U,Y )fn+1−−→ HomΛ(U,Cn)

fn−−→ ·· ·

· · · f2−−→HomΛ(U,C1)f1−−→ radΛ(U,X)→ 0

is exact for every U ∈ C .(ii) The sequence

0→ HomΛ(X ,U)− f1−→HomΛ(C1,U)

− f2−→ ·· ·− fn−→HomΛ(Cn,U)

− fn+1−→ radΛ(Y,U)→ 0

is exact for every U ∈ C .

23

Then the following are equivalent.(a) X is indecomposable.(b) Y is indecomposable.

If any of the two equivalent conditions (a) and (b) are also satisfied, then thesequence (2.3) is called an n-almost split sequence.

We can also define higher-dimensional versions of the Auslander–Reitentranslations. Let C ⊆ modΛ be an n-cluster tilting subcategory. Define thefunctors

τn := τΩn−1 : modΛ→modΛ,

τ−n := τ

−Ω−(n−1) : modΛ→modΛ,

called the n-Auslander–Reiten translations. Then we have the following high-er-dimensional analogue to Theorem 2.1.10.

Theorem 2.2.4. [24] Let C ⊆modΛ be an n-cluster tilting subcategory.(a) For any indecomposable nonprojective Λ-module X ∈ C there exists an

n-almost split sequence as in (2.3) such that Y ∼= τn(X).(b) For any indecomposable noninjective Λ-module Y ∈ C there exists an

n-almost split sequence as in (2.3) such that X ∼= τ−n (Y ).

In particular, the functor τn gives a bijection from the set of isomorphismclasses of indecomposable nonprojective Λ-modules in C to the set of isomor-phism classes of indecomposable noninjective Λ-modules in C , with τ−n as aninverse.

2.2.2 n-cluster tilting and global dimensionLet Λ be an n-representation-finite algebra and let d = gl.dim(Λ). If n > dthen Exti(DΛ,Λ) = 0 for all i ≥ 0 and so Λ is semisimple. Hence we mayassume that n≤ d.

In particular, if d = 1, then n = d = 1 and so Λ is a representation-finitehereditary algebra. These algebras are well-understood. In particular, if thefield k is algebraically closed, then Gabriel’s theorem [19] classifies them upto Morita equivalence as path algebras of quivers with a so-called Dynkin dia-gram, as an underlying graph. In this case we also have

add(M) = add

(⊕i≥0

τ−i(Λ)

).

In complete analogue, in the case n = d > 1 we have the following theorem.

24

Theorem 2.2.5. [26, Proposition 1.3] Let Λ be a d-representation-finite alge-bra where d = gl.dim(Λ) and let M be a d-cluster tilting Λ-module. Then

add(M) = add

(⊕i≥0

τ−id (Λ)

).

From this point of view, d-representation-finite algebras generalize repre-sentation-finite hereditary algebras. Indeed the case n = d is one of the mostfruitful in higher-dimensional Auslander–Reiten theory and has been studiedby many authors, see for example [28, 20, 21, 29, 13]. Notice that if M ∈modΛ is a d-cluster tilting module and X ∈ add(M) is an indecomposablenonprojective Λ-module, then τ

−d (X) is an indecomposable noninjective Λ-

module in add(M). In particular Ω−(d−1)(X) is noninjective and so Ω−d(X)is nonzero. Since gl.dim(Λ) = d, it follows that Ω−d(X) must be injective. Inparticular we have that add(M) is closed under Ω−d and dually is closed underΩd . In general, there is a class of n-cluster tilting subcategories called nZ-cluster tilting subcategories which was first explicitly introduced in [27] and ischaracterized by being closed under Ωn and Ω−n. The name nZ-cluster tiltingcomes from the fact that if C ⊆modΛ is an nZ-cluster tilting subcategory, thengl.dim(Λ) ∈ nZ (with the possibility gl.dim(Λ) = ∞). A module M ∈ modΛ

such that add(M) is an nZ-cluster tilting subcategory is called an nZ-clustertilting module. Many families of examples of nZ-cluster tilting modules havebeen recently constructed in [33].

For an arbitrary n and d = ∞, there have been constructions of n-clustertilting modules mainly for selfinjective algebras, that is algebras where Λ is aninjective Λ-module, see for example [16, 14].

One of the main motivations of this thesis, especially in Papers I and II,is to study the existence of n-cluster tilting modules for arbitrary n ≤ d. Inparticular we construct many examples where n - d.

25

3. Summary of papers

In this section we present the main results of the papers which comprise thisthesis. The main shared motivation between them is the construction of n-cluster tilting subcategories and modules with different properties. Note thatan algebra Λ may be n-representation-finite for different values of n. Hencewe may start by studying 1-representation-finite algebras, which are well-understood through classical Auslander–Reiten theory, to explore the exis-tence of n-cluster tilting modules for n > 1. It turns out that representation-di-rected algebras are especially receptive to such methods.

In Paper I we provide a useful characterization of representation-directedalgebras whose module categories admit an n-cluster tilting module in termsof classical Auslander–Reiten theory and homological algebra. We proceed byclassifying n-cluster tilting subcategories of acyclic Nakayama algebras withhomogeneous relations by an arithmetic condition. Using these methods, inthe same paper we classify d-representation-finite Nakayama algebras whered is the global dimension of the underlying algebra.

In Paper II we present a method of gluing two suitable representation-direct-ed algebras to construct a new representation-directed algebra whose modulecategory admits an n-cluster tilting subcategory. Under suitable assumptionswe show that we can control the global dimension of the new algebra and sowe prove that for any given d and n large enough there exists an n-representa-tion-finite algebra of global dimension d.

In Paper III we first generalize the methods of Paper II in a suitable wayto obtain a notion of infinite gluing. This does not give rise to a finite-dimen-sional algebra, but to the quotient category kQ/R of the path category of aninfinite quiver Q by an ideal R. The module category mod(kQ/R) admitsan n-cluster tilting subcategory. Under certain symmetry assumptions, we ap-ply an orbit construction to kQ/R and, applying a theorem of Darpö–Iyama,we obtain a representation-finite algebra whose module category admits ann-cluster tilting module. This algebra is not representation-directed in gen-eral. As an application, we construct many new families of algebras whosemodule categories admit n-cluster tilting modules with a variety of interestingproperties.

