Fakult¨at f¨ur
Elektrotechnik, Informatik und Mathematik
Institut f¨ur Mathematik
Dissertation
Representation Theory
of
EI-categories
Karsten Dietrich
March 2010
Betreuer: Prof. Dr. Henning Krause
Abstract
In this thesis we investigate the category of finite-dimensional modules over an EI-category
algebra. More precisely, we analyze the representation type for this class of algebras in
the first part. It will be shown that a representation-finite EI-category is an amalgam of a
representation-finite poset and a collection of representation-finite groups. We will then see
that the representation type depends on the characteristic of the ground field. Furthermore,
we give a necessary criterion for an EI-category with two objects to be representation-finite.
Under additional assumptions on the automorphism groups of the objects we give a full
classification of the representation-finite EI-categories with two objects. In the second part
we present a new proof for the existence of an upper bound for the finitistic dimension of
an EI-category algebra. Inspired by this proof we define a new class of algebras, which we
call algebras with a directed stratification. We prove a result on the finitistic dimension of
these algebras. This reduces the finitistic dimension conjecture to a class of algebras which
we can describe combinatorially in terms of their Gabriel-quiver.
Zusammenfassung
In dieser Arbeit untersuchen wir die Kategorie der endlichdimensionalen Moduln ¨uber einer
EI-Kategorienalgebra. Genauer analysieren wir im ersten Teil den Darstellungstyp dieser
Klasse von Algebren. Es wird gezeigt, dass eine darstellungsendliche EI-Kategorie sich
aus einer darstellungsendlichen halbgeordneten Menge und darstellungsendlichen Gruppen
zusammensetzt und dass der Darstellungstyp von der Charakteristik des zugrundeliegen-
den K¨orpers abh¨angt. Dar¨uber hinaus beweisen wir ein notwendiges Kriterium f¨ur die
Endlichkeit des Darstellungstyps einer EI-Kategorie mit zwei Objekten. Unter zus¨atzlichen
Voraussetzungen an die Automorphismengruppen, geben wir eine vollst¨andige Klassifika-
tion der darstellungsendlichen EI-Kategorien mit zwei Objekten an. Im zweiten Teil der
Arbeit pr¨asentieren wir einen neuen Beweis f¨ur die Existenz einer oberen Schranke f¨ur die
finitistische Dimension einer EI-Kategorienalgebra. Dieser Beweis motiviert die Definition
einer neuen Klasse von Algebren, die wir Algebren mit einer gerichteten Stratifizierung nen-
nen. F¨ur diese Algebren beweisen wir ein Resultat ¨uber die finitistische Dimension. Dieses
Resultat reduziert die finitistische Dimensionsvermutung auf eine Klasse von Algebren, die
wir kombinatorisch mithilfe ihres Gabriel-K¨ochers beschreiben k¨onnen.
2
Contents
1 Introduction 4
2 Preliminaries 8
2.1 Definition of EI-categories and examples . . . . . . . . . . . . . . . . . . . . 8
2.2 Simple and projective modules . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Induction and restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 EI-categories of finite representation type 15
3.1 Finite representation type for finite groups, quivers and posets . . . . . . . 15
3.2 The Gabriel-quiver of a finite dimensional algebra . . . . . . . . . . . . . . . 18
3.3 Coveringtheory.................................. 19
3.4 Relative projectivity, vertices and sources . . . . . . . . . . . . . . . . . . . 21
3.5 Theendotrivialcase ............................... 24
3.6 EI-categories with two objects . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6.1 The easiest example . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.6.2 The characteristic plays a role . . . . . . . . . . . . . . . . . . . . . . 30
3.6.3 Free action implies infinite type . . . . . . . . . . . . . . . . . . . . . 31
3.7 EI-category algebras with two simple modules . . . . . . . . . . . . . . . . . 35
3.8 Two objects and cyclic automorphism groups . . . . . . . . . . . . . . . . . 38
4 The finitistic dimension of EI-category algebras 40
5 The finitistic dimension of algebras with a directed stratification 43
5.1 Trivial extensions of abelian categories and finitistic dimension . . . . . . . 43
5.2 Basic notions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 Projective resolutions and the main result . . . . . . . . . . . . . . . . . . . 49
5.4 Relation to known results and examples . . . . . . . . . . . . . . . . . . . . 52
5.4.1 Relation to recollements . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.4.2 A non-trivial example . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6 Outlook 58
References 60
1 Introduction
The study of EI-categories and their representations has its origin in the work of tom
Dieck [31] and L¨uck [26] in the late 1980s. These authors used representations of EI-
categories in algebraic K-theory and it took some years before EI-categories attracted
attention from mathematicians working with representations of finite-dimensional algebras.
Webb [32] investigated the question under which circumstances EI-categories are standardly
stratified (in the sense of [10, 11]) and his student Xu recently worked on the cohomology
theory of EI-categories and related aspects of their representation theory, compare [33–35].
Also fusion systems, transporter categories and other categories constructed from the set
of subgroups of a given group are EI-categories and the object of recent research in the
framework of p-local finite groups, see for instance [8].
An EI-category Cis a category in which every endomorphism is an isomorphism. For a
fixed base ring kthe associated category algebra is denoted by kC. It has as basis the set
of morphisms in Cwith multiplication induced by composition of morphisms. Hence, the
category algebra of a finite EI-category is a simultaneous generalization of several important
constructions in representation theory, such as the group algebra of a finite group, the path
algebra of a finite quiver without oriented cycles or the incidence algebra of a finite poset.
Although representations of finite groups and representations of quivers are, up to a
certain degree, well-understood and lie at the heart of modern representation theory, the
two theories are in some sense orthogonal to each other. It is therefore a natural question
to ask for a general theory which contains representations of finite groups and finite quivers
as special cases. One way to obtain this goal is the analysis of representations of finite
EI-categories.
In this thesis we will focus on two central questions that always arise naturally for a new
class of algebras, namely: What are the algebras of finite representation type and does the
finitistic dimension conjecture hold for this class of algebras? The latter is also motivated by
work of Grodal and Smith [19], where the projective dimensions of certain modules over EI-
category algebras play a role for the description of algebraic models for finite G-spaces that
appear in algebraic topology. Furthermore, the finitistic dimension of standardly stratified
algebras is always finite, while for stratified algebras the finitistic dimension conjecture is
still open. By work of Webb [32] we know that an EI-category algebra is always stratified
but not standardly stratified in general. Here, we follow Cline, Parshall and Scott [10,11]
for the definition of stratified and standardly stratified algebras.
4
Introduction 5
A classification of the representation-finite EI-category algebras would be a new way of
generalizing the results for groups and quivers. Bautista, Gabriel, Roiter, Salmeron and
others developed a theory for representation-finite algebras in general in the 1980s, but they
always assumed that the Gabriel-quiver of the algebra in question is known. In particular,
they assumed that every simple module has dimension one. The computation of the quiver
of an algebra is in general a non-trivial task. Therefore, the results of the mentioned authors
are often not applicable for the treatment of algebras which are not given in terms of quivers
with relations.
Throughout this thesis kwill denote a commutative ring with unit and often an alge-
braically closed field. For a finite category Cit is a well-known fact due to Mitchell [27]
that the category of modules over the category algebra kCis equivalent to the category of
k-linear representations of C, i.e. the functor category Fun(C,Mod k). Thus, we can work
with representations of Cto derive results for the module category of kC, which is often an
advantage for explicit computations.
The characterizations of representation-finite path algebras, incidence algebras and group
algebras of finite quivers, finite groups and finite posets respectively are classical results in
the representation theory of finite-dimensional algebras. Since all these classes of algebras
appear as special cases of EI-category algebras, a representation-finite EI-category algebra
has to satisfy the conditions for representation-finiteness for these three classes of algebras
simultaneously.
For a finite EI-category Cwe introduce a new category b
Cwhich is the ’endotrivialization’
of C, i.e. all endomorphisms are made trivial and morphisms x→ythat lie in the same
(Aut(x)×Aut(y))-orbit are identified. As a first step we show that if kCis a representation-
finite EI-category algebra, then kb
Cis a representation-finite incidence algebra. Furthermore,
it is an easy observation that for a representation-finite EI-category algebra kCevery group
algebra kAut(x) for x∈Ob Cis representation-finite. The question that needs to be
answered for a characterization of all representation-finite EI-category algebras is the fol-
lowing: Which combinations of representation-finite groups and representation-finite posets
give representation-finite EI-categories (always with respect to the ground field k)? It turns
out that this question is not easy to answer in general. Therefore, we will restrict ourselves
to EI-categories with two objects, which is the easiest class of EI-categories that has not
yet been investigated. The analysis of the representation type of EI-categories with two
objects provides us with necessary criteria for finite representation type.
The first observation is that the representation type of an EI-category algebra kCsignif-
icantly depends on the characteristic of the ground field. Despite this dependence on the
characteristic we will show that the category algebra kCof an EI-category Cwith two non-
isomorphic objects xand y, where Aut(x)×Aut(y) acts freely on the morphism set C(x, y),
is of infinite representation type for any algebraically closed field k. This result gives a nec-
essary criterion for EI-category algebras to be of finite representation type. Unfortunately,
Introduction 6
we are not able to give a full classification of these algebras, even under the assumption
that the underlying category Chas only two objects. Under additional assumptions on the
automorphism groups of the objects of the EI-category C, one can compute the Gabriel-
quiver of the associated category algebra explicitly and then use results about quivers with
relations to determine the representation type. In that way we can for instance characterize
all representation-finite EI-category algebras with two simple modules. It turns out that
for EI-categories with two objects the representation type is governed by the group action
of the automorphism groups of the two objects on the set of morphisms between the two
objects.
The (little) finitistic dimension of an algebra Adefined as
fin.dim(A) = sup {proj.dim M|M∈mod A, proj.dim M < ∞ }
has been introduced by Bass [5] in 1960 and is an important invariant of the module
category, which, roughly speaking, measures the complexity of mod A. Bass proposed the
question whether this finitistic dimension is always finite as a ’problem’ and for finite-
dimensional algebras it is nowadays known as the finitistic dimension conjecture (while for
commutative rings it is easily seen to fail). Up to now there is no proof for this conjecture,
but also no counterexample. The finitistic dimension has been calculated for several classes
of algebras and turned out to be finite in these cases.
L¨uck [26] gave an upper bound for the finitistic dimension of an EI-category algebra kC(k
a field), namely he showed fin.dim kC ≤ `(C), where `(C) denotes the length of the category
C, i.e. the maximal length of a chain of non-isomorphisms in C. We will present a new
proof for this upper bound using recent results of Xu [33], that describe the structure of
projective resolutions of modules over EI-category algebras. The intrinsic structure of an
EI-category Cthat guarantees the finiteness of fin.dim kCis the natural poset structure on
the set of isomorphism classes of objects defined by [x]≤[y]⇔ C(x, y)6=∅together with
the finiteness of the finitistic dimension of all automorphism groups of objects in C, which
are group algebras.
Inspired by this observation we define a new class of finite-dimensional algebras which
we call algebras with a directed stratification. According to our definition an algebra A
has a directed stratification of length nif there exist idempotents e1, . . . , enin Awith
1 = e1+· · · +enand ejAei= 0 for all i > j. This class of algebras contains EI-category
algebras as a special case and we can describe the structure of projective resolutions of
modules in the same way as Xu did for EI-category algebras and finally prove that an
algebra Awith a directed stratification given by e1, . . . , enhas finite finitistic dimension if
and only if all algebras eiAeifor i= 1, . . . , n have finite finitistic dimension. As a matter
of fact, this result could also be obtained by induction from a result of Fossum, Griffith,
Reiten [14] and Fuller, Saorin [15], but these authors used trivial extensions of abelian
categories for the proof, which is a rather abstract machinery. With our proof we gain a
Introduction 7
deeper insight into the structure of projective resolutions of modules over algebras with a
directed stratification. Furthermore, we give a combinatorial description of the algebras
that do not admit a non-trivial directed stratification (i.e. of length >1) in terms of their
Gabriel-quiver. Finally, we construct examples which show that this reduction technique
for the finitistic dimension conjecture is not a special case of other well-known results for
the conjecture.
Outline
In Chapter 2 we collect the fundamental definitions of category algebras and EI-categories
together with basic facts about their representation theory. We also recall the description
of all simple and projective modules over EI-category algebras and introduce induction and
restriction functors for representations of small categories. The third chapter contains our
results on the representation type of EI-category algebras that have partly been mentioned
above as well as short surveys on techniques we need for our proofs like covering theory
and the Gabriel-quiver of a finite-dimensional algebra. In Chapter 4 we give a new proof
for L¨uck’s upper bound for the finitistic dimension of EI-category algebras. This motivates
the fifth chapter, where we develop the whole theory of projective resolutions for modules
over algebras with a directed stratification and prove the result on the finitistic dimension
for these algebras. Finally, we conclude this thesis with chapter 6 in which we name open
problems and some ideas how one could attack them in the future.
Acknowledgements
First and foremost I wish to express my thanks to my supervisor Henning Krause. In
particular, I would like to thank Henning for providing me with an interesting project, for his
guidance and for many helpful discussions in the last two years. I also thank the Paderborn
representation theory group, the various guests and many other people from the institute in
Paderborn for all the mathematical and non-mathematical help. I am particularly grateful
to Jan M¨ollers, Sven-Ake Wegner and Reiner Hermann for proofreading parts of this thesis.
Finally, I would like to thank Magdalena and my family. Their unconditional support has
been of utmost importance for me during the time of my PhD.
2 Preliminaries
In this chapter we collect the definitions of category algebras, EI-categories and basic facts
about their representation theory. Throughout this thesis all modules will be left modules.
2.1 Definition of EI-categories and examples
We begin with the classical definition of a category algebra.
Definition 2.1. Let Cbe any category and ka commutative ring with identity. Then we
define the category algebra kCto be the free k-module whose basis is the set of morphisms
in C. The multiplication on two basis elements fand gof kCis defined as follows
f·g=(f◦gif fand gcan be composed in C,
0 otherwise.
We are mostly interested in the category mod kCof finitely generated left modules over
category algebras. Under certain assumptions, this category can be identified with the
category of k-linear representations of Cwhich is defined as follows.
Definition 2.2. Let Cbe a small category and ka commutative ring. A representation of
Cover kis a covariant functor M:C → Mod kfrom Cinto the category of k-modules. To-
gether with natural transformations of functors this gives an abelian category with enough
projective and injective objects. This category will be denoted by RepkC.
The following elementary observation relates the concepts of representations of Cand
modules over kC.
Proposition 2.3 (Mitchell, [27]).Let Cbe a category with finitely many objects and ka
commutative ring. Then the categories RepkCand Mod kCare equivalent.
Later on we will deal with category algebras over a field k. In this case the equivalence
in the Proposition restricts to an equivalence mod kC → repkC= Fun(C,mod k) from the
category of finite-dimensional modules to the category of finite-dimensional representations.
