# Foundation of the Representation Theory of Artin Algebras

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Contemporary Mathematics Volume 00, XXXX

Foundation of the Representation Theory of Artin Algebras,

Using the Gabriel-Roiter Measure

Claus Michael Ringel

Abstract. These notes are devoted to a single invariant, the Gabriel-Roiter measure of ﬁnite length modules: this invariant was introduced by Gabriel (under the name ‘Roiter measure’) in 1972 in order to give a combinatorial interpretation of the induction scheme used by Roiter in his 1968 proof of the ﬁrst Brauer-Thrall conjecture. It is strange that this invariant (and Roiter’s proof itself) was forgotten in the meantime. One explanation may be that both Roiter and Gabriel pretend that their considerations are restricted to algebras of bounded representation type which are shown to be of ﬁnite representation type, thus restricted to algebras of ﬁnite representation type. But, as we are going to show, this invariant is of special interest when dealing with algebras of inﬁnite representation type! And there may be a second explanation: in the early seventies, it was possible to calculate this invariant only for few examples, whereas nowadays there is a wealth of methods available. Looking at such examples, we are convinced that the Gabriel-Roiter measure has to be considered as a very important invariant and that it can be used to lay the foundation of the representation theory of artin algebras.

The notes are based on lectures given at Hirosaki (2003), Queretaro (2004), Hangzhou and Beijing (2005) looking at some basic questions in the representation theory of artin algebras. The main emphasis in the Hirosaki lectures was to develop a direct approach to the representation theory of artin algebras, using the GabrielRoiter measure, independent of Auslander-Reiten theory. Indeed, the GabrielRoiter measure should be considered as being elementary: only individual modules are considered (as abelian groups with a set of prescribed endomorphisms), whereas the notions of Auslander-Reiten theory relate from the beginning the given module to the whole module category. In the later lecture series (at Queretaro, Hangzhou and Beijing), some useful connections between the Gabriel-Roiter measure and Auslander-Reiten theory have been included. In the present notes we do not hesitate to mingle the diﬀerent approaches whenever it seems suitable.

2000 Mathematics Subject Classiﬁcation. Primary: 16G10, 16G20, 16G60, 16G70 Secondary: 16D10, 16D70..

Key words and phrases. Brauer-Thrall conjectures. Artin algebras. Finite dimensional associative algebras. Quivers and their representations. Representation type. Large indecomposable modules. Direct sums of ﬁnitely generated modules. Relative Σ-injectivity..

c XXXX American Mathematical Society

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CLAUS MICHAEL RINGEL

The presentation centers on some old topics in representation theory, and we hope to convince the reader of the usefulness of the Gabriel-Roiter measure when dealing with results of the following kind:

• The ﬁrst Brauer-Thrall conjecture established by Roiter and Auslander, which asserts that inﬁnite representation type implies unbounded representation type.

• Any module for an artin algebra of ﬁnite representation type is a direct sum of ﬁnitely generated modules.

• Auslander’s theorem asserting that artin algebras of inﬁnite representation type have indecomposable modules which are not ﬁnitely generated.

Of course, it is not surprising that the Gabriel-Roiter measure is useful when dealing with the Brauer-Thrall conjectures, since it was introduced in this context. But it seems that the relationship to the other two topics was not realized before.

This report has three parts. The ﬁrst part develops general properties of the Gabriel-Roiter measure, the second part deals with what we call the “take-oﬀ part” of the module category of an artin algebra, the third one with the “landing part”. With respect to the Gabriel-Roiter measure, the category mod Λ for any artin algebra Λ is divided into three diﬀerent subcategories: the take-oﬀ part, the central part and the landing part. Of highest concern should be the central part — however, at present, there is not yet a single result available in print concerning the central part.

We do not touch at all the dual construction, the Gabriel-Roiter “comeasure”. Concerning the Gabriel-Roiter comeasure, and the interrelation between the measure and the comeasure, we propose to look at [R5]: the rhombic picture seems to provide a fascinating description of the module category, but also here we have to wait for future research in order to get a full understanding of what is going on.

The core of the notes is self-contained, but for some of the proofs we refer to [R5]. There are several remarks and examples (usually marked by a star: Example*, Remark*) which involve notions which are unexplained. They may be helpful for some (or most) of the readers, but can be skipped in a ﬁrst reading.

Notation: N1 = {1, 2, . . . } denotes the positive natural numbers. Given two integers a ≤ b, let [a, b] be the set of integers z with a ≤ z ≤ b.