3.1 Paper ILet Λ be an algebra and C be an n-cluster tilting subcategory of modΛ.It readily follows from the definition of n-cluster tilting subcategories that

26

add(Λ),add(DΛ) ⊆ C and so we obtain a necessary condition for a subcat-egory of modΛ to be n-cluster tilting. Another necessary condition is thefollowing. Denote by CP and CI the sets of isomorphism classes of indecom-posable nonprojective respectively noninjective Λ-modules in C . Then by [25,Theorem 2.8] the functors τn and τ−n induce mutually inverse bijections

CP CI .τn

τ−n

Using Proposition 3.1 in Paper I, it also follows that if M ∈ CP, N ∈ CI andi ∈ 1, . . . ,n− 1, then Ωi(M) and Ω−i(N) are indecomposable and so weobtain two more necessary conditions. In general these conditions are notsufficient for the existence of an n-cluster tilting subcategory; for example ifΛ is selfinjective, they are all trivially satisfied for C = add(Λ) but add(Λ) isnot an n-cluster tilting subcategory in general. However, it turns out that in thecase of representation-directed algebras they are indeed sufficient and so as afirst result we have the following theorem.

Theorem 3.1.1. Let Λ be a representation-directed algebra and let C ⊆modΛ

be a full subcategory closed under direct sums and summands. Then C is ann-cluster tilting subcategory if and only if the following conditions hold.

(i) add(Λ)⊆ C .(ii) τn and τ−n induce mutually inverse bijections

CP CI .τn

τ−n

(iii) Ωi(M) is indecomposable for all M ∈ CP and 0 < i < n.(iv) Ω−i(N) is indecomposable for all N ∈ CI and 0 < i < n.

Theorem 3.1.1 gives an easy way to check if the module category of a repre-sentation-directed algebra admits an n-cluster tilting subcategory by reducingthe problem to computations of syzygies, cosyzygies and (classical) Auslan-der–Reiten translations.

Example 3.1.2. Let Q be the quiver

1 2

3

4

5

6 .

27

Then the Auslander–Reiten quiver of the algebra A = kQ/R2Q is

5

35

3

6

46

4

234

24

23

2

12

1

where one can check using Theorem 3.1.1 that the additive closure of the en-circled modules is a 2-cluster tilting subcategory.

An immediate consequence of Theorem 3.1.1 is that if the module categoryof a representation-directed algebra Λ admits an n-cluster tilting subcategoryC , then C is unique and given by

C = add

(⊕i≥0

τ−in (Λ)

). (3.1)

Moreover, the fact that conditions (iii) and (iv) of Theorem 3.1.1 are necessaryfor any algebra, give us the following restriction on the shape of the quiverof a bound quiver algebra whose module category admits an n-cluster tiltingsubcategory.

Proposition 3.1.3. Let Q be a finite connected quiver with m vertices, letΛ = KQ/R where R is an admissible ideal and let n≥ 2. Let k be a vertex inQ0, which is a sink or a source such that the full subquiver of Q with vertex setQ0 \k is disconnected. Then modΛ admits no n-cluster tilting subcategory.

We then proceed by applying Theorem 3.1.1 in the case of a bound quiveralgebra Λ = kQ/R where the underlying graph of Q is a Dynkin diagram oftype A. By Proposition 3.1.3 it follows that modΛ does not admit an n-clustertilting subcategory if Q is not linearly oriented. Hence we may assume thatQ = Am.

Even using Theorem 3.1.1, it turns out that it is too difficult to classify alltriples (m,R,n) such that R is an admissible ideal of kAm and mod(kAm/R)admits an n-cluster tilting subcategory. However, if R =Rl

Amfor some l, this is

possible. We denote by Λm,l the algebra kAm/RlAm

and we prove the followingtheorem.

Theorem 3.1.4. modΛm,l admits an n-cluster tilting subcategory if and onlyif one of the following conditions holds.

(i) l = 2 and m = nk+1 for some k ≥ 0, or

28

(ii) n is even and m = n2 l +1+ k(nl− l +2) for some k ≥ 0.

A Nakayama algebra is a bound quiver algebra kQ/R where Q∈Am, Am.Let Λ be a Nakayama algebra and let d = gl.dim(Λ). We show in Proposition5.1 of Paper I that if Λ is d-representation-finite, then Λ=Λm,l for some m≥ 1,l ≥ 2. In view of Theorem 3.1.4, to classify all d-representation-finite algebrasit is enough to compute the global dimension of Λm,l . In Proposition 5.2(d) ofPaper I we prove that

gl.dim(Λm,l) =

⌊m−1

l

⌋+

⌈m−1

l

⌉, (3.2)

where bxc denotes the largest integer k with k≤ x and dxe denotes the smallestinteger k with x ≤ k. Combining all these results, we obtain the followingclassification of d-representation-finite Nakayama algebras.

Theorem 3.1.5. Let Λ be a Nakayama algebra and gl.dimΛ= d. Then modΛ

admits a d-cluster tilting subcategory C if and only if Λ = Λm,l and l | m−1or l = 2. Moreover, in that case, C = add(Λ⊕DΛ) and d = 2 m−1

l .

We note that by combining Theorem 3.1.4 and (3.2), we obtain algebras ofglobal dimension d whose module categories admit an n-cluster tilting subcat-egory such that n - d. For example, the module category modΛ18,3 admits a4-cluster tilting subcategory and has global dimension 11.

3.2 Paper IIAs we already noted, algebras of global dimension d whose module categoriesadmit a d-cluster tilting module are the most studied case in higher-dimen-sional Auslander–Reiten theory. In Paper II we are interested in studying thefollowing question.

Question 1. Let (n,d) be a pair of positive integers (n,d) with n < d. Doesthere exist an algebra Λ of global dimension d such that modΛ admits ann-cluster tilting subcategory?

We call an algebra satisfying the conditions of Question 1 an (n,d)-repre-sentation-finite algebra. We answer Question 1 positively for d ≥ 2n when nis even and for d ≥ n when n is odd by explicitly constructing an (n,d)-rep-resentation-finite algebra. Our construction is based on Theorem 3.1.1 and agluing procedure. Let us also note that similar gluing constructions have beenmade before (see for example [22, 35, 12]) although not in connection to theexistence of n-cluster tilting subcategories.