We are particularly interested in a very special class of small categories, namely in the
class of finite EI-categories.
8
Definition of EI-categories and examples 9
Definition 2.4. An EI-category is a category Cin which every endomorphism is an iso-
morphism. If Cis a finite EI-category, the associated k-algebra
kC=
X
f∈Mor C
λffλf∈k
is a finitely generated unital k-algebra, sometimes called the associated EI-algebra. The unit
element is Px∈Ob C1xand obviously the elements {1x|x∈Ob C } form a set of pairwise
mutually orthogonal idempotents in kC. These idempotents are in general not primitive.
Example 2.5. EI-categories arise as important examples in at least in two branches of
mathematics, namely representation theory of finite dimensional algebras and algebraic
topology. We will present examples from both branches, starting with representation theory.
(1) Let Gbe a finite group and let Gbe the category with one object xand End(x) = G.
Then Gis an EI-category and kG=kG.
(2) Let Qbe a finite quiver without oriented cycles and Qits path category. Then Qis
an EI-category with kQ =kQ.
Another important class of EI-algebras is the class of incidence algebras associated to
finite partially ordered sets (short: finite posets).
Definition 2.6. Let (X, ≤) be a finite poset, i.e. a finite set equipped with a binary relation
≤which is reflexive, antisymmetric and transitive. The incidence algebra A(X) (over k) of
Xconsists of all incidence functions
A(X) = {f:X×X→k|f(x, y) = 0 if x 6≤ y}
with pointwise summation and scalar multiplication. Moreover, we define the product of
two such functions fand gas
(f∗g)(x, y) = X
x≤z≤y
f(x, z)g(z, y).
The Kronecker delta function δ(x, y) is the two-sided identity of A(X).
To every finite poset Xwe can also associate a finite category CXin the following way:
•Ob CX=X;
•For x, y ∈Xthere is a morphism x→yin CXif and only if x≤yand we require
that for any two objects x, y all morphisms x→yare equal.
In other words, CXis the bound path category of a quiver with relations and we could
alternatively define a poset to be a finite category Pwith the property that |P(x, y)| ≤ 1
for all x, y ∈Ob P. The incidence algebra defined above is then isomorphic to kP.
Definition of EI-categories and examples 10
Example 2.7. Let X={a, b, c, d }be equipped with the following partial order
a≤(b
c)≤d.
Then the category CXis the path category of the quiver
d
b
γ@@
c
δ
^^=
=
=
=
=
=
=
=
a
α
__>
>
>
>
>
>
>
>β
??
bound by the relation γα =δβ.
The three examples mentioned above are the most important and classical examples of
EI-categories in representation theory of finite-dimensional algebras. Now we present some
examples from algebraic topology such as fusion systems that have recently been studied
by Broto, Levi and Oliver [8] in the context of p-local finite groups, orbit categories that
play a prominent role in the theory of finite G-spaces, or transporter categories. All these
categories are constructed from a set of subgroups of a given group in the following way.
Definition 2.8. Let Sbe a set of subgroups of a finite group G.
(1) The transporter category TShas as objects the elements of Sand the morphisms
are Hom(H, K) = NG(H, K) = g∈GgHg−1⊆K. For the case H=Kthe
set of endomorphisms is the normalizer subgroup of Hin Gand therefore all the
endomorphisms are isomorphisms.
(2) The orbit category OSis the category whose objects are the coset spaces G/H for
H∈ S and the morphisms in OSare the G-equivariant mappings. Those are all
epimorphisms and hence, every endomorphism is an isomorphism.
(3) The Frobenius category FSassociated to S(or the fusion system) plays an important
role for the definition of a p-local finite group. Its objects are the elements of Sand
the morphisms are group homomorphisms H→Kthat are given by conjugation with
an element of G. In this category every morphism is a monomorphism, which implies
that FSis an EI-category.
To an arbitrary EI-category Cwe will associate another category b
Cwith only identity
endomorphisms, which reflects the global structure of C. This category b
Cwill play an
important role in the analysis of the representation type of EI-categories later on.
Definition 2.9. Let fand gbe two morphisms in a finite, skeletal EI-category C. Then
Definition of EI-categories and examples 11
we define a relation ∼on the set of morphisms of Cas follows.
f∼g:⇔f=f00h1f0and g=f00h2f0for some f00, f0∈Mor Cand endomorphisms h1, h2
This is clearly a reflexive and symmetric relation. We will consider the transitive hull of this
relation and denote it again by ∼. This relation is also compatible with the composition of
morphisms in C. Therefore, we get a new category b
C:= C/∼which is by construction an
endotrivial category (in particular EI).
Roughly speaking, b
Cis constructed from Cby making all endomorphisms trivial and
identifying all morphisms x→yin the same (Aut(x)×Aut(y))-orbit. By construction
it has the following important universal property. Suppose that Cis an EI-category and
F:C → D any functor to an endotrivial category D. Then this functor Ffactors via
a unique functor through the quotient functor G:C → b
C, i.e. the following diagram is
commutative.
CG//
F
b
C
∃!
D
Example 2.10. (1) If Gis a finite group and C=Gthe associated EI-category, then b
G
consists of one object xand the only morphism is the identity 1x.
(2) If Qis a finite quiver without oriented cycles and C=Qthe path category, then
b
C=C.
(3) If Cis the EI-category associated to a finite poset (X, ≤), we get b
C=C.
Remark 2.11. (1) An EI-category Cis not uniquely determined by the category b
Cto-
gether with its automorphism groups. To recover the entire structure of Cone needs
to know the composition of morphisms which is the same as the whole structure of C.
Nevertheless, the category b
Cis of great importance for us. As an example consider
the EI-category
C:x
f99
i1
i2
))
i355
i4
AAy,
satisfying the relations f4= 1xand i1f=i2,i2f=i3,i3f=i4,i4f=i1and the
EI-category
C0:a
g99
h//b,
with the relations g4= 1aand gh =h. Then both b
Cand b
C0are the path category of
A2and Cand C0have the same automorphism groups. However, they are not equiv-
Simple and projective modules 12
alent and the associated category algebras have completely different representation-
theoretic properties. We will later see that Cis representation-infinite and C0has
finite representation type over any algebraically closed field k.
(2) By construction of b
Cwe have the quotient functor G:C → b
Cwhich is the identity on
objects and surjective on morphisms. This functor induces a fully faithful embedding
of mod kb
Cinto mod kC.
2.2 Simple and projective modules
In this part we give an explicit description of the projective modules and the simple modules
over an EI-category algebra kC. From now on we assume that kis a field (or a complete
discrete valuation ring in order to have the Krull-Schmidt property), if not stated otherwise.
We start with an important observation: If Cis an EI-category, then one has a natural
preorder defined on the set of objects Ob C, given by
x≤y⇔ C(x, y)6= 0.
This preorder clearly induces a partial order on the set of isomorphism classes of objects of
C.
In [26] L¨uck gave a characterization of all indecomposable projective and simple modules
over EI-category algebras. These results can also be obtained using work of Auslander [4],
but in the following formulation they are due to L¨uck.
Proposition 2.12 (L¨uck, [26]).Let Cbe an EI-category. Then any finitely generated
projective kC-module is isomorphic to a direct sum of indecomposable projectives of the
form kC · e, with e∈kAut(x)being a primitive idempotent for some x∈Ob C.
For an object xin Cwe denote by [x] the isomorphism class of x. With this notation
there is the following theorem.
Theorem 2.13 (L¨uck, [26]).Let Cbe an EI-category. For each object x∈Ob Cand every
simple kAut(x)-module Vthere is a simple kC-module Msuch that [x]is exactly the set of
objects on which Mis non-zero and M(x) = V. Conversely, if Mis a simple kC-module,
then there is a unique isomorphism class of objects [x]on which Mis non-zero and each
M(x)is a simple kAut(x)-module. Thus, there is a bijection between the isomorphism
classes of simple kC-modules and the pairs ([x], V )where xis an object in Cand Va
simple kAut(x)-module.
With this theorem it is natural to denote a simple kC-module by Sx,V if it corresponds
to the pair ([x], V ) and to write Px,V for its projective cover. Note that the structure of
Px,V is determined by its value at x.
Induction and restriction 13
Remark 2.14. If Cand Dare equivalent EI-categories, then their associated module-
categories mod kCand mod kDare equivalent as well.
Thus, we may throughout assume without loss of generality, that all EI-categories are
skeletal and therefore the set of objects (not only the isomorphism classes) carries the
natural structure of a finite poset.
2.3 Induction and restriction
Induction and restriction functors play an important role in modular representation theory
of finite groups, for example to classify the group algebras of finite representation type. This
concept can be carried over to the context of category algebras where we replace subgroups
of a given finite group by subcategories of a finite category. In the general setting the
definition of the restriction is the following.
Definition 2.15. Let kbe a commutative ring and Cand Dbe two small categories and
µ:D → C a covariant functor. Then we define the restriction along µto be the functor
Resµ: repkC → repkDwhich sends a representation Mof Cto the representation M◦µof
D.
This functor also has its counterpart on the level of modules over the category alge-
bras, which we also denote by Resµ: mod kC → mod kDand it sends a module M=
Lx∈Ob CM(x) to the module ResµM=Ly∈Ob DM(µ(y)).
A functor µas in the definition extends naturally to a k-module homomorphism µ:kD →
kC, but this map is in general not an algebra homomorphism. The cases when this happens
are characterized in the following proposition.
Proposition 2.16. A functor µ:D → C extends linearly to an algebra homomorphism
µ:kD → kCif and only if µis injective on Ob D. In this case, the induced functor
followed by 1kD, i.e. 1kD· ↓kC
kD: mod kC → mod kDis exactly Resµ.
In this work we want to use this concept for the case where Dis a (full) subcategory of
a finite category Cand we take for µ=ιthe inclusion functor. In this case the restriction
Resι: mod kC → mod kDis determined by the algebra homomorphism ι:kD → kC, hence
by ι:D → C. Therefore, we will not distinguish Resιand ↓kC
kDand write ↓kC
kDand Resι
as ↓C
D, which is the usual notation in representation theory, for example in the articles of
Xu. In the representation-theoretic setting the restriction ↓C
Dhas a left adjoint, which is
the induction ↑C
D=kC ⊗kD−: mod kD → mod kC. These induction and restriction functors
will play a crucial role in almost all situations we will consider.
In the framework of restriction and induction functors we will later need the following
definitions.
Induction and restriction 14
Definition 2.17. Let Cbe an EI-category.
(1) Let xbe an object in C. Then we define C≤xto be the full subcategory of Cconsisting
of all objects y∈Ob Cwith C(y, x)6=∅. Similarly we define C≥x.
(2) An ideal in Cis a full subcategory Dof Csuch that for any object xin Dwe have
that C≤x⊆ D. A coideal in Cis a full subcategory Eof Csuch that C≥x⊆ E for any
x∈ E.
(3) Let Mbe a kC-module. The M-minimal objects are the objects x∈Ob Csuch that
M(x)6= 0 and for any y∈Ob Cwith y6≃ xand C(y, x)6=∅one has M(y) = 0.
Analogously one defines M-maximal objects.
(4) Let again Mbe a kC-module. We put CMto be the full subcategory consisting of all
y∈Ob Cwith C(x, y)6=∅for some M-minimal object xin C.
It is clear by definition that any kC-module Mis determined by its values on CM. We
are now going to point out a nice property of ideals, namely that in this case the restriction
preserves projectives.
Proposition 2.18 (Xu, [33] Lemma 3.1.6).Let Dbe an ideal in an EI-category C. Then
the restriction functor ↓C
Dpreserves projective (left-)modules
If one would deal with right modules instead of left modules, then the restriction to
coideals would preserve projectives.
3 EI-categories of finite representation type
Since the concept of EI algebras generalizes the concept of group algebras of finite groups
and of path algebras of finite quivers or more generally of finite posets, it is a natural
question to ask for a classification of the EI-category algebras of finite representation type.
In 3.24 we will classify all endotrivial representation-finite EI-categories, which, roughly
speaking, gives us the global shape of representation-finite EI-categories. Afterwards, we
will turn our attention to EI-categories with two objects, since they are the easiest categories
which are not group algebras. Even for this class it turns out that a classification of finite
representation type is not easy. Nevertheless, we will derive some necessary criteria for finite
representation type and compute various examples. In section 3.7 and section 3.8 we will put
stronger conditions on the automorphism groups of the objects in our EI-categories and are
then able to classify the representation-finite categories under this additional assumptions.
It will become clear how involved this classification gets, even for very small categories.
In the beginning of this chapter we will recall the classification of representation-finite
group algebras, quivers and posets and afterwards present basic definitions and results on
the Gabriel-quiver of a finite-dimensional algebra as well as a quick survey about covering
theory for bound path algebras of quivers. In the introductory sections there will be no
proofs in order to keep this part short and streamlined. Let us start with the most important
definition for this chapter.
Definition 3.1. Let Abe a finite-dimensional algebra over any field k. We say that
Ais representation-finite or of finite representation type if there are only finitely many
isomorphism classes of indecomposable A-modules.
3.1 Finite representation type for finite groups, quivers and
posets
The classification of representation-finite quivers and group algebras of finite groups is given
by the following theorems.
Theorem 3.2 (Gabriel [18]).Let Qbe a finite and connected quiver and kany field. Then
the path algebra kQ is of finite representation type if and only if the underlying graph of Q
15
Finite representation type for finite groups, quivers and posets 16
is a simply laced Dynkin diagram, i.e. one of the following graphs.
An:• • • ... •(n≥1) E6:• • • • •
•
Dn:• • .... •
•
•?
?
?(n≥4) E7:• • • • • •
•
(n=number of vertices)E8:• • • • • • •
•
Theorem 3.3 (Highman [22], Kasch, Kneser and Kupisch [24]).Let Gbe a finite group
and let kbe an algebraically closed field of characteristic pdividing the order of G. Then
the group algebra kG is of finite representation type if and only if the Sylow p-subgroups Gp
of Gare cyclic.
If the characteristic does not divide the order of G, a theorem of Maschke states that
Ais semisimple, in particular representation-finite. A good reference for both theorems is
for example the book of Assem, Simson and Skowronski [3]. Also the representation-finite
posets have been classified by Loupias (and independently by some russian mathematicians)
and we want to recall the results here. A complete list of all representation-finite posets
may be created using the criterion we will present. One can find such a list in the diploma
thesis of H. K¨uchenhoff, who was a student of Gabriel, from 1982. Another good reference
for representations of posets in general is the book of Simson [30], but one should note
that he has a different definition of representation-finite posets. Therefore, we will mainly
stick to the notations of Loupias since they are convenient for our purposes. From now on
assume that (I, ≤) is a connected finite poset.
Definition 3.4. Let Iand Jbe two finite partially ordered sets and f:I→Ja surjective
morphism of ordered sets, i.e. f(x)≤f(y) if x≤y. If f−1(j) is connected for every j∈J,
then we call Jacontracted set of I.