29

From now on we assume that all algebras mentioned in this section are rep-resentation-directed. Before we describe our construction, let us introduce auseful piece of notation. Let Λ be an algebra and C and V be subcategoriesof modΛ. We set C\V to be the additive closure of all indecomposable Λ-modules X ∈ C such that X 6∈ V .

Let A = kQA/RA be a representation-directed algebra where QA is a quiverof the form

a1 a2 a3 · · · ah−1 ah ,α1 α2 α3 αh−2 αh−1Q′A

and no path of the form αi · · ·αi+k is in RA. In this case the indecomposableprojective module P(1) corresponding to the vertex 1 is called a left abutmentof height h. Notice that by definition we have that P(i) is a left abutments ofheight h+ 1− i for all 1 ≤ i ≤ h. Moreover, we have P(i) ⊆ P(i− 1). Weuse this relation to define a partial order on the set of isomorphism classes ofleft abutments by setting [P] ≤ [Q] if P is isomorphic to a submodule of Q.We refer to left abutments as maximal if they are maximal with respect to thispartial order. We also set

Pab = addP ∈ add(A) | P is a left abutment of A .

It is shown in Proposition 2.13 of Paper II that left abutments can also becharacterized in terms of a subquiver of the Auslander–Reiten quiver Γ(A) ofA; they correspond to a full subquiver P4 of Γ(A) of the form

,

where there are h vertices in the bottom row, [P] is the top vertex of this triangleand there are no other arrows with target in P4.

Let W be a maximal left abutment of A of height hW . A fracture T = T (W ) isan A-module with exactly hW non-isomorphic indecomposable summands, alllying in W4, and such that Ext1A(T,T ) = 0 (in other words a fracture of W canbe identified with a tilting kAhW -module). If P is another left abutment withP≤W , we choose a decomposition T (W ) = T (P)⊕Q in such a way that all in-decomposable summands of T (P) lie in P4 and no indecomposable summandof Q lies in P4.

30

A left fracturing T L of A is defined to be a direct sum of fractures, onefor each maximal left abutment W . Given a left fracturing T L, we set PL :=

add

add(A)\Pab ,T L

.

Dually we can define right abutments and characterize them by a full sub-quiver 4I isomorphic to P4 but where now there are no arrow with source in4I . We set

I ab = addI ∈ add(D(A)) | I is a right abutment of A .

Fractures of right abutments and right fracturings are defined similarly. Given

a right fracturing T R we set I R := add

add(D(A))\I ab ,T R

. A fracturing

of A is a pair (T L,T R) where T L is a left fracturing of A and T R is a rightfracturing of A.

Let C ⊆modA be a subcategory. We denote by CPL (respectively CI R) acomplete class of isomorphism classes of indecomposable A-modules in C\PL

(respectively C\I R). We have the following definition.

Definition 3.2.1. Let n≥ 2. Assume that A is a representation-directed algebrawith a fracturing (T L,T R) and let C be a subcategory of modΛ. Then C iscalled a (T L,T R,n)-fractured subcategory if the following conditions hold.

(i) PL ⊆ C .(ii) τn and τ−n induce mutually inverse bijections

CPL CI R .τn

τ−n

(iii) Ωi(M) is indecomposable for all M ∈ CPL and 0 < i < n.(iv) Ω−i(N) is indecomposable for all N ∈ CI R and 0 < i < n.

The assumption n≥ 2 is included for a technical reason, but note that (1,d)-representation-finite algebras are well known to exist for any d so this is nota restriction (for instance, the acyclic Nakayama algebra Λ1,d+1 is (1,d)-representation-finite). By comparing with Theorem 3.1.1 it follows that a(T L,T R,n)-fractured subcategory is an n-cluster tilting subcategory if and onlyif T L ∼= A and T R ∼= D(A).

Now let B = kQB/RB be a representation-directed algebra where QB is thequiver

b1 b2 b3 · · · bh−1 bhβ1 β2 β3 βh−2 βh−1 Q′B ,

31

and no path of the form βi · · ·βi+k is in RB. We define the gluing B P .I A of Aand B over P = PA(1) and I = IB(h) to be the algebra Λ = KQΛ/RΛ whereQΛ is the quiver

1 2 3 · · · h−1 h,λ1 λ2 λ3 λh−2 λh−1Q′A Q′B , (3.3)

and RΛ is generated by all elements in RA and RB and all paths starting fromQ′A and ending in Q′B, under the identifications αi = βi = λi. In Corollary 2.38of Paper II we show that the Auslander–Reiten quiver Γ(Λ) of Λ is then thequiver Γ(Λ) = Γ(B) ∏

4Γ(A), where the righthand side denotes the amalga-mated sum under the identification P4 =4I . In this sense we may say thatthe representation theory of Λ can be described using the representation theoryof A and B. In particular, we have the following reformulation of Proposition2.30 of Paper II.

Proposition 3.2.2. Let M = (Mi,µα) ∈modΛ be an indecomposable Λ-mod-ule viewed as a representation. Let MA = (MA

i ,µAα) be the A-module obtained

by setting MAi = Mi for all i∈ (QA)0 and µA

α = µα for all α ∈ (QA)1. Similarlydefine the B-module MB. Then either MA is indecomposable and MB = 0, orMB is indecomposable and MA = 0, or MA and MB are both indecomposableand MA

i = MBi = 0 for all i 6∈ 1, . . . ,h.

Let M ∈ modA. We may view M as a representation of QA bound by RA.We denote by M∗ the representation of QΛ bound by RΛ obtained by extendingM to QΛ by putting 0 in vertices and arrows of QΛ \QA. In particular M∗ is aΛ-module and MA

∗ = M. Similarly we define M∗ for a B-module M.From the above we can compute (maximal) left abutments of Λ. In par-

ticular, if PΛ = PΛ(r) is a (maximal) left abutments of Λ for some r ∈ (QΛ)0,then if r ∈ (QB)0 we have that PB(r) is a (maximal) left abutment of B and ifr 6∈ (QB)0 we have that PA(r) is a (maximal) left abutment of A. Set

r := r ∈ (QΛ)0 | PΛ(r) is a maximal left abutment.