Theorem 3.5 (Loupias 1974, [25]).Let Ibe a finite partially ordered set. Then Iis of
finite representation type if and only if it has no subset and no contracted set, which is
given by one of the Hasse-diagrams (or their duals) in the list below. If there is a just a
line between two points the orientation is arbitrary.
e
An:•
•
..... •
•
?
?
?
•
. . . . .
•
?
?
?
(n≥0) e
Dn:• • ... •
•
•
?
?
?
•?
?
?
•
(n≥4)
Finite representation type for finite groups, quivers and posets 17
e
E6:• • • • •
•
•
(n+ 1 = number of vertices)
e
E7:• • •••••
•
e
E8:• • • • • • • •
•
e
D1:• • •
•
•e
A4:• •
•
•
??
?
?
?
?
?
__?
?
?
?
?
R1:• •
•
•
• • • •
•
??
__?
?
?
?
?
??
__?
?
?
?
?R2:• •
•
•
• • • •
•
??
?
?
?
?
???
?
?
?
?
?
R3:• • • •
•
•
• •
•
??
__?
?
?
???
__?
?
?
?R4:• • •
•
•
• • • •
•
??
__?
?
?
???
__?
?
?
?
R5:• •
•
•
•
• • • •
??
__?
?
?
?
?
WW/////////GG
GG
R6:• • •
•
•
•
• • •
??
__?
?
?
?
?
GG
WW////
WW////
R7:• • •
•
•
•
•
• •
GG
GG
WW/////////
WW////
WW////GG
The Gabriel-quiver of a finite dimensional algebra 18
From this result it is possible to produce the Hasse-diagrams of all representation-finite
partially ordered sets as it was done by K¨uchenhoff.
The leading question throughout this chapter is the following: Is there any general concept
which contains the classification results from above as special cases and which gives a
characterization of all representation-finite EI-category algebras?
3.2 The Gabriel-quiver of a finite dimensional algebra
To each finite-dimensional connected algebra Aover a field kone can associate a finite
quiver Γ(A) and, in case the field kis algebraically closed, a theorem of Gabriel yields
that Ais Morita-equivalent to kΓ(A)/I for some admissible ideal I. In this section we
will briefly recall the most important definitions and results in this context, because we
will use them intensively in our treatment of finite representation type for EI-categories.
For more details for this whole section one may consult the book of Assem, Simson and
Skowronski [3, Chapter II].
Definition 3.6. Let Qbe a finite quiver and Rthe ideal generated by all arrows in the
path algebra kQ. A two-sided ideal Iin kQ is called admissible if there exists n≥2 such
that Rn⊆I⊆R2. If Iis an admissible ideal, then the pair (Q, I) is called a bound quiver
and kQ/Iis said to be a bound path algebra.
Definition 3.7. Let Abe a basic connected finite-dimensional algebra over a field kand
e1, . . . , ena complete set of primitive orthogonal idempotents of A. The Gabriel-quiver of
A, denoted by Γ(A), is defined in the following way:
(i) The vertices of Γ(A) correspond bijectively to the idempotents e1, . . . , en;
(ii) For two vertices a, b ∈Γ(A)0, the arrows α:a→bare in bijective correspondence
with the vectors of a k-basis of the k-vector space ea(rad A/ rad2A)eb.
One can easily verify that this quiver does not depend on the choice of the primitive
idempotents in the definition. One should note here that every finite-dimensional k-algebra
Ais Morita-equivalent to a basic algebra A0and from the representation-theoretic point
of view it makes no difference whether we deal with Aor A0. One can also define Γ(A)
in a way that only depends on the category mod Aand not on the structure of Aitself as
follows: The vertices 1, . . . , n of Γ(A) are in bijective correspondence with the isomorphism
classes S1, . . . , Snof simple A-modules with dimkExt1
A(Si, Sj) vertices from ito j. For
explicit computations the first definition in terms of idempotents is often more applicable.
The central result in this context is the following theorem of Gabriel.
Theorem 3.8 (Gabriel).Let Abe a basic and connected finite-dimensional algebra over
an algebraically closed field k. Then there exists an admissible ideal Iin kQ such that
A∼
=kΓ(A)/I.
Covering theory 19
Note that the ideal Iis in general not unique. See [3, II.2.2] for an example of an algebra
Awith two admissible ideals satisfying the conditions of the theorem. We will now illustrate
how one can compute the quiver of an EI-category with one example.
Example 3.9. We consider the EI-category Cwhich will also be treated in Proposition 3.27.
That is: Let Cbe an EI-category with two objects xand ysuch that End(x) = hfi∼
=Z2and
End(y) = hgi∼
=Z2. Furthermore, we require that C(x, y) = {i1, i2, i3, i4}with i1◦f=i2,
i3◦f=i4,g◦i1=i3and g◦i2=i4.Cmay be illustrated as follows.
C:x
f99
i1
i2
((
i366
i4
AAyg
ee
To compute the quiver of the (basic) algebra kCwe have to distinguish between the case
where char(k) = 2 and the case char(k)6= 2.
Suppose char(k) = 2. Then 1xand 1yform a complete list of orthogonal, primitive
idempotents of A:= kC. We compute the radical to be rad(A) = hi1,1x+f, 1y+giand
therefore rad2(A) = {is+it|s6=t}. Hence, the quiver Γ(A) has two vertices correspond-
ing to 1xand 1yeach of this vertices having a loop corresponding to 1x+fand 1y+g
respectively. Finally, there is one arrow from 1xto 1ycorresponding to the class of i1in
1y(rad(A)/rad2(A))1xand the quiver of Ais
Γ(A) : ◦
α99
β//◦γ
ee.
With the assignments from above, we get an algebra epimorphism kΓ(A)→Awhose kernel
is the admissible ideal (α2, γ2) and hence A∼
=kΓ(A)/(α2, γ2).
If char(k)6= 2, then kCis hereditary as the path algebra of
◦//
@
@
@
@
@
@
@◦
◦
??
~
~
~
~
~
~
~
~//◦,
which is a quiver with underlying Eucledian graph ˜
A3.
3.3 Covering theory
In [7] Bongartz and Gabriel introduced covering theory for finite-dimensional algebras.
Their work had been inspired by work of Riedtmann [28]. We will briefly recall some
definitions, but mainly try to explain the use of covering theory for our purposes with an
Covering theory 20
example.
Definition 3.10. Let Aand Bbe two k-linear categories. A k-linear functor F:A → B
is called covering functor if the induced maps
a
F y=b
A(a, y)→ B(Fa, b) and a
F y=b
A(y, a)→ B(b, Fx)
are bijective for every a∈ A and b∈B.
Every covering functor gives rise to a push-down functor Fλ: Repk(A)→Repk(B), where
Fλ(M)(b) = `F a=bM(a). The definition on morphisms is fairly obvious.
A theorem of Gabriel [16] states that, under certain assumptions, the push-down functor
preserves indecomposability. We will use this method for quivers with relations and loops
at some vertices. We will explain how this procedure works in the following example, which,
in our opinion, explains sufficiently well how a cover of the given quiver is constructed for
all cases we are interested in.
Consider the Jordan-quiver
Q:a
α99,
subject to the relation α2= 0. This is the quiver of the group algebra of the cyclic group
of order 2 in characteristic 2. For the infinite quiver
Q:. . . γ//x−1γ//x0
γ//x1
γ//. . . ,
bound by γ2= 0, we get a functor F:Q→Qdefined by F(xi) = afor all iand F(γ) = α,
which is a covering functor (here we take for Qand Qthe k-linear category spanned by their
associated path categories). In this case the push-down functor preserves indecomposability
(and also sends almost-split sequences to almost-split sequences). To get all indecomposable
representations of Q/α2, we pick a finite piece of Q/γ2and knit its Auslander-Reiten quiver.
Then we use the push-down to get indecomposable representations of Q/α2and see, that
this procedure is 2-periodic and we immediately get all indecomposable representations
“downstairs”, namely two representations of dimension 1 and 2, respectively.
Analogously, one constructs the cover for all quivers with relations that appear in this
thesis. The example we will see most frequently is the one of a quiver of the form
Γ : ◦
α99
β//◦γ
yy,
subject to some admissible relations. The universal cover of this quiver is then the infinite
quiver which is displayed below, bound by the same relations as the original quiver. The
covering basically coincides with the one in the last example and the push-down functor
again preserves indecomposability. In the language of Gabriel, this is a Galois-covering.
Relative projectivity, vertices and sources 21
.
.
.
α
.
.
.
γ
◦
α
β//◦
γ
◦
α
β//◦
γ
◦
α
β//◦
γ
.
.
..
.
.
Figure 3.1: The universal cover of Γ
The fact we will heavily use is that, if the covering quiver of Q/I contains a relation-free
part which is representation-infinite, then Q/I is representation-infinite since we can push-
down the infinite family of indecomposable representations. Once again we refer to the
literature for a more detailed treatment of this fact.
3.4 Relative projectivity, vertices and sources
The main tools for the classification of group algebras of finite representation type are
restriction and induction functors as well as the concepts of relative projectivity, vertices,
sources and defect groups. For EI-category algebras Xu [33] developed a theory of vertices
and sources, which is very similar to the one for finite groups. In this subsection we will
first briefly present the main definitions and results of this theory and then give an abstract
classification of EI-category algebras of finite representation type in terms of vertices, which
is a more or less immediate Corollary of Xu’s results.
Definition 3.11. Let Cbe an EI-category and Da subcategory of C. We call a kC-module
Mrelatively D-projective (or projective relative to D) if the canonical surjective kC-module
homomorphism ε=εM:M↓C
D↑C
D=kC ⊗kDM→M, a ⊗m7→ a·msplits.
Again as in the group case, we have several equivalent characterizations of relative pro-
jectivity.
Proposition 3.12 (Xu [33, Proposition 3.2.2]).Let Cbe a finite EI-category, D ⊆ C a
subcategory and MakC-module. Then the following conditions are equivalent:
(1) Mis relatively D-projective;
(2) Mis a direct summand of M↓C
D↑C
D;
Relative projectivity, vertices and sources 22
(3) Mis a direct summand of N↑C
Dfor some kD-module N;
(4) If 0→M0→M→M00 →0is an exact sequence of kC-modules which splits in
mod kD, then it splits in mod kC.
Relative projectivity of a module Mis closely related to M-minimal objects in the fol-
lowing way.
Lemma 3.13 (Xu [33, Lemma 3.2.5]).Let Mbe a kC-module and D ⊆ C a subcategory.
(1) If Mis relatively D-projective, then Ob Dcontains all M-minimal objects.
(2) If Mis relatively D-projective, then the module M(x)is relatively AutD(x)-projective
as a kAutC(x)-module for any M-minimal object x.
For the case of full subcategories of an EI-category C, relative projectivity of a module
yields that this module is already generated by its values on the full subcategory relative
to which it is projective in the following sense.
Proposition 3.14 (Xu [33, Proposition 3.3.1]).Let Dbe a full subcategory of an EI-
category Cand MakC-module which is relatively D-projective. Then Mis generated by
its values on D, i.e. M↓C
D↑C
D∼
=M.
The next theorem guarantees that for a representation-finite EI-category every connected,
full subcategory has to be of finite representation type.
Theorem 3.15 (Xu [33, Theorem 3.3.2]).Let Cbe an EI-category, Da connected, full sub-
category and Nan indecomposable kD-module. Then the kC-module N↑C
Dis indecomposable
and relatively D-projective.
This theorem can be compared with Green’s indecomposability theorem for representa-
tions of finite groups (see [2]). Its inverse holds as well:
Theorem 3.16 (Xu [33, Theorem 3.3.4]).Let Mbe an indecomposable kC-module which
is relatively D-projective for a connected, full subcategory D ⊆ C. Then M↓C
Dis an inde-
composable kD-module.
The two previous theorems give an equivalence of categories by means of the following
definition.
Definition 3.17. Let D ⊆ C be a connected, full subcategory of an EI-category C. We
denote by mod kCDthe full subcategory of mod kCconsisting of those kC-modules which
are relatively D-projective.
Proposition 3.18 (Xu [33, Proposition 3.3.6]).The functors ↓C
Dand ↑C
Dinduce quasi-
inverse equivalences
mod kCD
↓C
D//mod kD
↑C
D
oo.
Relative projectivity, vertices and sources 23
The next result is necessary to develop the theory of vertices and sources in our framework.
Proposition 3.19 (Xu [33, Corollary 3.3.8]).For any indecomposable kC-module Mthere
exists a smallest ideal g
VMin Crelative to which Mis projective.
Definition 3.20. The full subcategory VM=g
VM∩ CMis called the vertex of M.
It is clear by definition that the vertex of an indecomposable kC-module is always a
connected, full subcategory of C. We say that a subcategory Dof Cis convex if, whenever
there is a sequence of morphisms xα
−→ yβ
−→ zin Cwith x, z ∈Ob D, then both αand β
are in Mor D. With this definition there are three equivalent characterizations of the vertex
of a kC-module.
Proposition 3.21 (Xu [33, Proposition 3.3.12]).Let Mbe an indecomposable kC-module
and D ⊆ C a connected and full subcategory of C. Then the following statements are
equivalent:
(1) Dis the vertex of M;
(2) Dis the smallest ideal in CMrelative to which Mis projective;
(3) Dis the smallest full and convex subcategory of Crelative to which Mis projective.
If Mis again an indecomposable kC-module and VMits vertex, then M↓C
VM↑C
VM
∼
=M
and M↓C
VMis indecomposable. Therefore, Mis (up to isomorphism) determined by the
indecomposable kVM-module M↓C
VM. For that reason, we call M↓C
VMthe source for M.
Using the whole theory Xu developed, we derive the following easy Proposition which is
an abstract characterization of EI-categories of finite representation type.
Proposition 3.22. Let Cbe a finite, connected EI-category. Then the algebra kCis of finite
representation type if and only if kVMis of finite representation type for any indecomposable
module Min mod kC.
Proof. If Cis of finite representation type, then every full, connected subcategory is of fi-
nite representation type by Theorem 3.15. Conversely, suppose that every kVMis of finite
representation type. Since Cis finite, there can only be finitely many vertices of indecom-
posable kC-modules, each having only finitely many indecomposable representations up to
isomorphism. An indecomposable kC-module Mis (up to isomorphism) determined by its
source M↓C
VM, which is an indecomposable kVM-module. Together, this yields that there
are only finitely many isomorphism classes of indecomposable kC-modules.
Remark 3.23. (i) In representation theory of finite groups it is known that for a finite
group Gand a field kof characteristic p > 0 dividing |G|every kG-module is projective
relative to the Sylow p-subgroup Pof G. For that reason, the representation type
of kG is governed by the representation type of kP. In contrast to that, for an
EI-category algebra kCit almost always happens that the smallest full subcategory
The endotrivial case 24
relative to which a kC-module Mis projective is the category Citself. Thus, the
theory of vertices and sources for EI-categories does not help much for an explicit
characterization of finite representation type.
With existing methods it is also not possible to generalize the concept of a defect
group to an equivalent concept for EI-category algebras since the conjugation with
group elements of a group G(as basis vectors in kG) is essential to obtain a reasonable
theory for defect groups. For EI-categories there is no concept similar to conjugation.
(ii) Restriction and induction are defined for an arbitrary subcategory Dof an EI-category
C. Most of the results of Xu only work for the case where Dis a full subcategory of C.
The problem with arbitrary subcategories is, that the computation of the induction of
a module is in general very complicated. If Dis a full subcategory, then the induction
of every indecomposable kD-module to kCis indecomposable. This is not true for
arbitrary subcategories (e.g. subgroups of a finite group) and for that reason it is
not clear how one can get a reduction technique for the representation type of an
EI-category in terms of arbitrary subcategories.
3.5 The endotrivial case
For any EI-category Cwe have seen that there is a fully faithful embedding mod kb
C →
mod kC, where b
Cis the ’endotrivialization’ of Cwhich has been defined above. The existence
of this embedding implies that, if Cis representation-finite, then b
Cis representation-finite as
well. Therefore, it is natural to ask for a description of all representation-finite EI-categories
with only trivial endomorphisms. This description characterizes the global structure of any
representation-finite EI-category.
Theorem 3.24. Let Cbe an EI-category with only trivial endomorphisms. Then Cis of
finite representation type if and only if kCis Morita-equivalent to an incidence algebra of
finite representation type.
Proof. Let Cbe endotrivial and of finite representation type. We have to show that kCis
Morita-equivalent to an incidence algebra of finite type. The other direction in the theorem
is trivial. Again, we may (up to Morita-equivalence of kC) assume that Cis skeletal.
Since Cis representation-finite, there are no objects x, y ∈ C with two distinct morphisms
f, g :x→y. Otherwise one gets a fully faithful embedding of the category of representations
of the 2-Kronecker into mod kC. Since Cis skeletal its set of objects carries a natural
structure of a finite poset defined by
x≤y:⇔ ∃f∈ C(x, y).
Using, that C(x, y)≤1 for all x, y ∈Ob C, we get that kCis the incidence algebra associated
EI-categories with two objects 25
to the finite poset (Ob C,≤) and the claim follows.
Corollary 3.25. Let Cbe a finite and skeletal EI-category and ka field such that kCis
representation-finite. Then b
Cis a finite poset of finite representation type.
Remark 3.26. Suppose Cis an endotrivial EI-category. Then its set of objects naturally
carries the structure of a finite partially ordered set, but in general it will not be the case
that Cis the category associated to this poset. For instance, take Cto be the category
d
b
γ@@
c,
δ
__>
>
>
>
>
>
>
>
a
α
__>
>
>
>
>
>
>
>β
??
~
~
~
~
~
~
~
without any relations. Then the category associated to (Ob C,≤) is Cmodulo the relation
γα =δβ.
3.6 EI-categories with two objects
The easiest class of EI-categories for which the representation type has not yet been
investigated is the class of EI-categories with two non-isomorphic objects. Clearly, a
representation-finite EI-category Cwith two objects has to satisfy b
C=A2and both group
algebras attached to the two objects have to be of finite representation type. Therefore,
we may illustrate a possibly representation-finite EI-category Cwith two non-isomorphic
objects as follows.
x
GxfGy//y
Here fis one representative of the class of morphisms from xto ywith respect to our
relation ∼and Gx,Gydenote the endomorphism groups of xand yrespectively. In other
words, fis one representative of the unique orbit of the group action of Gx×Gyon C(x, y).
One should note that the representation type of an endotrivial category does not depend
on the characteristic of the ground field k. In contrast to that, the representation type of
group algebras depends on char(k) and the same is true for EI-category algebras which are
not group algebras, as we will see later.
An arbitrary EI-category algebra kCis representation-infinite if there exists at least one
full subcategory D ⊆ C such that the algebra kDis representation-infinite. Therefore,
the treatment of EI-categories with two objects yields necessary criteria for an EI-category
algebra to be representation-finite.
We assume that kdenotes an algebraically closed field.
EI-categories with two objects 26
3.6.1 The easiest example
The following example, which we will treat in every detail, illustrates how involved the
characterization of representation-finite EI-category algebras might get. We consider an EI-
category Cwith two objects xand ysuch that End(x) = hfi∼
=Z2and End(y) = hgi∼
=Z2.
Under these assumptions there are 5 different EI-categories that can appear, namely:
(1) C(x, y) = {i}with i◦f=iand g◦i=i;
(2) C(x, y) = {i1, i2}with facting trivially on {i1, i2}and gpermuting the two mor-
phisms;
(3) C(x, y) = {i1, i2}with fpermuting i1and i2and gacting trivially;
(4) C(x, y) = {i1, i2}with fand gpermuting i1and i2;
(5) C(x, y) = {i1, i2, i3, i4}with i1◦f=i2,i3◦f=i4,g◦i1=i3and g◦i2=i4.
The second case has briefly been studied by Xu in [33], where he claims that this category
is of infinite representation type in characteristic 2, but the representations he constructed
turn out to be decomposable and (as we will see later) and the category is of finite repre-
sentation type. It will turn out that the representation type of an EI-category with two
objects is mainly governed by the group action of the endomorphism groups on the set of
morphisms between the two non-isomorphic objects.
The investigation of case (5), which is obviously the one with the most complicated
representation theory, leads to the following result.
Proposition 3.27. Let Cbe an EI-category with two objects xand ysuch that End(x) =
hfi∼
=Z2and End(y) = hgi∼
=Z2. Furthermore, we require that C(x, y) = {i1, i2, i3, i4}
with i1◦f=i2,i3◦f=i4,g◦i1=i3and g◦i2=i4.
C:x
f99
i1
i2
((
i366
i4
AAyg
ee
Then kCis of infinite representation type, no matter which base field kwe choose.
Proof. We prove the theorem by constructing an infinite family of indecomposable repre-
sentations of C:
For n∈Nlet Vn∈repkCbe defined by Vn(x) = k2n,Vn(y) = k2n−1and
Vn(f) =
0 1
1 0 ...0 1
1 0
, Vn(g) =
0 1
1 0 ...0 1
1 0 1
, Vn(i1) =
0 0
0 1
. .
. .
. .
0 1
.
This gives (together with the compositions of the maps we defined above) a representation
EI-categories with two objects 27
of C. We prove the indecomposability of Vn(which we will for simplicity denote by V) by
computing its endomorphism ring.
Let φbe an endomorphism of V, that means φ= (A, B) where A∈M2n(k) and B∈
M2n−1(k). First of all, A= (ai,j) has to commute with V(f) which is equivalent to the
condition that a2l−1,2t−1=a2l,2tand a2l−1,2t=a2l,2t−1for any 1 ≤l, t ≤n. Therefore, Ais
of the following form.
A=
a1,1a1,2a1,3a1,4. . . a1,2n−1a1,2n
a1,2a1,1a1,4a1,3. . . a1,2na1,2n−1
a3,1a3,2a3,3a3,4. . . a3,2n−1a3,2n
a3,2a3,1a3,4a3,3. . . a3,2na3,2n−1
. . . . . .
. . . . . .
. . . . . .
a2n−1,1a2n−1,2a2n−1,3a2n−1,4. . . a2n−1,2n−1a2n−1,2n
a2n−1,2a2n−1,1a2n−1,4a2n−1,3. . . a2n−1,2na2n−1,2n−1
Analogously, Bhas to commute with V(g) which is equivalent to the conditions b2l−1,2t−1=
b2l,2t,b2l−1,2t=b2l,2t−1,b2l,2n−1=b2l−1,2n−1and b2n−1,2t=b2n−1,2t−1for any 1 ≤l, t ≤
n−1, i.e. Bis of the following form.
B=
b1,1b1,2. . . b1,2n−3b1,2n−2b1,2n−1
b1,2b1,1. . . b1,2n−2b1,2n−3b1,2n−1
. . . . .
. . . . .
. . . . .
b2n−3,1b2n−3,2. . . b2n−3,2n−3b2n−3,2n−2b2n−3,2n−1
b2n−3,2b2n−3,1. . . b2n−3,2n−2b2n−3,2n−3b2n−3,2n−1
b2n−1,1b2n−1,1. . . b2n−1,2n−3b2n−1,2n−3b2n−1,2n−1
Additionally, Aand Bhave to satisfy the relation
BV (i1) = V(i1)A. (3.1)
The left hand side of this equation equals (0, B) where 0 denotes the zero vector in k2n−1.
The right hand side is easily seen to be V(i1)A=b
A, where we want b
Ato denote the matrix
one obtains from Aby erasing the first line.
Now we claim, that (3.1) implies A=λ·E2nand B=λ·E2n−1with λ:= a1,1=b1,1.
First of all, we note that both Aand Bonly have entries whose first index is odd and that
all diagonal entries of Aand Bare equal to λ. Now we compare the left hand side and the
right hand side of (3.1) columnwise, starting with the first column. This yields a2i−1,j = 0
EI-categories with two objects 28
for all i= 1, . . . , n, j = 1, . . . 2nwith 2i−16=jand j6= 2i+ 1 as well as b2l−1,t = 0 for
all l= 1, . . . , n, t = 1, . . . 2n−1 with t6= 2l−1 and t6= 2l+ 1. Furthermore, we infer that
a2i−1,2i+1 =a2l−1,2l+1 =b2j−1,2j+1 =b2t−1,2t+1 for all i, j, l, t = 1, . . . , n −1.
In other words the entries of Aand Bare zero if they do not lie on the diagonal or have
indices of the form 2i−1,2i+ 1, and the latter are all equal. The last column of (3.1) gives
0 = a2n−1,2n=b2n−3,2n−1which implies 0 = a2i−1,2i+1 =b2j−1,2j+1 for all i, j = 1, . . . , n−1
and the claim follows.
This is the first non-trivial example of an EI-category of infinite representation type,
where non-trivial means, that neither b
Cnor kEnd(x) (for some x∈Ob C) is of infinite
representation type. Indeed, even the characteristic of the ground field is arbitrary in
the proposition and therefore the group algebras associated to the two objects may be
semisimple, e.g. for char(k) = 0.
We will now dicuss all five cases from above by computing their Gabriel-quivers and
deducing their representation type from this quiver (with relations). Surprisingly, it will
turn out, that the characteristic of the ground field does not play any role in this special
cases. We will also get the result from above again by this discussion, but the construction
of an infinite family of non-isomorphic indecomposables is interesting on its own.
Proposition 3.28. Let Cbe the EI-category from case (1). Then kCis of finite represen-
tation type.
Proof. The computation of the Gabriel quiver of kCin characteristic 2 yields the quiver
Q:◦
α99
β//◦γ
ee.(3.2)
An easy calculation shows that kCis isomorphic to kQ/I, where Iis the admissible ideal
generated by the zero relations 0 = α2=γ2=βα =γβ. Therefore, kCis a string
algebra without any bands and those algebras are known to be of finite representation
type. Alternatively one can consult the list at the end of [7] and see that this algebra is of
finite representation type and also find its Auslander-Reiten quiver.
If char(k)6= 2, then the algebra kCis isomorphic to the path algebra kQ where Qis the
quiver
◦ ◦
◦//◦,
which is obviously of finite representation type.
The other cases are discussed analogously and we deduce the following results (with
respect to the numbering from the beginning of this subsection). Fix the quiver Qfrom
EI-categories with two objects 29
(3.2).
(2) kC ≃ kQ/hα2, γ2, βαiif char(k) = 2 and this algebra is of finite representation type,
again as a string algebra without any bands. If the characteristic is different from 2,
then kCis the path algebra of the representation-finite quiver
◦ ◦
◦
>>
}
}
}
}
}
}
}//◦.
(3) This situation is dual to the one above. In char(k) = 2 we have kC ≃ kQ/hα2, γ2, γβi,
which again is a string algebra without bands, and in char(k)6= 2 the algebra kCis
hereditary of finite representation type as the path algebra of the following quiver
◦
A
A
A
A
A
A
A◦
◦//◦.
(4) If char(k) = 2, then kC ≃ kQ/hα2, γ2, βα −γβi. This algebra is representation-finite,
which can be seen by knitting the AR-quiver using covering theory as we explained
before (or see for example [16]). If the characteristic is different from 2, then the
algebra kCis isomorphic to the path algebra of the quiver
◦//◦
◦//◦,
which is of finite representation type.
(5) For this case we have already seen that kCis of infinite representation type in any
characteristic. This also follows from the computation of the Gabriel quiver and the
attached relations. If char(k) = 2, then we get that kC ≃ kQ/hα2, γ2iwhich is a
string algebra with infinitely many bands and therefore it is of infinite representation
type. In any characteristic different from 2 the algebra kCis hereditary (as we know
from work of Xu) and in this particular case isomorphic to the path algebra of the
following quiver
◦//
@
@
@
@
@
@
@◦
◦
??
~
~
~
~
~
~
~
~//◦,
which is a quiver with underlying Eucledian graph ˜
A3.
EI-categories with two objects 30
3.6.2 The characteristic plays a role
In this subsection we present an example, which shows that the representation type of an EI-
category algebra kC, which does not come from a group, depends on the characteristic of the
field k. The reason for this is that the same is true for group algebras. The easiest example
where different characteristics of kyield different representation types is the following one.
Example 3.29. Consider the EI-category
C:x
g99
f1
&&
f2//
f388yh
ee,
with the relations g2= 1x, fig=fi,for i= 1,2,3, h3= 1yand hf1=f2, hf2=f3,
hf3=f1.
We will now show that the associated k-algebra kCis of finite representation type if and
only if char(k)6= 2. First of all, we suppose that char(k)6= 2,3. Then kCis hereditary (and
basic) and we compute the Gabriel quiver to be
◦//
A
A
A
A
A
A
A
0
0
0
0
0
0
0
0
0
0
0
0
0
0◦
◦ ◦
◦,
which is of finite representation type as the union of quivers with underlying graphs A1and
D4.
Now we assume that the characteristic of kis 3. Then the radical of kCis h1y−h, f1, f2, f3i
and rad2kC=h(1y−h)2, f1−f2, f2−f3iwhere h—idenotes the k-span. A complete list
of primitive, orthogonal idempotents is given by 1y,1
2(1x+g),1
2(1x−g). Therefore, kCis
isomorphic to the path algebra of the quiver
◦β//◦γ
ee
◦
,
bound by the relation γ3= 0. This bound path algebra is of finite representation type since
it is the union of A1with a bound path algebra whose module category can be embedded
into the module category of k(◦
ν//◦
δ
ooρ
ee)/hρ3, δρ, δν, δρνi. The latter is known to be
of finite type by work of Bautista, Gabriel, Roiter and Salmeron [6] and work of Bongartz
and Gabriel [7].