By the above it follows that if T LA and T L

B are left fracturings of A and B re-spectively, then for each r ∈ r we can define a left fracturing of Λ by setting

T (PΛ(r))Λ

:=

T (PB(r))∗ , if r ∈ (QB)0,

T (PA(r))∗ , otherwise,

and T RΛ

:=⊕

r∈r T (PΛ(r))Λ

. Dually, right fracturings T RA and T R

B of A and B giverise to a right fracturing T R

Λof Λ. We call (T L

Λ,T R

Λ) the gluing of the fracturings

(T LA ,T

RA ) and (T L

B ,TR

B ).

32

Above we have explained how to glue algebras over abutments and howto glue fracturings of algebras. The next step is to glue actual n-fracturedsubcategories. Let CA be a (T L

A ,TR

A ,n)-fractured subcategory of A and CB bea (T L

B ,TR

B ,n)-fractured subcategory of B. A candidate for the gluing of CAand CB is CΛ := addCA∗,CB∗. However, notice that by the definition ofn-fractured subcategories, we have that T (P) ∈ CA and T (I) ∈ CB. MoreoverT (P) ∈ P4, T (I) ∈ 4I and P4 = 4I when computing the Auslander–Reitenquiver of Λ. In a sense, this is the only place where CA∗ and CB∗ potentiallyoverlap. It follows that it is reasonable to assume that the fracture T (P) and thefracture T (I) agree, in other words we require that

T (P)∗ ∼= T (I)

∗ .

If P and I are maximal, then this assumption is actually sufficient for CΛ tobe a (T L

Λ,T R

Λ,n)-fractured subcategory. If P is not maximal, then T (P) is not

defined. Let W be the unique maximal left abutment of A with P≤W . Set T (P)

to be a module isomorphic to the direct sum of all indecomposable summandsof T (W ) that lie in P4. We additionally assume that all of the indecomposablesummands of T (W ) that are not in P4 are actually projective A-modules. IfI is not a maximal right abutment. then we make the dual assumptions anddefinitions for I as well. If T (P)

∗ ∼= T (I)∗ , then we prove the following.

Theorem 3.2.3. Let A,B,Λ,(T LA ,T

RA ),(T L

B ,TR

B ),(T LΛ,T R

Λ),n,CA,CB,CΛ be as

above. Then CΛ is an n-fractured subcategory of modΛ.

In particular, we have the following corollary.

Corollary 3.2.4. If CA and CB are n-cluster tilting subcategories and P and Iare simple modules, then CΛ is an n-cluster tilting subcategory and

maxgl.dim(A),gl.dim(B) ≤ gl.dim(Λ)≤ gl.dim(A)+gl.dim(B).

We illustrate with an example.

Example 3.2.5. Let A be as in Example 3.1.2. Then P = P(6) is a simple leftabutment of A and I = I(1) is a simple right abutment of A. Hence the gluingA P .I A is defined and is given by the bound quiver algebra Λ = kQ′/R2

Q′ whereQ′ is the quiver

12 22

32

42

52

62 = 11 21

31

41

51

61 .

33

Then the Auslander–Reiten quiver of the algebra Λ is

51

3151

31

61

4161

41

213141

2141

2131

21

1121

11=62

52

3252

32

4262

42

223242

2242

2232

22

1222

12

where the additive closure of the encircled modules is a 2-cluster tilting sub-category.

In some special cases, we can compute the exact global dimension of theglued algebra. In particular, we have the following theorem.

Theorem 3.2.6. Assume that CA⊆modA and CB⊆modB are n-cluster tiltingsubcategories. Assume also that P is a simple projective A-module with injec-tive dimension equal to gl.dim(A) and that B = Λkn+1,2 and I = IB(kn+ 1).Then Λ = B P .I A has gl.dim(Λ) = gl.dim(A)+ kn and CΛ ⊆ modΛ is an n-cluster tilting subcategory.

We now fix n and let n≤ d′ ≤ 2n−1. Assume that there exists a represen-tation-directed algebra A which is (n,d′)-representation-finite and there existsa simple projective A-module with injective dimension equal to d′. Let d ≥ nand write d = kn+d′. It follows by Theorem 3.2.6 that there exists an (n,d)-representation-finite algebra. So our aim is to construct an algebra A withthese properties. When n is odd, we succeed.

Proposition 3.2.7. Let n be odd and n ≤ d′ ≤ 2n− 1. Then there exists an(n,d′)-representation-finite Nakayama algebra A where the unique simple pro-jective A-module has injective dimension equal to d′.

The proof of Proposition 3.2.7 is given by explicitly constructing a class ofacyclic Nakayama algebra with the required properties. For n even we do notsucceed completely, but we have the following result.

Proposition 3.2.8. Let n be even and 2n ≤ d′ ≤ 3n− 1. Then there existsan (n,d′)-representation-finite Nakayama algebra A where the unique simpleprojective A-module has injective dimension equal to d′.

34

Actually we prove a slightly stronger result: we find an (n,d)-representa-tion-finite Nakayama algebra for the even integers d in [n,2n] as well. Col-lecting everything together, we have the following theorem.

Theorem 3.2.9. Let n be a positive integer and d ≥ n.(a) If n is odd, then there exists an (n,d)-representation-finite algebra.(b) If n is even, and d is even or d ≥ 2n, then there exists an (n,d)-represen-

tation-finite algebra.

Theorem 3.2.9 answers Question 1 almost completely. In particular, wehave that Question 1 is answered affirmatively for n odd and if n is even thenthere are at most n

2 integers d such that an (n,d)-representation-finite algebradoes not exist.

3.3 Paper IIIIn Papers I and II we develop methods of constructing n-cluster tilting subcat-egories for representation-directed algebras. In Paper III we show how we canexpand these methods to construct n-cluster tilting subcategories for algebraswhich are not necessarily representation-directed.

Although the ideas of Paper III are not particularly complicated, the appli-cation of these ideas turns out to need a lot of technical work. First we developthe theory in the more abstract case of n-cluster tilting subcategories of abeliancategories. Our results are then applications of this theory in the case of gluingof n-fractured subcategories of finite-dimensional algebras.