EI-categories with two objects 31
The last case is the one where char(k) = 2. Here the radical of kCis h1x+g, f1, f2, f3i
and its square is zero. A complete list of primitive, orthogonal idempotents is given by
1x,1
3(1y+h+h2),1
3(1y+εh +ε2h2),1
3(1y+ε2h+εh2), where εdenotes a primitive third
root of unity in k. Therefore, we deduce for the Gabriel quiver of kCthe quiver
◦
Γ : ◦
α99
β1??
~
~
~
~
~
~
~
~β2//
β3
@
@
@
@
@
@
@
@◦,
◦
and kCis isomorphic to kΓ/hα2, β1α, β2α, β3αi. This bound quiver is of infinite represen-
tation type since its universal cover contains the 4-subspace quiver without relations. It
is also not difficult to write down a 1-parameter family of indecomposables of dimension
vector (3,1,1,1).
The discussion of the first case in the example can be generalized to the following Propo-
sition, which provides us with a situation where we can prove that the EI-category algebras
in question always have finite representation type.
Proposition 3.30. Let Cbe a finite, skeletal EI-category with 2 objects xand ysuch that
C(x, y) = {f}and the two groups End(x)and End(y)are abelian. Let kbe an algebraically
closed field whose characteristic neither divides the order of End(x)nor the order of End(y).
Then kCis of finite representation type.
Proof. Let n:= |End(X)|and m:= |End(Y)|.
First of all we note that the algebra kCis hereditary (see Theorem 4.2.4 in [33]) and
basic since the two groups are abelian. Therefore, it is isomorphic to the path algebra of
its Gabriel quiver.
Since the groups End(x) and End(y) are assumed to be abelian of order not divisible by
the characteristic of our field, it is known that kChas exactly m+nisomorphism classes
of simple modules (see [29] and [26]) and all the simples do not have self-extensions. This
implies, that the Gabriel quiver Γ(kC) has m+nvertices. Furthermore, we know that the
k-dimension of kCis m+n+ 1. This yields that we have exactly one arrow in Γ(kC) and
the claim follows.
3.6.3 Free action implies infinite type
In this subsection we prove the fact that free action of the automorphism groups of an
EI-category Cwith two objects xand yon C(x, y) implies infinite representation type in
EI-categories with two objects 32
any characteristic. This is the most general result we will achieve in our treatment of EI-
categories with two objects. The proof is carried out by considering various cases. The
most interesting cases will be presented as Lemmata starting with the following one.
Lemma 3.31. Let Cbe an EI-category with two non-isomorphic objects x, y and abelian
endomorphism groups End(x)and End(y)of order ≥2such that the group action of
End(x)×End(y)on C(x, y)is free and transitive. Let kbe an algebraically closed field
which characteristic neither divides the order of End(x)nor the order of End(y). Then kC
is of infinite representation type.
Proof. Analogous to the proof in the last subsection, we have that kCis isomorphic to the
path algebra of its Gabriel-quiver which has m+nvertices. Since dimkkC=m+n+m·n
there are m·narrows in Γ(kC). Therefore, the underlying graph of the Gabriel quiver is
not Dynkin.
If both group algebras are not semisimple, we can prove the assertion without constructing
representations or computing the Gabriel quiver.
Lemma 3.32. Let Cbe an EI-category with two objects xand yand let kbe an algebraically
closed field of positive characteristic pdividing both |End(x)|and |End(y)|. Further, we
assume that End(x)×End(y)acts freely on C(X, Y ). Then kCis of infinite representation
type.
Proof. For simplicity we write G:= End(x) and H:= End(y). We will prove the theorem
by constructing a fully faithful embedding F: mod k(G×H)→mod kC. The construction
of this functor is rather obvious. For M∈mod k(G×H) let F(M)(x) = M=F(M)(y)
together with the natural actions of Gand Hon M given by G×idYand idX×H, respec-
tively. Furthermore, we let F(M)(C(x, y)) = G×H. This is indeed a representation of kC.
Now let µ:M→M0be a morphism in mod k(G×H). We put F(µ) = (µ, µ) which gives a
morphism F(M)→F(M0) of kCmodules. Finally this functor is fully faithful and k-linear
by construction and it is known that k(G×H) (in this particular framework) is of infinite
representation type.
Remark 3.33. If Cis a skeletal EI-category with two objects x, y such that Aut(x)×
Aut(y) acts freely on C(x, y), then we can also localize the category Cwith respect to the
set of morphisms S:= C(x, y) in the set of Gabriel and Zisman [17]. This gives a new
category C[S] and the category of representations of C[S] is equivalent to the subcategory
of representations of Cconsisting of all representations Vfor which V(f) is invertible for
all fin C(x, y). Then, it is easy to see that this category contains mod(Aut(x)×Aut(y))
as a full subcategory as we have seen in the previous proof.
EI-categories with two objects 33
One should note that the localization of an EI-category is not again an EI-category in
general, but if we assume that the EI-category has no parallel morphisms (i.e. |C(x, y)| ≤ 1
for all x, y ∈Ob C), then every localization is again EI.
For cyclic groups the computation of the Gabriel quiver is rather easy, which gives the
next lemma.
Lemma 3.34. Let Cbe an EI-category with two non-isomorphic objects xand ysuch that
the action of End(x)×End(y)on C(x, y)is free and End(x)and End(y)are cyclic of order
≥2. Then kCis of infinite representation type for any algebraically closed field k.
Proof. In the case of cyclic endomorphism groups one can easily compute the Gabriel quiver
of kCand derive the relations on it such that kCis isomorphic to this bound path algebra.
One has to distinguish between the cases where the orders of both groups are divided by
the characteristic, only one of them or none. In all the three cases one ends up with infinite
representation type. We are not going to present the details here, the computations are
exactly the same as in the examples we discussed above.
Up to now we have seen several special cases of EI-categories with two objects and two
non-trivial endomorphism groups with free action which are representation-infinite. We are
now in the position to prove the general statement
Theorem 3.35. Let Cbe a skeletal EI-category with two objects xand ysuch that the
groups End(x)and End(y)are non-trivial and their product End(x)×End(y)acts freely on
C(x, y). Then kCis of infinite representation type for any algebraically closed field k.
Proof. The claim has already been proven for the case where both group algebras are non-
semisimple, for the case where both groups are cyclic and for the case of two semisimple
abelian group algebras. To prove the theorem we still have to distinguish between different
cases. For simplicity denote G:= End(X) and H:= End(Y).
(a) Suppose that kG and kH are semisimple. We are going to construct an infinite family
(Vλ)λ∈k?of pairwise non-isomorphic indecomposable representations of C. Let λ∈k?
and define Vλ(X) = M,Vλ(Y) = Nwhere Mis a kG-module, NakH-module, both
having dimension at least 2. Furthermore we have to choose one linear map M→N
which we want, for some fixed basis, to be given by the matrix
Aλ:=
1 0 0 0 . . .
λ100. . .
0 0 0 0 · · ·
.
.
..
.
..
.
..
.
.· · ·
,
where all the dots stand for zeros. For the choice of Mand Nwe again have to
distinguish different cases.
EI-categories with two objects 34
(i) Suppose that neither Gnor His abelian. Then we can choose Mto be a simple
kG-module and Nto be a simple kH-module, both of dimension ≥2. Then,
since Mand Nare indecomposable, it is clear that every Vλis indecomposable.
We will now show that for λ6=µwe have Vλ6∼
=Vµ. To see that, suppose
that (φ, ψ) : Vλ→Vµis an isomorphism of kC-modules. This implies that
φ∈EndkG(M)∼
=kand ψ∈EndkH(N)∼
=k, which means φ=α·1Mand
ψ=β·1N. In addition, φand ψhave to be compatible with the action of the
matrix Aλ(defined above). This gives the equations α=βand αλ =βµ and
hence α=β= 0 which contradicts the assumption that (φ, ψ) is an isomorphism.
(ii) Suppose that His non-abelian and Gis abelian (the other way around is dual).
Choose Nas above and put M=k2with G-action given by the matrix a0
0b
where a6=band both are non-zero. In other words we want Mto be the direct
sum of two non-isomorphic one-dimensional simple kG-modules. In this case we
have that EndkG(M) = k0
0kand we deduce that Vλis indecomposable. As
above we see that Vλ6∼
=Vµfor λ6=µ.
(b) The last case that has to be treated (again by subdivision into different cases) is the
case where one of the group algebras kG and kH is semisimple while the other is
not and not both of them are abelian. We will deal with the case where kG is not
semisimple and kH is semisimple, the other case can be proven analogously.
We may assume, that the Sylow p-subgroup Dof Gis cyclic (p= char(k)), since
otherwise kG and hence kCis of infinite representation type and we have nothing
to prove. By standard results from representation theory of finite groups we can
then choose an indecomposable kG-module Msuch that its restriction M↓Dhas a
p-dimensional direct summand on which Dacts by the matrix
S=
1 1
1.
. .
. .
.1
1
.
If now His not abelian we choose, as above, a simple kH-module Nwith dim N≥2
and for λ∈k?we denote by Aλthe same matrix as above. Then MAλ//Nis an
indecomposable representation of C. We should now show that Vλ6∼
=Vµfor λ6=µ.
Suppose that ((bi,j),(ci,j)) is an isomorhpism of representations Vλ→Vµ. Then the
matrix (bi,j) has to commute with the G-action on M, in particular with a matrix
of the shape S0
0?, where ?is any matrix. This gives the conditions b2,1= 0 and
b1,1=b2,2. An endomorphism of Nas a kH-module is just a scalar multiple of the
EI-category algebras with two simple modules 35
identity, i.e. (ci,j) = c·1N. Finally, the following diagram has to commute.
M
(bi,j )
Aλ//N
c
MAµ//N
This yields the conditions b1,2= 0, c=b1,1=b2,2and λ·c=µ·c, which give that
c= 0 and hence Vλ6∼
=Vµ. If the group His abelian we replace the module Nfrom
above by a 2-dimensional kH-module which is the direct sum of two non-isomorphic
one-dimensional simple kH-modules and get the claim by the same computations as
we have just done.
To finish the proof we should consider the case where kG is semisimple and kH is
not and not both are abelian. In this case the argument is the same as in the case we
have treated above, only the computations are a little bit different.
3.7 EI-category algebras with two simple modules
As we have seen, the classification of EI-categories of finite representation type gets very
complicated, even with the assumption that the category has only two objects. The distin-
guishing mark for finite or infinite representation type seems to be the nature of the group
actions of the automorphism groups on the morphism sets between distinct objects. In this
section we will give a classification of all representation-finite EI-category algebras with
only two simple modules. This work is motivated by work of Bongartz and Gabriel [7] who
classified all representation-finite k-categories with two simples and radical of codimension
2. We will compute the Gabriel quiver of a given EI-category algebra with two objects and
then use the list of Bongartz and Gabriel. For the convenience of the reader we collect
those representation-finite and representation-infinite bound path algebras of quivers with
two vertices that we will need later on. For a complete list one may consult [6, page 242].
Every algebra (given by quiver and relations), that is a quotient or dual to a quotient of
an algebra in the following list is of finite representation type.
◦ν//◦ρ
ee(1) 0 = ρ2ν=ρ5,
◦
σ99
ν//◦ρ
ee(2) 0 = νσ =ρν =σt=ρt, t ≥2,
(3) 0 = νσ =σt=ρ2, t ≥2,
(4) νσ =ρ2ν, 0 = σ2=ρ3,
(5) νσ =ρν, 0 = σ2=ρt, t = 2r≥2,
(6) νσ =ρν, 0 = σ2=ρ5,
EI-category algebras with two simple modules 36
◦ν//◦
γ
ooρ
ee(7) 0 = ρ3=γρ =γν =νγ,
(8) 0 = ρ2=νγ =γρν.
As mentioned above, this list is not a complete list of representation-finite k-categories with
two simples, but it is sufficient for our purposes. For every algebra from this list, one can
prove representation-finiteness using covering theory in the way we have explained it in the
beginning of this chapter.
Conversely, the algebras in the following list are all representation-infinite and they are in
some sense minimal with that property. For details we refer to [6].
◦ν//◦ρ
ee(9) 0 = ρ2ν=ρ6,
(10) 0 = ρ3ν=ρ4,
◦
σ99
ν//◦ρ
ee(11) 0 = νσ =σ2=ρ2ν=ρ4,
(12) 0 = νσ =σ3=ρ2ν=ρ3.
Since we want to decide how many simples an EI-category algebra kChas and the simples
of kCare given by the simple modules over the group algebras kAut(x) for x∈Ob C, the
following well-known result from representation theory of finite groups is useful.
Lemma 3.36 (see for example [2]).Let Gbe a finite group and kan algebraically closed
field of characteristic p. Then the number of simple kG-modules equals the number of
conjugacy classes of elements in Gwhose order is not divisible by p.
The elements of a finite group Gof order not divisible by some prime pare called p-
regular, the remaining ones are called p-singular. Every element of Gcan be written as a
product of a p-singular and a p-regular element. Using this fact, we observe the following
easy but important statement.
Corollary 3.37. Let Gbe a finite group such that the group algebra kG has only one simple
module. Then Gis either the trivial group or a p-group for p= char(k).
Proof. From the lemma we know that Ghas only one p-regular element, namely the unit
1G. If pdoes not divide the order of G, then if follows that G={1G}. Suppose that
p| |G|and let x∈Gbe any element. Then we write x=zy as a product of a p-regular
element zand a p-singular element y. By assumption we have z= 1Gand therefore xis
p-singular. Hence, every element has order divisible by pand Gis a p-group.
This corollary together with the list from above implies the following result.
EI-category algebras with two simple modules 37
Theorem 3.38. Let Cbe a skeletal EI-category and kan algebraically closed field such
that kChas two simple modules. Then kCis of finite representation type if and only if it
satisfies one of the following conditions.
(1) Chas one object x, the group Aut(x) = Mor Chas two conjugacy classes of p-regular
elements and the Sylow p-subgroup of Gis cyclic.
(2) Chas two objects xand y, the natural action of the group Aut(x)×Aut(y)on C(x, y)
has at most one orbit, the Sylow p-subgroups of Aut(x)and Aut(y)are cyclic or the
groups are trivial and one of the follwing conditions holds.
(a)C(x, y) = ∅;
(b)|Aut(x)|·|Aut(y)| ≤ 3;
(c)|Aut(x)|·|Aut(y)|= 4 and |C(x, y)| ≤ 2;
(d)|Aut(x)|·|Aut(y)| ≥ 5and |C(x, y)|= 1.
Proof. Chas at most two objects since every object gives at least one simple kC-module.
If Chas only one object, then kCis a group algebra and, if it is representation-finite with
only two simples, it has to satisfy condition (1).