Without going into the more intricate details, the main motivation is thefollowing. Let A and B be two representation-directed algebras and P and I beabutments such that Λ1 = B P .I A is defined. As we have seen in Paper II, wemay identify modA and modB with certain subcategories of modΛ. By furthergluing Λ1 we may construct a sequence of algebras Λi such that their modulecategories satisfy modΛi ⊆ modΛi+1. Moreover, if the module categories ofall of the glued algebras admit n-fractured subcategories which are compatibleas described in the summary of Paper II, then every Λi comes equipped withan n-fractured subcategory. As a first step, can we make sense of a potentiallimit of this sequence of algebras and subcategories?

To answer the above question, we reformulated the above situation in thefollowing way. We start with a directed tree G with vertices VG and arrowsEG. We denote by IG the set of finite and connected subgraphs of G, partiallyordered by inclusion. To each vertex v ∈VG we associate a representation-di-rected algebra Λv = kQv/Rv and to each arrow e : u→ v in EG we associate apair of abutments (Pe, Ie) such that the algebra Λ〈u,v〉 := Λu

Pe .Ie Λv is defined.This data should satisfy certain axioms. First we assume that if e1 : u1→ v ande2 : u2→ v are two arrows, then neither Pe1 ≤Pe2 nor Pe2 ≤Pe1 holds. Secondly

35

we assume the dual condition for arrows with the same target also holds. Thereis a third condition (omitted here) that is included to avoid certain degeneratecases. If these three conditions are satisfied then the data (Λv,Pe, Ie)v∈VG,e∈EGdefines a gluing system on G, see Definition 4.8 of Paper III. By the aboveaxioms it follows that for every H ∈ IG, performing all the gluing operationsdefined on H in any order gives the same algebra. We denote this algebra byΛH = kQH/RH . It follows that Qv ⊆ QH and that Rv ⊆RH for every v ∈ H,where we identify vertices and arrows in Qv with vertices and arrows in QH asin (3.3).

Let L = (Λv,Pe, Ie)v∈VG,e∈EG be a gluing system on a directed tree G. Ifeach modΛv admits a (T L

v ,TR

v ,n)-fractured subcategory Mv such that all thegluings induced by the gluing system are compatible as in the setting of Theo-rem 3.2.3, then we call (Mv)v∈VG an n-fractured system of L . An n-fracturedsystem is called complete if two more conditions hold. Firstly for every vertexv ∈ VG and every maximal left abutment W of Λv such that the fracture T (W )

vis not projective, there exists an arrow e : u→ v such that P = Pe ≤W andall non-projective summands of T (W )

v lie in P4. Secondly the dual of the firstcondition holds.

Assume that (Mv)v∈VG is a complete n-fractured system of a gluing systemL = (Λv,Pe, Ie)v∈VG,e∈EG . Then for every H ∈ IG we may consider the algebraΛH , which now comes equipped with a (T L

H ,TR

H ,n)-fractured subcategory MH .In particular, for all H,K ∈ IG with H ⊆ K we have QH ⊆ QK and RH ⊆RK .We may then consider the quiver

Q =⋃

H∈IG

QH .

In general Q is not a finite quiver and so the path algebra kQ is not well-defined. However, the path category kQ is defined. Moreover, since RH ⊆RKfor H ⊆ K, it is not difficult to see that R :=

⋃H∈IG

RH is an ideal of the pathcategory kQ. Hence the k-linear category kQ/R is well-defined. It turnsout that the category kQ/R is locally bounded. A corollary of this is thatrepresentations of kQ/R may always be computed in a large enough but finitesubquiver of Q. In this sense, this infinite gluing behaves locally as a finitegluing.

The first main result of this observation is the following.

Theorem 3.3.1. Let n ≥ 2, let L = (Λv,Pe, Ie)v∈VG,e∈EG be a gluing systemon a directed tree G and let (Mv)v∈VG be a complete n-fractured system of L .Then mod(kQ/R) admits an n-cluster tilting subcategory M .

At this point we have constructed an n-cluster tilting subcategory, albeitnot for the module category of a finite-dimensional algebra. Under certainconditions we can construct a gluing system in such a way that the category

36

kQ/R and the n-cluster tilting category M ⊆ mod(kQ/R) admit a certainsymmetry. This symmetry can be described in the language of group actionsand orbit categories; for definitions we refer to Section 5.2 of Paper III. In thisway we construct an orbit category (kQ/R)/Z such that mod((kQ/R)/Z)can be identified with the module category mod Λ a finite-dimensional algebraΛ = kQ/R. Using [14, Corollary 2.14], the symmetry of M induces an n-cluster tilting subcategory M ⊆mod(Λ). We illustrate with an example.

Example 3.3.2. Let A = kQ/R2Q be as in Example 3.1.2. Let P = PA(6) and

I = IA(1). Let G be the infinite graph

−1 0 1

We set Av = A for every vertex v ∈VG, and Pe = P and Ie = I for every arrowe ∈ EG. This data defines a gluing system and moreover by Example 3.1.2the module category of each algebra Av admits a 2-cluster tilting subcategorywhich is compatible with the gluing as in Theorem 3.2.3. This gives riseto a 2-fractured system, which is trivially complete since every 2-fracturedsubcategory is an actual 2-cluster tilting subcategory.

In Example 3.2.5 we performed the gluing of A with A along the abutmentsP(6) and I(1). Repeating this process infinitely many times, we obtain theinfinite quiver Q given by

63 = 12 22

32

42

52

62 = 11 21

31

41

51

61 = 10 20

30

40

50

60 = 1−1

It also easily follows that R = R2Q and so the category mod(kQ/R2

Q) admits

a 2-cluster tilting subcategory M . Then the quiver Q admits a Z-action byletting k ∈ Z map a vertex im to the vertex im+k. Taking the quotient by thisZ-action amounts to identifying the vertices im for all m and the vertices 6mand 1m for all m. Then we get the quiver Q given by

1 2

3

4

5

and it follows by Corollary 5.11 of Paper III that the module category of thealgebra A = kQ/R2

Q admits a 2-cluster tilting subcategory. Indeed, the Aus-

37

lander–Reiten quiver of A is

5

35

3

1

41

4

234

24

23

2

12

1

where the encircled modules form a 2-cluster tilting subcategory. Notice thatthe simple module 1 appears twice and that A is not representation-directed.

As an application of the theory developed in this paper we give many ex-amples of n-cluster tilting subcategories. In Proposition 6.2 of Paper III weclassify starlike algebras with radical square zero relations whose module cat-egories admit an n-cluster tilting subcategory. We also prove the followingresult.

Theorem 3.3.3. For any s, t ∈ Z≥0 there exists a bound quiver algebra Λ =kQ/R such that Q has s sources and t sinks and such that modΛ admits ann-cluster tilting subcategory.

Proposition 3.1.3 shows that if a bound quiver algebra Λ = kQ/R admitsan n-cluster tilting, then there are no sinks or sources on Q such that removingthem would make Q disconnected. On the other hand, Theorem 3.3.3 showsthat there is no bound on the number of sinks or sources of Q that do notexhibit this behaviour.

Moreover, we can use Theorem 3.3.3 to prove that to any directed tree wecan assign a gluing system and an n-fractured system.

Theorem 3.3.4. Let G be a directed tree. Then there exists a gluing systemL = (Λv,Pe, Ie)v∈VG,e∈EG on G and a complete n-fractured system (Mv)v∈VGof L .

38

4. Possible future research

The theory developed in Papers I-III gives a way of constructing many exam-ples of algebras whose module categories admit n-cluster tilting subcategories.However, there are two considerable limitations at this point. First, all of thealgebras constructed are representation-finite. Secondly, it is unclear how widethe class of algebras constructed by these methods is. In this section we ad-dress these two issues by offering some possible future research directions.

4.1 n-cluster tilting subcategories for algebras which arenot representation-finite

The main idea for research in this direction is to find a small family of exam-ples of algebras whose module categories admit n-cluster tilting subcategoriesand are representation-finite and then to try and extrapolate from those to caseswhich may not be representation-finite. A concrete example is starlike alge-bras. Starlike algebras with radical square zero relations whose module cate-gories admit n-cluster tilting subcategories are classified in Proposition 6.2 ofPaper III. One next step would be to classify the ones with radical cube zerorelations, or at least the ones with radical cube zero relations that are repre-sentation-finite. The hope is that it might then be possible to have a conjectureabout a subclass of starlike algebras modulo powers of the radical whose mod-ule categories admit an n-cluster tilting subcategory. For a high enough powerof the radical, these algebras are not representation-finite anymore. Prelimi-nary experimentation shows that this might indeed be quite possible. It shouldbe noted that different methods should be used to show the existence of n-cluster tilting subcategories in this way; a possibility might be to use inductivemethods similar to [26].

4.2 Classification of algebras with radical square zerorelations whose module categories admit n-clustertilting subcategories

There are a few reasons to believe that such a classification could be doable.First, classification of radical square zero algebras satisfying a certain propertyhas been done before. For example Gabriel classified radical square zero rep-resentation-finite algebras in [19] using the separated quiver. Adachi provided

39

an analogous classification of τ-rigid finite algebras in [1], again by usingthe separated quiver. Studying the separated quiver of radical square zeroalgebras whose module categories admit an n-cluster tilting subcategory mightprovide some indication about their classification. Moreover, in this case, thenecessary conditions (iii) and (iv) of Theorem 3.1.1 for the existence of ann-cluster tilting subcategory become relevant. From these one can infer thatif the module category of a bound quiver algebra kQ/R2

Q admits an n-clustertilting subcategory, and if Q is a tree, then any vertex of Q has at most twooutgoing and at most two ingoing arrows.

There is one more reason to expect such a classification to be possible.One crucial observation in this situation is what happens when a simple mod-ule which is neither projective nor injective is included in the n-cluster tiltingsubcategory. In such a case we may perform an “ungluing” operation and sep-arate the quiver at the vertex corresponding to that simple module. This breaksdown the starting algebra to two algebras whose module categories both admitan n-cluster tilting subcategory. One strategy to attack this problem is to finda list of building blocks from which one can construct all radical square zeroalgebras whose module categories admit an n-cluster tilting subcategory viagluing.

40

5. Svensk sammanfattningv (Summary inSwedish)

5.1 BakgrundEn algebra är en mängd vars element kan adderas och multipliceras, och därvarje element kan multipliceras med en skalär (till exempel ett komplext tal). Idenna avhandling beskrivs algebror med koger och relationer. Ett koger bestårav en ändlig mängd punkter och en ändlig mängd pilar mellan punkterna. Enväg av längd l är en följd av l pilar så att varje pil slutar där nästa pil börjar.Vi tilldelar dessutom vare punkt en trivial väg av längd 0. Vägalgebrans ele-ment är formella linjärkombinationer av vägar. Multiplikation av vägar ges avkonkatenering när det är möjligt och annars noll. En relation är en linjärkom-bination av vägar av längd minst två med samma start- och slutpunkt, vilketska tolkas som att denna linjärkombination identifieras med noll.

För en algebra beskriven av ett koger med relationer fås en representationeller modul över algebran genom att tilldela varje punkt ett vektorrum ochvarje pil en linjär avbildning på ett sånt sätt att relationerna uppfylls då varjepil ersätts med motsvarande linjära avbildning. Morfismer mellan moduler äravbildningar som är kompatibla med modulstrukturerna. Moduler och modul-morfismer bildar algebrans modulkategori. Representationsteori kan förståssom studiet av denna kategori.

Ett av de viktigaste resultaten i representationsteori är Auslander–Reiten-teori, som utvecklades under sjuttiotalet. För vissa klasser av algebror gerAuslander–Reitenteori en fullständig bild av modulkategorin. En högre ver-sion av Auslander–Reitenteori utvecklades under nollnolltalet för att studeraalgebror med mer komplicerade modulkategorier. Iden är att fokusera på enlämplig delkategori av modulkategorin. Sådana delkategorier kallas n-klustertiltingdelkategorier.

n-kluster tiltingdelkategorier är i allmänhet svåra att hitta eller konstruera.Denna avhandling handlar om att finna n-kluster tiltingdelkategorier för olikaklasser av algebror med intressanta egenskaper.