Suppose that Chas two objects xand y. The assumption that kChas two simples implies
that both kAut(x) and kAut(y) have one simple module. Hence, they are either trivial or
p-groups. In case of two trivial automorphism groups, Cis the path category of the Dynkin
quiver A2which is representation-finite. If one of Aut(x) and Aut(y) is a p-group it has
to be representation-finite which means that it is a cyclic p-group. Assume that Aut(x) is
a cyclic p-group and Aut(y) is trivial. Then the computation of the Gabriel-quiver of kC
yields, that kCis isomorphic to the following path algebra with relations or its dual:
◦
α%%β//◦, αm= 0 = βαn, m =pr, n |m.
According to the Bongartz-Gabriel list, this algebra is representation-finite only for the
following values of mand n
•m= 2, n= 1,2;
•m= 3, n= 1,3;
•m= 4, n= 1,2;
•m≥5, n= 1.
Any of these cases fulfils one of the conditions from (2).
Analogously, we assume that both Aut(x) and Aut(y) are cyclic p-groups. In this case kC
is isomorphic to the following path algebra with relations or its dual:
◦
α%%β//◦γ
yy, γt=αm=0=γsβ=βαn, m =pr, t =pl, n |m, s |t.
Again we consult the list of Bongartz-Gabriel and find that (up to duality) only the following
values for m, n, s and tgive a representation-finite algebra:
Two objects and cyclic automorphism groups 38
•m= 2 = sand n= 1, t = 2 or n= 2, t = 1 or n=t= 1;
•m, s ≥3 and n=1=t.
Again this fits into our assertion and no other cases can occur, which finishes the proof.
Remark 3.39. For the representation-finite EI-categories with two simples one can com-
pute the Auslander-Reiten quiver, since they are either hereditary of Dynkin type or occur
in the list of Bongartz-Gabriel. For the latter case one again uses covering-theory to knit
the Auslander-Reiten quiver of the covering and then pushes everything down to the algebra
itself.
3.8 Two objects and cyclic automorphism groups
The last special case of EI-categories we will consider is the one of EI-categories Cwith
two objects xand ysuch that Aut(x) is a cyclic p-group and Aut(y) is a cyclic q-group for
two distinct primes pand q. For this class of EI-categories the characterization of finite
representation type can be obtained in the same way as for EI-category algebras with two
simples modules. We will only treat the case where one group algebra is semisimple and
the other is not. The case with two semisimple group algebras is rather trivial.
Proposition 3.40. Let Cbe an EI-category with two non-isomorphic objects xand ysuch
that G:= Aut(x)is a cylic p-group and H:= Aut(y)is a cyclic q-group for two distinct
primes pand q. Let kbe an algebraically closed field of characteristic p. Then kCis
representation-finite if and only if C(x, y)(or C(y, x)) consists of one G×H-orbit and Cor
its dual satisfies one of the following conditions.
(1) Hacts trivially on C(x, y)and the category C0=C/Aut(y)(’trivialize’ all endomor-
phisms of y) is representation-finite with two simples;
(2) q= 2,C(x, y) = {f1, f2}and Gacts trivially on C(x, y);
(3) |C(x, y)|= 1, i.e. both Gand Hact trivially on C(x, y).
Proof. The proof works in the same fashion as the proof in the last section. Therefore, we
will not provide many details here. First of all, if Hacts trivially, one computes the quiver
of kCto be the union of
Q:◦
α99
β//◦
with |H|−1 isolated points. Hence, the representation type depends on Qand the relations
for which this quiver is representation-finite have been listed in the previous section.
If Hdoes not act trivially we get more arrows from left to right in the quiver of kC. If
this are more than three, the universal cover contains the 4-subspace quiver as a relation-
free part, hence kCis representation-infinite. The remaining representation-finite cases are
easily seen to be conditions (2) and (3).
Two objects and cyclic automorphism groups 39
Remark 3.41. In the case of EI-categories with only two simple representations the repre-
sentation type was really governed by the size of the group algebras. If the product of their
orders is big enough, only trivial actions yield finite type. In contrast to this, we could take
Hto be arbitrarily big and still have that condition (2) from the proposition is satisfied.
4 The finitistic dimension of EI-category
algebras
The finitistic dimensions of a ring Λ provide a measure for the complexity of the module
category of Λ. They are defined as
fin.dim(Λ) = sup {proj.dim M|M∈mod(Λ),proj.dim M < ∞ } ,
Fin.dim(Λ) = sup {proj.dim M|M∈Mod(Λ),proj.dim M < ∞ } .
There are at least two canonical questions that arise in studying these invariants, namely:
Are these two dimensions finite for any ring Λ and do they coincide? For noetherian rings
both questions have to be answered in the negative, but for finite-dimensional algebras
there is no counterexample up to now. In 1960 Bass published the two questions for finite-
dimensional algebras as “problems” and they are nowadays known as the finitistic dimension
conjectures.
The little finitistic dimension, fin.dim, is known to be finite for certain classes of algebras,
for example for algebras with representation dimension at most 3, monomial algebras or
algebras with radical cube zero. One may consult [37] for a survey on this conjecture and
other homological conjectures (not including the result of Igusa and Todorov from [23]
concerning the relation of the representation dimension and the finitistic dimension).
In [26] L¨uck gave an explicit upper bound for the finitistic dimension of an EI-category
algebra. This result seems not to be well-known even among specialists and we will give a
new and fairly elementary proof of this result in the language of representation theory. To
be able to do this, we use results of Xu [33] on the structure of projective resolutions of
modules over EI-category algebras, which we will briefly recall and prove in the beginning
of this chapter.
The main theorem of this chapter is the following.
Theorem 4.1 (L¨uck, [26, Proposition 17.31] ).Let Cbe an EI-category and kCits associated
unital k-algebra. Then the global dimension of kCis finite if and only if |Aut(x)|is invertible
in kfor any x∈Ob Cand
Fin.dim(kC)≤`(C),
where `(C)is the maximal length of a chain of non-isomorphisms in C.
40
The finitistic dimension of EI-category algebras 41
For the proof we need some preparations. The following characterization of projective
resolutions of kC-modules is the main tool to compute the finitistic dimension of EI-category
algebras.
Lemma 4.2 (Xu, [33] Lemma 4.1.1).Let Mbe a kC-module and PMits projective cover.
Then PMis supported on CMand for any M-minimal object xthe module P(x)is the
projective cover of M(x).
Proof. By the characterization of the indecomposable projective modules and by definition
of CMit is clear that the support of PMis contained in CM. Now let xbe an M-minimal
object in C. The full subcategory C≤xis an ideal in Cand its intersection with CMis just
x. Now by Proposition 2.18 the kAut(x)-module PM(x) is a projective module and admits
a surjection onto M(x). Thus, what remains to be shown is the minimality of PM(x). If
PM(x) would not be the projective cover of M(x), then by the universal property of the
projective cover, there would exist projective modules P1and P2such that PM=P1⊕P2
with P1(x) being the projective cover of M(x) and P2(x)6= 0 such that, if π:PM→Mis
the defining essential epimorphism, then π↓C
Dsends P2(x) to zero. The module P2has an
indecomposable projective direct summand P0
2. Now, since P0
2(x) = kAut(x)·efor some
primitive idempotent e∈kAut(x) and P0
2=kC · e, it follows that πsends P0
2to zero. This
is a contradiction to the minimality of PM.
This Lemma gives us the following description of the minimal projective resolution of a
kC-module M.
Corollary 4.3. Let Mbe a kC-module and PMa minimal projective resolution. Then PM
is supported on CMand for any M-minimal x∈Ob Cwe have that PM(x)is a minimal
projective resolution of M(x).
With these preparations we are now in the position to prove Theorem 4.1.
Proof of Theorem 4.1.Let Mbe a kC-module which is of finite projective dimension and
consider a minimal projective resolution
PM: 0 →Pn→Pn−1→ · · · → P1→P0→M→0.
Then PMis supported on CMas we have seen above. Now let xbe an M-minimal object in
C. By Corollary 4.3 PM(x) is a minimal projective resolution of M(x) as kAut(x)-module.
As a module over a group algebra of a finite group, the module M(x) is either projective or
of infinite projective dimension. The latter case is impossible since PMis a finite resolution.
This implies that P1(x) = 0 and P1is supported on CM\ {M-minimal objects}. Applying
this argument inductively to any of the Pi, we get that n≤`(C) and hence the claim.
Finally we will present some examples to illustrate this result.
The finitistic dimension of EI-category algebras 42
Example 4.4. (1) Let Cbe an EI category with only one object. Then kCis the group
algebra of a finite group G. For this case it is well known that the finitistic dimension
is zero and `(C) equals zero as well.
(2) Suppose that Cis the path category of a finite quiver without oriented cycles, then
kCis hereditary and therefore fin.dim(kC) = gl.dim(kC)≤1. Thus, in this case the
given bound is not optimal.
(3) Let
C:X
g77
f//Y1Y
ff
be the EI-category given by the relations g2= 1Xand fg =f. Further, suppose that
kis a field of characteristic 2. Then the indecomposable projective representations of
Care exactly
PX:k2
A66
(1 1) //k(1)
eewhere A= ( 0 1
1 0 ) and
PY:0
(0) 99
(0) //k(1)
ee.
Now it is clear that this algebra is of infinite global dimension since the group algebra
kAut(x) is not semisimple. Its finitistic dimension equals 1 since it is easy to see,
that the representation
M:k2
A66
(0) //0(0)
eewith A= ( 0 1
1 0 )
has projective dimension 1 and there is no module with projective dimension greater
than one (if the projective dimension is finite) by our theorem.
5 The finitistic dimension of algebras with a
directed stratification
In this chapter we introduce the notion of a directed stratification for a finite-dimensional
algebra A. This definition is inspired by the study of EI-category algebras, where we have
already seen that the finitistic dimension is always finite. The proof of this finiteness of
fin.dim for EI-category algebras reduces the problem to the finitistic dimension of the group
algebras of the automorphism groups. This concept will be generalized to algebras with a
directed stratification in the second subsection, where we show that finiteness of the finitis-
tic dimension of such an algebra only depends on the finitistic dimensions of the strata. As
a matter of fact, our reduction technique is a corollary (using induction) of a much more
general theorem of Fossum, Griffith and Reiten in [14], which they obtained in the context
of trivial extensions of abelian categories. We will briefly present their result in the first
section of this chapter. Nevertheless, the approach we will present in terms of represen-
tations of a category which is associated to an algebra with a directed stratification gives
a very convenient combinatorial description of the projective resolutions of the modules.
Furthermore, our reduction technique will prove to be of broader applicability for the com-
putation of concrete examples. This reduction reduces the finitistic dimension conjecture
to the class of algebras, which are minimal in the sense that they do not admit a non-trivial
directed stratification. We will give a combinatorial description of these algebras in terms
of their Gabriel-quiver. Finally, in the last section, we relate our result to other known
results for the finitistic dimension, for example to results of Happel [20], Cline, Parshall
and Scott [9–11] and Huisgen-Zimmermann [36].
5.1 Trivial extensions of abelian categories and finitistic
dimension
As mentioned in the introduction of this chapter, Fossum, Griffith and Reiten developed
the theory of trivial extensions of abelian categories and derived remarkably beautiful and
general results on the finitistic dimension. Let us begin with the definition of trivial exten-
sions.
Definition 5.1. Let Abe an abelian category and F:A → A an additive endofunctor of
A. We construct new additive categories FoAand AnFas follows.
43
Trivial extensions of abelian categories and finitistic dimension 44
The objects of AnFare morphisms α:FA →Afor some A∈Ob Asuch that α◦Fα = 0.
Let α:FA →Aand β:FB →Bbe objects in AnF, then a morphism γ:α→βis a
morphism γ:A→Bsuch that the following diagram commutes.
FA
α
F γ //FB
β
Aγ//B
The composition in AnFis just composition in A.
Similarly we define the category FoAwith objects α:A→FA and morphisms defined
in an analogous way as for AnF.
Example 5.2. (i) Suppose Ris a ring and Man R-bimodule. The category A= Mod R
is abelian and we have at least two natural functors associated with M, namely the
tensor product F=M⊗R−and the internal Hom G= HomR(M, −). These two
functors give possibilities to define trivial extensions of Rby M. One can also define
it to be the ring whose additive group is the direct sum R⊕Mwith multiplication
(r, m)·(r0, m0)=(rr0, mr0+rm0).
Denote this ring by RnM(or MoR). One can show that GoA,AnFand
Mod(RnM) are isomorphic.
(ii) Suppose Aand Bare abelian categories and F:A → B an additive functor. The
category Map(FA,B) is the category whose objects are triples (A, f, B) where A∈
Ob A, B ∈Ob Band f:FA →B. The morphisms are pairs (α, β) of morphisms in
A×Bsuch that the following diagram commutes.
FA
f
F α //FA0
f0
Bβ//B0
The functor Finduces a functor e
F:A×B → A×B by e
F(A, B) = (0, FA) and
e
F(α, β) = (0, Fα) and the categories Map(FA,B) and (A×B)ne
Fare isomorphic.
In the case of the second example Fossum, Griffith and Reiten obtained the following
result.
Theorem 5.3 (Fossum, Griffith, Reiten, [14]).Let Aand Bbe abelian categories with
enough projectives and F:A→Ba right exact functor. Let M:= Map(FA,B). Then the
following inequalities hold.
(1) Fin.dim B ≤ Fin.dim M ≤ 1 + Fin.dim A+ Fin.dim B,
Basic notions and properties 45
(2) If Fis exact, then Fin.dim M ≥ Fin.dim A,
(3) max(gl.dim A,gl.dim B)≤gl.dim M ≤ 1 + gl.dim A+ gl.dim B.
In particular, this result applies to the setting of triangular matrix algebras: Let Rand
Sbe rings and Man R-S-bimodule. The associated triangular matrix algebra is defined
as Λ = R0
M S . This is related to the trivial extensions and the result above, since here we
have Mod Λ ∼
=Map(FMod R, Mod S), where F=M⊗R−. The corollary is the following.
Corollary 5.4. Let R, S, M and Λbe as above, M6= 0. Then
(1) Fin.dim S≤Fin.dim Λ ≤1 + Fin.dim R+ Fin.dim S,
(2) If Mis a flat R-module, then Fin.dim Λ ≥Fin.dim R,
(3) max(gl.dim R, gl.dim S, proj.dimSM+ 1) ≤gl.dim Λ
and gl.dim Λ ≤max(gl.dim R+ proj.dimSM+ 1,gl.dim S).
Now, as another special case of this corollary, we get the following lemma.
Lemma 5.5 (Fossum, Griffith, Reiten [14] and Fuller, Saorin [15]).Let Abe any ring and
e, f two non-zero idempotents in Awith 1 = e+fand eAf = 0. Then
max(gl.dim eAe, gl.dim fAf)≤gl.dim A≤gl.dim eAe + gl.dim fAf + 1,
and the same inequalities hold for Fin.dim.