5.2 Sammanfattning av artiklarI artikel I ger vi en karaktärisering av n-kluster tiltingdelkategorier för repre-sentationsriktade algebror baserad på klassisk Auslander–Reitenteori och ho-mologisk algebra. Som en tillämpning klassificerar vi acykliska Nakayama

41

algebror med homogena relationer vars modulkategorier har n-kluster tilt-ingdelkategorier. Vi ger även en formel för den globala dimensionen av så-dana algebror. Som konsekvens klassificerar vi Nakayama algebror av globaldimension d < ∞ vars modulkategorier har d-kluster tiltingdelkategorier.

I artikel II undersöker vi existensen av algebror av global dimension d varsmodulkategorier har n-kluster tiltingdelkategorier. Vi konstruerar många ex-empel genom att använda representationsriktade algebror. Givet två repre-sentationsriktade algebror A och B tillsammans med en projektiv A-modul Poch en injektiv B-module I som uppfyller vissa villkor, så konstruerar vi enny representationsriktad algebra Λ := B P .I A på ett sådant sätt att represen-tationsteorin för Λ kan beskrivas fullständigt i termer av representationsteori-erna för A och B. Vi introducerar n-brutna delkategorier, vilka generalisern-kluster tiltingdelkategorier för representationsriktade algebror (som karak-täriserades i artikel I). Därefter visar vi hur en n-kluster tiltingdelkategori förΛ kan konstrueras genom att använda n-brutna delkategorier för A och B. Somen tillämpning av denna konstruktion visar vi för n udda och d ≥ n att det ex-isterar en algebra med global dimension d vars modulkategori har en n-klustertiltingdelkategori. För n jämnt visar vi samma resultat för alla d som uppfyllerd ≥ 2n.

I artikel III introducerar vi ett nytt sätt att konstruera n-kluster tiltingdelkate-gorier av abelska kategorier. Vår metod tar som input ett riktat system av abel-ska kategorier Ai tillsammans med vissa delkategorier och ger under rimligaantaganden en n-kluster tiltingdelkategori av ett tillåtligt mål till det riktadesystemet. Vi tillämpar denna metod på ett riktat system av modulkategoriermodΛi av representationsriktade algebror Λi och får på så sätt en n-kluster tilt-ingdelkategori M av en modulkategori mod C av en lokalt begränsad Krull–Schmidtkategori C . I vissa fall kan vi även konstruera en tillåtlig Z-aktionpå C . Genom att övergå till motsvarande kategori av banor och tillämpa ettresultat av Darpö–Iyama får vi en n-kluster tiltingdelkategori av mod(C /Z).Vi visar att mod(C /Z) är ekvivalent med modulkategorin av en ändligtdimen-sionell algebra som vi även beskriver. Därefter klassificerar vi stjärnlika alge-bror med radikalkvadrat noll vars modulkategorier har n-kluster tiltingdelkat-egorier. Som tillämpning av våra resultat konstruerar vi flera familjer av rep-resentationsändliga algebror vars modulkategorier har n-kluster tiltingdelkat-egorier.

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6. Acknowledgements

First of all I would like to express my gratitude to my supervisor Martin Her-schend. Martin, if I am the actor here, you are the director. Only I have neverseen such a free-rein director before. Thank you for always considering myopinion and trying to understand what I was thinking; most of the time youdid it better than me. Thank you for all the help, support and encouragement.I will miss our meetings.

I want to thank Walter Mazorchuk from whom I’ve learned a lot. Thankyou for all the seminars you organized and all the mathematical insights youhave shared during them and for teaching all these fun courses I have takenfrom you since I was a Master student. They gave me the final push towardsAlgebra.

I want to express my thanks to the administrative stuff at the department,especially Inga-Lena Assarsson, Elisabeth Bill, Lina Flygersted, Fredrik Lan-nergård and Mattias Landelius. You have always helped me in a fast andefficient way every time I needed something.

I am indebted to my fellow PhD students for the great environment thatthey have created. Love, thank you for being helpful and friendly. Filipe,thank you for organising game nights at my house and for your great sense ofhumour. Jakob, you have taught me a lot and have always been the first personfor me to turn to and ask for advice and instructions (you see, I did it again!).Andrea, my older academic brother, thank you for all our conversations; it hasalways been fun to be around you, and I’ve always learned something new andinteresting.

I wish to thank Chrysostomos Psaroudakis for being my opponent twice,for inviting me to Thessaloniki and for giving me support. I really appreciateyour help and advice.

I also want to thank Peter Jørgensen for giving me the chance to work withhim and Martin on an interesting problem near the start of my PhD. Our worktogether gave me valuable insight into how research mathematics is done.

I would also like to thank Apostolos Thoma from the University of Ioanninawho was for me a first example not only of how teaching should be done butalso of how to respect and value the students’ opinion.

If I am here now, it is because of everything that has come before. Thereare moments and people that have shaped me and, each one in their own way,supported me.

I am thankful to Giorgos, my elementary school teacher for always doinghis best to make his students, first and foremost, good people. Since today

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my closest friends are from these years, whatever he did worked. Thank youArdit, Christos, Konstantinos, Memos and Spiros for all the fun and supportfor almost 25 years.

I want to express my gratitude to the Infinity club and especially Dimitris,Manos, Nikos and Vaggelis for teaching me two things: how to work withothers and why proofs are nice to have. Thank you so much for making Sundayafternoons fun and productive.

The theatrical student group of Ioannina, ΘE.Σ.Π.I. made my stay at Ioan-nina unforgettable. I thank everyone there from the bottom of my heart forthe beautiful years. They taught me practical things that I use to this day evenwhen doing mathematics. Special thanks to Konstantina, Sotiris and Thanosfor all the fun on- and off-stage.

My friends in Sweden, Babis, Dimitris, Fotini, Kostas, Laia, Panos, andVaggelis have made my time both here and in other places of the world funand enjoyable. Thank you all for your support, company and for always beingthere when needed.

I want to thank my parents and my sister for the support and encouragementI have received from them all these years. I may not say it often but it meansa lot to me.

Finally, thank you Elisavet for making everything easy. Thank you forsharing your energy, your hopes, your life with me.

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References

[1] Takahide Adachi. Characterizing τ-rigid-finite algebras with radical squarezero. Proceedings of the American Mathematical Society, 144, January 2014.

[2] Ibrahim Assem, Daniel Simson, and Andrzej Skowronski. Elements of theRepresentation Theory of Associative Algebras: Volume 1: Techniques ofRepresentation Theory. Elements of the Representation Theory of AssociativeAlgebras. Cambridge University Press, 2006.

[3] Maurice Auslander. Representation Theory of Artin Algebras I.Communications in Algebra, 1(3):177–268, 1974.

[4] Maurice Auslander. Representation Theory of Artin Algebras II.Communications in Algebra, 1(4):269–310, 1974.

[5] Maurice Auslander. Functors and morphisms determined by objects. InRepresentation theory of algebras (Proc. Conf., Temple Univ., Philadelphia,Pa., 1976), pages 1–244. Lecture Notes in Pure Appl. Math., Vol. 37, 1978.

[6] Maurice Auslander and Idun Reiten. Stable equivalence of dualizingR-varieties. Advances in Mathematics, 12(3):306–366, 1974.

[7] Maurice Auslander and Idun Reiten. Representation theory of Artin algebras.III. Almost split sequences. Communications in Algebra, 3:239–294, 1975.

[8] Maurice Auslander and Idun Reiten. Representation theory of Artin algebras.IV. Invariants given by almost split sequences. Communications in Algebra,5(5):443–518, 1977.

[9] Maurice Auslander and Idun Reiten. Representation theory of Artin algebras.V. Methods for computing almost split sequences and irreducible morphisms.Communications in Algebra, 5(5):519–554, 1977.

[10] Maurice Auslander and Idun Reiten. Representation theory of Artin algebras.VI. A functorial approach to almost split sequences. Communications inAlgebra, 6(3):257–300, 1978.

[11] Maurice Auslander, Idun Reiten, and Svere Olaf Smalø. Representation theoryof Artin algebras, volume 36 of Cambridge Studies in Advanced Mathematics.Cambridge University Press, Cambridge, 1995.

[12] Viktor Bekkert, Flávio U. Coelho, and Heily Wagner. Tree Oriented Pullback.Communications in Algebra, 43(10):4247–4257, 2015.

[13] Aaron Chan, Osamu Iyama, and René Marczinzik. Auslander–Gorensteinalgebras from Serre-formal algebras via replication. Advances in Mathematics,345:222–262, 2019.

[14] Erik Darpö and Osamu Iyama. d-representation-finite self-injective algebras.Advances in Mathematics, 362:106932, 2020.

[15] Tobias Dyckerhoff, Gustavo Jasso, and Yanki Lekili. The symplectic geometryof higher Auslander algebras: Symmetric products of disks. arXiv e-prints,page arXiv:1911.11719, November 2019.

45

[16] Karin Erdmann and Thorsten Holm. Maximal n-orthogonal modules forselfinjective algebras. Proceedings of the American Mathematical Society,136(9):3069–3078, 2008.

[17] Sergey Fomin and Andrei Zelevinsky. Cluster Algebras I: Foundations. Journalof the American Mathematical Society, 15(2):497–529, 2002.

[18] Sergey Fomin and Andrei Zelevinsky. Cluster algebras II: Finite typeclassification. Inventiones Mathematicae, 154(1):63–121, October 2003.

[19] Peter Gabriel. Unzerlegbare Darstellungen I. Manuscripta Mathematica,6(1):71–103, March 1972.

[20] Martin Herschend and Osamu Iyama. n-representation-finite algebras andtwisted fractionally Calabi–Yau algebras. Bulletin of the London MathematicalSociety, 43(3):449–466, 2011.

[21] Martin Herschend and Osamu Iyama. Selfinjective quivers with potential and2-representation-finite algebras. Compositio Mathematica, 147(6):1885–1920,2011.

[22] Kiyoshi Igusa, María Inés Platzeck, Gordana Todorov, and Dan Zacharia.Auslander algebras Of finite representation type. Communications in Algebra,15(1-2):377–424, 1987.

[23] Osamu Iyama. Auslander correspondence. Advances in Mathematics,210(1):51–82, 2007.

[24] Osamu Iyama. Higher-dimensional Auslander–Reiten theory on maximalorthogonal subcategories. Advances in Mathematics, 210(1):22–50, 2007.

[25] Osamu Iyama. Auslander–Reiten theory revisited. In Trends in representationtheory of algebras and related topics, EMS Series of Congress Reports, pages349–397. European Mathematical Society, Zürich, 2008.

[26] Osamu Iyama. Cluster tilting for higher Auslander algebras. Advances inMathematics, 226(1):1–61, 2011.

[27] Osamu Iyama and Gustavo Jasso. Higher Auslander Correspondence forDualizing R-Varieties. Algebras and Representation Theory, 20(2):335–354,Apr 2017.

[28] Osamu Iyama and Steffen Oppermann. n-representation-finite algebras andn-APR tilting. Transactions of the American Mathematical Society,363(12):6575–6614, 2011.

[29] Osamu Iyama and Steffen Oppermann. Stable categories of higher preprojectivealgebras. Advances in Mathematics, 244:23–68, 2013.

[30] Osamu Iyama and Michael Wemyss. A New Triangulated Category for RationalSurface Singularities. arXiv e-prints, page arXiv:0905.3940, May 2009.

[31] Osamu Iyama and Michael Wemyss. On the Noncommutative Bondal–OrlovConjecture. arXiv e-prints, page arXiv:1101.3642, January 2011.

[32] Osamu Iyama and Michael Wemyss. Maximal modifications andAuslander–Reiten duality for non-isolated singularities. Inventionesmathematicae, 197(3):521–586, 2014.

[33] Gustavo Jasso, Julian Külshammer, Chrysostomos Psaroudakis, and SondreKvamme. Higher Nakayama algebras I: Construction. Advances inMathematics, 351:1139–1200, 2019.

[34] Gustavo Jasso and Sondre Kvamme. An introduction to higherAuslander–Reiten theory. Bulletin of the London Mathematical Society,

46

51(1):1–24, 2019.[35] Jessica Lévesque. Nakayama oriented pullbacks and stably hereditary algebras.

Journal of Pure and Applied Algebra, 212(5):1149–1161, 2008.[36] Claus Michael Ringel. Representation theory of Dynkin quivers. Three

contributions. Frontiers of Mathematics in China, 11(4):765–814, Aug 2016.

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