This result can be used inductively to obtain our result for algebras which admit what
we call a directed stratification. We will see in the following sections that our approach will
have the advantage that we gain new insight into the structure of the projective resolutions.
5.2 Basic notions and properties
In [11] Cline, Parshall and Scott introduced the notion of a stratifying ideal as well as the
notion of stratified and standardly stratified algebras. Since our class of algebras fits into
this framework we will recall their definitions.
Definition 5.6. An ideal Jin an algebra Ais called stratifying if the following conditions
are satisfied.
(i) J=AeA for some idempotent e∈A,
(ii) Multiplication induces an isomorphism Ae ⊗eAe eA ˜→J,
(iii) ToreAe
n(Ae, eA) = 0 for all n > 0.
It was also observed by Cline, Parshall and Scott that an ideal Jin Ais stratifying if and
only if the derived functor i?:D+(A/J)→D+(A) induced by the exact inflation functor
i?: mod A/J →mod Ais a full embedding.
Basic notions and properties 46
Astratification of Aof length nis a chain
0 = J0⊂J1⊂ · · · ⊂ Jn=A
of ideals with the property that Ji/Ji−1is a stratifying ideal in A/Ji−1. The stratification
is called (left-)standard if Ji/Ji−1is projective (as left A/Ji−1-module).
The class of algebras which we will define and deal with in this chapter somehow sits
between stratified algebras and standardly stratified algebras. We will make this precise in
Remark 5.10.
Definition 5.7. Let Abe a finite-dimensional algebra over some field k. Then we say that
Ahas a directed stratification of length nif there exist pairwise orthogonal idempotents
e1, . . . , enin Awith Pn
i=1 ei= 1Asuch that eiAej= 0 for all i<j.
One should note that we do not require the idempotents to be primitive. It is clear that
every algebra admits a directed stratification of length 1 given by its identity element, but
in this case the theory we will develop will give us nothing new.
Example 5.8. (1) Let Cbe a finite and skeletal EI-category with nobjects and A=kC
its category algebra. Then we have a partial order defined on the set of objects of
Cwhich gives us a directed stratification of length ngiven by the idempotents 1Xi,
where Xi, i = 1, . . . , n are the objects of Cand the numbering respects the partial
order.
(2) Let Qbe any finite quiver without oriented cycles and Iany admissible ideal in kQ.
Then A=kQ/I admits a directed stratification of length |Q0|given by the primitive
idempotents εiwith a suitable numbering.
We will see more examples in the last part of this chapter.
As a matter of fact, one can identify the module category of an algebra Awith a directed
stratification given by e1, . . . , enwith the category of representations of a certain category
Awhich we will define now.
Definition 5.9. Let Abe as above. Then the associated category Ais defined as follows.
The objects x1, . . . , xnof Aare in bijective correspondence with the idempotents e1, . . . , en
that define our stratification and the morphisms xi→xjare in bijective correspondence
with a k-basis of ejAei. Under the assumption that Ais a finite-dimensional algebra, the
category Ais finite.
By means of this definition, Ais the category algebra kAof A.
In this setting we have seen, that the categories repk(A) and mod Aare equivalent. For
this reason, we will switch frequently between the concepts of representations and modules
without any further explanation. For instance for an A-module Mwe write M(xi) for its
evaluation at the object xias a functor.
Basic notions and properties 47
Remark 5.10. With the above characterization of Aas the category algebra of A, we may
apply a result of Webb [32, Proposition 2.2] which almost immediately gives that any algebra
with a directed stratification is also stratified in the sense of Cline, Parshall and Scott. Pre-
cisely, if Ahas a directed stratification given by e1, . . . , en, then we take Ji=A(Pn
l=n−iel)A.
These are indeed stratifying ideals by Webbs theorem and therefore give a stratification of
Aof length n. Another result of Webb [32, Theorem 2.5] characterizes the standardly
stratified EI-category algebras to be exactly those, that are given by an EI category Cin
which for every morphism α:x→ythe group StabAut(y)(α) = {θ∈Aut(y)|θα =α}has
order invertible in k. Therefore, we may for instance take the category algebra kCof the
following EI category
C:y
1y99
α//x
1x
h
ZZ, h2= 1x, hα =α.
If khas characteristic 2, then the category algebra kCis not standardly stratified but it
clearly admits a directed stratification given by the idempotents 1x,1y.
Hence, an algebra with a directed stratification is always stratified but in general not
standardly stratified. This is an interesting point since the finitistic dimension conjecture
is known to hold for standardly stratified algebras by work of ´
Agoston, Happel, Luk´acs and
Unger [1] while it is still open for stratified algebras.
We can describe the algebras Awhich do not admit a non-trivial directed stratification
(i.e. of length ≥2) in a very convenient way by conditions that should be rather easy to
check in concrete examples. This characterization is given by the following Proposition.
Proposition 5.11. Let Abe a finite-dimensional k-algebra, where kis any field. Denote
by Qits Gabriel-quiver. Then Adoes admit a non-trivial directed stratification if and only
if there exist disjoint subsets Q0
0and Q00
0of Q0satisfying the following conditions.
(1) Q0=Q0
0∪Q00
0,
(2) For any i∈Q0
0and j∈Q00
0there is no path from ito jin Q.
Proof. First assume that Aadmits a directed stratification. We may without loss of gen-
erality assume that it has length 2 and hence is given by two idempotents eand fwith
eAf = 0. Denote by Γ the Gabriel-quiver of eAe and by Γ0the Gabriel quiver of fAf.
Then we set Q0
0= Γ0and Q00
0= Γ0
0. Since we have 1 = e+fcondition (1) is satisfied and
eAf = 0 implies the second condition.
For the converse implication assume that conditions (1) and (2) hold. Then put e=
Pi∈Q0
0eiand f= 1 −e=Pj∈Q00
0ej. By definition we have 1 = e+fand eAf = 0 follows
from condition (2).
Basic notions and properties 48
Our main result in this chapter will then reduce the finitistic dimension conjecture to
exactly the class of algebras mentioned in the proposition above.
The following result describes the simple and the projective A-modules if Aadmits a
directed stratification. It is completely analogous to the one given by L¨uck in [26] for
EI-category algebras and would again also follow from work of Auslander in [4].
Proposition 5.12. Let Abe an algebra with a directed stratification given by idempotents
e1, . . . , en. Then for every simple A-module Sone has eiS6= 0 for exactly one ei. In other
words as a representation Sis supported on exactly one object Xiand S(Xi)is a simple
eiAei-module. Their projective covers (i.e. all the indecomposable projective A-modules) are
of the form Ae for some primitive idempotent e∈eiAeifor some i∈ { 1, . . . , n }.
Proof. The assertion on the indecomposable projective modules is obvious since 1 = Pn
i=1 ei.
Then we decompose every eiand infer that the summands have to be in eiAei.
Let Sbe a simple A-module and choose eiwith eiS6= 0. Then consider the submodule U
of Sgenerated by eiS. Since the idempotents e1, . . . , endefine a directed stratification we
have ejU= 0 whenever j < i. Let Nbe the submodule of Sgenerated by all the ejSwith
j > i. Again, since Ahas a directed stratification, it follows that eiN= 0, which, together
with the fact that Sis simple, implies that N= 0 and therefore ejS= 0 for every j6=i. If
eiSwould not be a simple eiAei-module, then S=eiSwould have a non-trivial submodule
since it is itself an eiAei-module.
With this Proposition it is natural to use the same notation as for EI-category algebras
and denote the simple A-modules by Sx,V where xis an object of Aand Va simple exAex-
module (here the idempotent excorresponds to the object x) and to let Px,V denote the
projective cover of Sx,V .
The main tool to understand the structure of projective resolutions of modules over
algebras with a directed stratifications will be the use of restriction functors as introduced
in Chapter 2. We will use the same definitions as for EI-category algebras like ideals etc.
and see that the whole theory can be carried over with slightly more complicated proofs.
The following definition is completely analogous to the one for EI-categories.
Definition 5.13. Let Abe an algebra with a directed stratification and let Abe the
associated finite category.
(1) Let xbe an object in A. Then we define A≤xto be the full subcategory of Aconsisting
of all objects y∈Ob Awith A(y, x)6=∅. Similarly we define A≥x.
(2) An ideal in Ais a full subcategory Bof Asuch that for any object xin Bwe have
that A≤x⊆ B.
(3) Let Mbe an A-module. The M-minimal objects are the objects x∈Ob Asuch that
M(x)6=∅and for any y∈Ob Awith A(y, x)6=∅one has M(y) = 0.
Projective resolutions and the main result 49
(4) Let Magain be an A-module. We put AMto be the full subcategory consisting of
all y∈Ob Awith A(x, y)6=∅for some M-minimal object xin A.
Proposition 5.14. Let Abe an algebra with a directed stratification, Athe associated
category and Ban ideal in A. Then the restriction ↓A
Bpreserves projectives.
Proof. We construct an exact right adjoint F: rep B → rep Aof the restriction functor in
the following way. For M ∈ rep Band any morphism f:M→N in rep Blet
FM(x) = (M(x) if x∈Ob B,
0 otherwise, F(f)x=(fxif x∈Ob B,
0 otherwise.
This defines an exact functor. Now let M ∈ rep Aand N ∈ rep B. Then we define a
morphism Ψ : Homrep B(M ↓A
B,N)→Homrep A(M, FN) via
Ψ(f)x=(fxfor x∈Ob B,
0 otherwise.
Thanks to Bbeing an ideal, this gives a k-linear map which is easily seen to be an isomor-
phism. Therefore, we get that Fis the desired exact right adjoint of the restriction.
5.3 Projective resolutions and the main result
In this section we will analyze the structure of projective resolutions for modules over
algebras with a directed stratification. It will turn out, that they can be described in the
same fashion as the ones for modules over EI-category algebras, only the proofs become a
little bit more involved.
Theorem 5.15. Let Abe an algebra with a directed stratification and Athe associated
category. Let Mbe an A-module and P=PMits projective cover. Then Pis supported on
AMand for any M-minimal object xin Athe module P(x)is a projective cover of M(x)
as an exAex-module.
Proof. (i) Clearly, we have P=Ly,U Py,U for some objects yin Aand simple eyAey-
modules U. What we have to show is that no y0with y0/∈ AMappears in that
direct sum. Let us assume the contrary and suppose that there is an object y0with
A(y0, x)6= 0 for an M-minimal object xthat appears in the direct sum decomposition
of P. Then, for any xwith A(y0, x)6= 0 and any f∈ A(y0, x) the following diagram
Projective resolutions and the main result 50
has to commute
Py0,U0(x)πx//M(x)
Py0,U0(y0)
Py0,U0(f)
OO
πy0
//M(y0) = 0,
OO
,
where π:P→Mis the defining essential epimorphism. By the characterization
of the projective A-modules we have PfIm Py0,U0(f) = Py0,U0(x), which gives that
πx= 0 (for any such x). This is a contradiction to the minimality of Pand we have
proven the first assertion.
(ii) Since A≤xis an ideal and A≤x∩ AM={x}, it follows from Proposition 5.14 that
P(x) is projective. Thus, we only have to show that P(x) is the projective cover of
M(x).
Suppose P(x) would not be the projective cover of M(x). Then P(x) = Q0⊕Q00 where
Q0, Q00 are projective and Q0is the projective cover of M(x), whereas πx(Q00) = 0.
Since xis M-minimal, we have that Q0=P0(x) and Q00 =P00(x) for some projective
A-modules P0and P00. Denote again by πthe defining essential epimorphism P→M.
Now, using that Pis supported on AMand a similar diagram as in the first part of
our proof, we get that π(P00) = 0 which contradicts the minimality of P.
Corollary 5.16. Let Abe an algebra with a directed stratification and Athe associated
category. Suppose that Mis an A-module and Pa minimal projective resolution of M.
Then for any M-minimal object xin Awe have that P(x)is a minimal projective resolution
of M(x)as an exAexmodule.
With this characterization of projective resolutions of A-modules we get the following
theorem, which, roughly speaking, states that the finitistic dimension of an algebra with a
directed stratification is determined by the finitistic dimension of the strata.
Theorem 5.17. Let Abe an algebra with a directed stratification and Athe associated
category.
(1) Ahas finite finitistic dimension if and only if exAexhas finite finitistic dimension for
any object xof A. In this case fin.dim A≤Px∈Ob Afin.dim exAex+|Ob A| − 1.
(2) Ais of finite global dimension if and only if exAexis of finite global dimension for
any object xof A. In this case gl.dim A≤Px∈Ob Agl.dim exAex+|Ob A| − 1.
Proof. We only prove part (1). The proof of (2) is completely analogous.
Let e1, . . . , enbe the idempotents, that give the directed stratification of A. First suppose
that the algebra eiAeihas infinite finitistic dimension for some i= 1, . . . , n. Then there
exists an indecomposable eiAei-module Nof projective dimension at least dfor any natural
Projective resolutions and the main result 51
number d≥1. Let Pbe a minimal projective resolution of Nas an A-module. By definition,
the object xof Acorresponding to eiis N-minimal. Hence, P(x) is a minimal projective
resolution of Nas an eiAei-module, which therefore is of length at least d. Thus, we infer
that Nregarded as an A-module has projective dimension at least dand Ahas infinite
finitistic dimension.
Now assume that for i= 1, . . . , n every eiAeihas finite finitistic dimension and denote
by xithe object in Acorresponding to ei. Let Mbe an A-module of finite projective
dimension. We consider a minimal projective resolution of M:
P: 0 →Pm→Pm−1→ · · · → P1→P0→M→0.
The object x1∈Ob Ais M-minimal for every A-module M. Therefore, P(x1) is a
minimal projective resolution of M(x1) as e1Ae1-module. Hence, Pd(x1) = 0 for all
d > fin.dim e1Ae1. Denote by sthe largest integer for which Ps(x)6= 0. Then, for
any d>s, the module Pdis supported on Ob A\{x1}and the object x2is N-minimal,
where N= ker(Ps→Ps−1). With the same argument as above we infer that Ps+d(x2)=0
for all d > fin.dim e2Ae2. Now the claimed inequality follows by induction.
The following corollary is equivalent to the theorem from above.
Corollary 5.18. Let Abe a finite-dimensional k-algebra with a directed stratification of
length 2given by idempotents e, f ∈A(i.e. 1 = e+fand eAf = 0). Then the following
statements hold.
(1) Ahas finite finitistic dimension if and only if eAe and fAf have finite finitistic
dimension. In this case fin.dim A≤fin.dim eAe + fin.dim fAf + 1 .
(2) Ahas finite global dimension if and only if eAe and fAf have finite global dimension.
In this case gl.dim A≤gl.dim eAe + gl.dim fAf + 1.
Remark 5.19. (i) As we have seen in the first part of this chapter, Corollary 5.18 has
already been obtained by Fossum, Griffith and Reiten and it also implies our theorem
by using induction. Nevertheless, our proof is different and provides us with inter-
esting new information about the structure of projective resolutions of A-modules, if
Ahas a directed stratification. Furthermore we will see that the iterated version of
Corollary 5.18 can easily be applied in examples that have not been studied so far.
(ii) Another immediate corollary of the theorem is the following well-known fact: Let Q
be a finite quiver without oriented cycles and Iany admissible ideal in kQ. Then
the algebra kQ/I has finite global dimension. The theorem applies to this setting
in the way that we take the natural directed stratification given by the primitive
idempotents εiin a suitable numbering. Then every stratum is just the ground field
kwhich has finite global dimension.
One interpretation of our result from above is to understand it as a technique to reduce
Relation to known results and examples 52
the finitistic dimension conjecture to a smaller class of algebras, namely those that do
not admit a non-trivial directed stratification. These algebras have been characterized in
Proposition 5.11.
5.4 Relation to known results and examples
5.4.1 Relation to recollements
In [20] Happel developed a reduction technique for the finitistic dimension conjecture (and
other homological conjectures) using recollements of the bounded derived categories. To
explain the relation of this result to our situation we first recall the definition of a recolle-
ment.
Definition 5.20. Let C,C0and C00 be triangulated categories. Then a recollement of C
relative to C0and C00 is given by six exact functors
C0i?=i!//C
i?
oo
i!
ooj?=j!//C00,
j?
oo
j!
oo
satisfying the following conditions.
(R1) (i?, i?),(i!, i!),(j!, j!) and (j?, j?) are adjoint pairs of exact functors,
(R2) j?i?= 0,
(R3) i?i?∼
=id,id ∼
=i!i!, j?j?∼
=id and id ∼
=j!j!,
(R4) for any X∈ C there exist triangles
j!j!X→X→i?i?X→j!j!X[1]
i!i!X→X→j?j?X→i!i!X[1].
With this concept of recollements Happel obtained the following result.
Theorem 5.21 (Happel, [20]).Let Abe a finite-dimensional algebra and assume that
Db(A)has a recollement relative to Db(A0)and Db(A00)for some finite-dimensional algebras
A0and A00. Then fin.dim A < ∞if and only if fin.dim A0<∞and fin.dim A00 <∞.
The structure of this result is similar to that of Corollary 5.18. Therefore, it is a natural
question to ask if the two reduction techniques are equivalent. To see that this is not
the case we have to translate the setting of algebras with a directed stratification into the
language of triangulated categories. Clearly, the triangulated categories that will appear
are the (bounded) derived module categories of the algebra itself and the algebras eiAei.
Relation to known results and examples 53
Let Abe a finite-dimensional algebra with a directed stratification of length 2 given by
idempotents eand fwith eAf = 0. Then J:= AfA is a stratifying ideal of A. With
B=A/J it is clear that B∼
=eAe. Following [11], we have partial recollement diagrams
D+(eAe)
i?//D+(A)
i!
oo
j?
//DD+(fAf)
j?
oo
D−(eAe)
i?//D−(A)
i?
oo
j?//D−(fAf).
j!
oo
If all the algebras involved have finite global dimension we get a full recollement of the
bounded derived categories. Cline, Parshall and Scott also proved that Ahas finite global
dimension if and only if both fAf and eAe have finite global dimension in this situation,
which is exactly the result we obtained with our characterization of the projective resolu-
tions of modules for A. The situation for the finitistic dimension is more complicated. The
following theorem provides a criterion for the above diagrams to become full recollement
diagrams for the bounded derived categories.
Theorem 5.22 (Cline, Parshall, Scott [9]).Let Abe a ring, Jan ideal in Aand B=A/J.
The functor i!=i?:Db(B)→Db(A)has a right adjoint i!satisfying i!i!= idDb(B)if and
only if
(1) Extn
A(BA, F)=0for all n > 0and all free right B-modules Fand
(2) proj.dim BA<∞.
We will now present an example of an algebra with a directed stratification in which
condition (b) is not satisfied. Consider the following EI category (and its category algebra
A=kC) in characteristic 2:
C:y
1y
g
XX
α//x
1x
h
ZZ, g2= 1y, h2= 1x, hα =α=αg.
In this example we choose f= 1xand e= 1y. The stratifying ideal is J=AfA =
h1x, h, αik. Thus, the algebra eAe is just the group algebra kAut(y). As an A-module, or
as a representation of C, the A-module Bis
k2
(1)
M
WW
0//0
0
0
YY,where M= ( 0 1
1 0 ).
Relation to known results and examples 54
The projective cover P1=PBof Bas an Amodule is
k2
(1)
M
WW
0//k
(1)
(1)
XX,again with M= ( 0 1
1 0 ).
The kernel Kof the essential epimorphism P1→Bis the representation
0
(0)
(0)
YY
0//k
(1)
(1)
XX
and we denote by P2the projective module
0
(0)
(0)
YY
0//k2
1
M
WW,with M= ( 0 1
1 0 ).
Finally, the minimal projective resolution of Bas an Amodule looks as follows
· · · → P2→P2→P2→P1→B→0.
In particular, the projective dimension of Bas an Amodule is infinite. Therefore we do
not have a recollement of Arelative to eAe and fAf, which means that we are not in the
position to apply Happels result. However, with our result on algebras with a directed
stratification it is obvious that Ahas finite finitistic dimension since the group algebras
kAut(x) = fAf and kAut(y) = eAe have this property.
5.4.2 A non-trivial example
To prove the finiteness of the finitistic dimension of an algebra Ain concrete examples,
there are at the moment (to the best knowledge of the author) four important classes of
algebras where the finiteness of fin.dim is known. First of all for algebras with radical
cube zero and for monomial relation algebras Huisgen-Zimmermann showed that fin.dim is
always finite. Those classes are very easy to detect, if the algebra in question is given by its
quiver and relations. Igusa and Todorov proved that the finitistic dimension of an algebra
Λ is finite if Λ has representation dimension at most 3. Here the representation dimension
Relation to known results and examples 55
of Λ is defined as
rep.dim Λ = inf {gl.dim End(M)|Mis a generator cogenerator in mod Λ }.
This dimension is very hard to calculate in general and therefore this result is often difficult
to apply in concrete examples or it is at least not applicable without tedious calculations.
For instance, it is not known how to calculate (or give an upper bound) for the representation
dimension of a group algebra of an arbitrary finite group.
The fourth class of algebras where the finitistic dimension conjecture is known to hold true
is the class of algebras Λ for which the category P∞(mod Λ) of modules of finite projective
dimension is contravariantly finite in mod Λ. The notion of contravariant finiteness goes
back to Auslander and Smalø while the result for the finitistic dimension is due to Auslander
and Reiten. For the convenience of the reader we recall the definition here.
Definition 5.23. A full subcategory Aof mod Λ is called contravariantly finite if each
module Min mod Λ has an A-approximation in the following sense: there exists a homo-
morphism f:A→Mfor some A∈ A such that every g∈HomΛ(B, M) with B∈ A
factors through f. This property may be illustrated by the following diagram.
Af//M
B
g
OO
∃
ff
If the category P∞(mod Λ) is contravariantly finite in mod Λ, then Auslander and Reiten
proved that fin.dim Λ <∞. The property of P∞being contravariantly finite has been
investigated by Happel and Huisgen-Zimmermann in [21] and they observed that this prop-
erty is rather ’unstable’. They also give an elementary criterion for P∞(mod Λ) not being
contravariantly finite in mod Λ for a bound path algebra Λ = kQ/I. To state the theorem
we need the following notation. Let Λ = kQ/I be a bound path algebra and p:e1→e2
some path in kQ. If Mis a Λ-module we write fp:e1M→e2Mfor the linear map cor-
responding to p. With this convention the theorem of Happel and Huisgen-Zimmermann
goes as follows.
Theorem 5.24 ( [21]).Let Λ = kQ/I. Suppose that e1and e2are vertices of Qand
p, q ∈kQ \Ipaths from e1to e2with Λp∩Λq= 0. Moreover, suppose that
(1) the cyclic module Λ(p, q)generated by (p, q)∈Λ2has finite projective dimension and
that one of the following conditions is satisfied: either
(2) whenever M∈ P∞(mod Λ), then fp(e1M\rad(Λ)M)∩fq(e1rad(Λ)M) = ∅; or,
(20)whenever M∈ P∞(mod Λ), then Ker(fp)⊆Ker(fq)and Ker(fp)⊆e1rad(Λ)M.
An easily recognizable situation in which the hypothesis of the criterion as well as both
Relation to known results and examples 56
conditions are satisfied is the following one: pis an arrow e1→e2,q∈kQ \Ia path
from e1to e2of positive length, different from psuch that rad(Λ)p=0=qrad(Λ), and
proj.dim(Λq)<∞whereas proj.dim(Λe2/rad(Λ)e2) = ∞.
After this definitions and preparations, we can now present an example to which none of
the above methods applies (perhaps besides calculation of rep.dim).
Example 5.25. Let Qbe the following quiver:
3
δ2
>
>
>
>
>
>
>
1
α//
β//2
γ
δ1
@@
ε1
>
>
>
>
>
>
>5
ρ
qq
4
ε2
@@
Then let A=kQ/I where Iis the ideal generated by
γ2, γβ, ε1β, δ1β, ε1γ, δ1γ, ε2ε1−δ2δ1, ρε2, ρδ2, ρ5.
Ais by definition not monomial and does not satisfy rad3(A) = 0. Furthermore this algebra
doesn’t have the property that P∞(mod A) is contravariantly finite in mod A. To see this,
we put q=αand p=β. Then rad(A)p= 0 = qrad(A) and Aq =Aα =Ae2is a
projective module. The module Ae2/rad(A)e2is of infinite projective dimension, since
it is (as a representation) non-zero only on the vertex 2 and as an e2Ae2-module it has
infinite projective dimension. Here we use that the idempotents e1, . . . , e5give a directed
stratification of Aand our results from the previous section.
The finiteness of fin.dim for Afollows on the one hand very easily from the fact that
e1, . . . , e5give a directed stratification of Aand on the other hand by considering e=
e1+· · · +e4and f=e5. These two idempotents give a directed stratification of Aof
length 2 and eAe as well as fAf have finite finitistic dimension, since they are monomial
relation algebras. Here we notice that the iterated version of the result of Fossum, Griffith
and Reiten, which we deduced with our methods, gives the finiteness of fin.dim Aalmost
immediately, while for a directed stratification of length two one has to be more careful and
use non-trivial results of other authors.
It is also interesting to point out that, for this example, there is no reasonable recollement-
situation in sight which would give us the finiteness of fin.dim Aimmediately. For instance,
if we take eand fas above, then the stratifying ideal is J=AfA and B:= A/J =eAe. In
this case we don not get a recollement of Db(A) relative to Db(eAe) and Db(fAf) because
Bas an A-module has the simple module Ae2/rad(A)e2as a summand and is therefore of
infinite projective dimension.
To sum up, we have constructed an example of an algebra Awhich is not monomial, does
Relation to known results and examples 57
not satisfy rad3(A) = 0, does not have the property that P∞(mod A) is contravariantly
finite in mod Aand does not give a recollement-situation which gives the finiteness of
fin.dim. Nevertheless, the finiteness of fin.dim Afollows immediately from our theorem
because we have a directed stratification with strata that are of finite finitistic dimension.
What we have not done is to calculate the representation dimension of Ain this example,
but in general the calculation of rep.dim is a very hard task. Here it is (at least for the
author) not obvious that Ahas representation dimension at most 3 and if it would be the
case, then one could imagine how one could construct arbitrarily complicated examples to
which our theorem applies and where one cannot calculate the representation dimension.
6 Outlook
In this final chapter we want to discuss some problems which we think are interesting for
further investigation and which we could not solve within this thesis. We also mention
techniques that might be used to attack the open problems.
The most interesting question that is still not answered is the following: What are the
representation-finite EI-category algebras?
We proved a necessary criterion for finite representation type, namely we showed that
if kCis representation-finite then, for any two objects xand yof Cfor which Aut(x) and
Aut(y) are non-trivial, the group Aut(x)×Aut(y) does not act freely on C(x, y). A sufficient
criterion for finite representation type, which may be applied to a large class of EI-category
algebras, is still missing. However, the computation of various examples and the results
for special classes of EI-categories that have been presented in this thesis lead us to the
following conjecture.
Conjecture 6.1. Let Cbe a finite EI-category and kan algebraically closed field such that
the group algebra kAut(x)is representation-finite for any object xof Cand with |C(x, y)| ≤ 1
for any two distinct objects xand yof C. Then the category algebra kCis representation-
finite.
EI-category algebras which satisfy the conditions of this conjecture are the easiest EI-
categories which are not group algebras or incidence algebras.
For the group algebra of a finite group Git is well-known that the group algebra kG is
semisimple if and only if char(k) does not divide |G|. In other words, the representation
theory gets more complicated if the characteristic of the ground field divides the group order.
Therefore, it is natural to expect a similar behaviour for EI-category algebras. An evidence
for this expectation is the fact that an EI-category algebra has finite global dimension if
and only if the characteristic of the field does not divide any of |Aut(x)|for x∈Ob C.
Again, together with our experience from the examples we computed, one may conjecture
the following:
Conjecture 6.2. Let Cbe a finite EI-category and kan algebraically closed field whose
characteristic does not divide any of |Aut(x)|for x∈Ob C. If the algebra kChas infinite
representation type, then k0Chas infinite representation type for any algebraically closed
field k0.
58
Outlook 59
Roughly speaking, this conjecture states that, if every group algebra kAut(x) for x∈Ob C
is semisimple, then we have the smallest number of indecomposable modules over kC.
Unfortunately, there are, up to now, only few techniques that one can apply to study
representations of EI-categories. Xu’s theory of vertices and sources looks promising for a
characterization of finite representation type at first sight. The problem is that one cannot
easily imitate the definition of a defect group for a finite EI-category, because there is no
analogue for the conjugation in a finite group. Nevertheless, it may be worth a try to work
in this direction to derive some interesting results. Another idea to use Xu’s results is to
drop the assumption that one always restricts to full subcategories of an EI-category. The
problem that arises here is that the computation of the induction is rather difficult if the
subcategory is not full.
For quivers with relations the construction of universal covers is often an easy way to de-
cide whether the given algebra is representation-finite or not. Covering theory as developed
by Bongartz and Gabriel works for arbitrary representation-finite algebras in principle, but
for the treatment of examples which are not given by quivers with relations it is often not
applicable. For instance, it is not known how one constructs the universal cover of a finite
group in general. It might therefore be interesting to develop a ’new’ covering theory for
EI-categories or more generally for small categories.
Finally, there is nothing known about the structure of the Auslander-Reiten quiver of an
EI-category algebra. Recently, several authors proved results on the shape of certain com-
ponents in the Auslander-Reiten quiver of a selfinjective algebra. Since the representation
theory of EI-categories is somehow related to selfinjective algebras one might expect similar
results in this framework.
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