0. Preliminaries

Let R be an arbitrary ring, we consider left R-modules and call them just modules. We will assume some basic results from module theory and homological algebra. All our consideration are related to ﬁnite length modules: we only deal with modules which are unions of modules of ﬁnite length. Recall that ﬁnite length modules are modules which are both artinian and noetherian; they have a ﬁnite chain of submodules which cannot be reﬁned (such a chain is called a composition series). The Jordan-Ho¨lder theorem asserts that the length of such a chain depends only on the module, it is called the length of the module. We denote the length of a module M by |M |. A module M is said to be indecomposable, provided it is non-zero and cannot be written as a direct sum M = M1 ⊕ M2 of two non-zero modules (a direct sum decomposition M = M1 ⊕ M2 is given by two submodules M1, M2 of M such that M1 + M2 = M and M1 ∩ M2 = 0.) It is obvious that any ﬁnite length module is a ﬁnite direct sum of indecomposable modules (and they are again of ﬁnite length). The Fitting Lemma tells us that the endomorphism ring of

FOUNDATION, USING THE GABRIEL-ROITER MEASURE

3

an indecomposable module of ﬁnite length is a local ring (the set of non-invertible endomorphisms forms an ideal), and this implies the Krull-Remak-Schmidt theorem: A decomposition of a ﬁnite length module into indecomposable modules is unique up to isomorphism.

0.1. Questions. Given a natural number d, are there indecomposable modules of length d? And how many such modules are there? In particular: are there inﬁnitely many isomorphism classes of indecomposable modules of length d? Let a(d) be the number of isomorphism classes of indecomposable modules of length d (recall: we mean left R-modules, the R being ﬁxed; of course we could write aR(d) in order to specify the ring). Thus, we consider the questions:

• When is a(d) = 0 ? • When is a(d) = ∞?

Example* 1. Let R be a discrete valuation ring, for example R = k[[T ]], the

power series ring in one variable, or R = Z(p) the ring of p-adic integers. The maximal ideal of R is a principal ideal, say generated by the element π. The module R/Rπd is indecomposable and of length d, and is, up to isomorphism the

only indecomposable module of length d. Thus we see that a(d) = 1 for any d ∈ N1.

Example 2. Let k be a ﬁeld and let R = Tn(k) be the ring of upper triangular

(n × n)-matrices with coeﬃcients in k. Then a(d) = n − d + 1, for 1 ≤ d ≤ n and 0

for n < d, thus altogether we obtain

n+1 2

isomorphism classes. Using “quivers” (as

we will often do), we can interprete R = Tn(k) as the path algebra of the linearly

ordered quiver of type An, with vertices 1 → 2 → · · · → n, and the indecomposables

of length d correspond bijectively to the intervals [i, i + d − 1] with 1 ≤ i ≤ n − d.

Remark*. More generally, Gabriel’s theorem asserts that the following quivers are of ﬁnite representation type (this means: d a(d) < ∞) and that the number of indecomposables is as follows (and in particular independent of the orientation):

....

type A D E E E .... ..

n

n

6

7

8

....

..............................................................................................................................................................................................................................................................................................................................

....

.... ..

n+1

a(d) n(n − 1) 36 63 120 d

.... ...

2

.

The types ∆ which occur here are just the simply laced Dynkin diagrams (which occur in Lie theory, but also in many other parts of mathematics); and the numbers below are known as the number of positive roots of ∆: in fact, there is a natural bijection between the indecomposable representations and the positive roots.

Example 3. The Kronecker quiver: it has two vertices x, y and two arrows, say going from x to y. Then one knows that a(n) = 2 for n odd, and a(n) ≥ 3 for n even. For n even, the number a(n) depends on the base ﬁeld k. In case we deal with representations over an inﬁnite ﬁeld k, then a(n) = ∞ for n even. (Here we have an example of what has been called strongly unbounded representation type: there are inﬁnitely many natural numbers d1 < d2 < . . . such that a(di) = ∞ for all i.)

The main problem of representation theory is to ﬁnd invariants for modules and to describe the isomorphism classes of all the indecomposable modules for which such an invariant takes a ﬁxed value. A typical such invariant is the length of a

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CLAUS MICHAEL RINGEL

module: the simple modules are those of length 1 (and there is just a ﬁnite number of such modules), the information concerning the indecomposable modules of length 2 is stored in the quiver (in case we deal with a ﬁnite dimensional algebra over some algebraically closed ﬁeld) or the “species” of Λ. Given any invariant γ, as a ﬁrst question one may look for values of ﬁnite type: these are those values v such that there are only ﬁnitely many isomorphisms classes of indecomposable modules M with γ(M ) = v. The invariant to be discussed here is the Gabriel-Roiter measure.

In most parts of these lectures we will assume that R is an artin algebra. This means that the center z(R) of R is artinian and that R is a ﬁnitely generated z(R)-module. Typical examples are the ﬁnite-dimensional k-algebras, where k is a ﬁeld, but also all ﬁnite rings. Some elementary properties of the Gabriel-Roiter measure hold for ﬁnite length modules over an arbitrary ring R, or, more generally, for arbitrary ﬁnite length categories (a ﬁnite length category is an abelian category such than any object is both artinian and noetherian, thus has a composition series; for any ring R, the category of all ﬁnite length R-modules is of course a length category).

0.2. The Brauer-Thrall conjectures. As we have mentioned the GabrielRoiter measure was introduced by Gabriel in [G] in order to clarify the intricate induction scheme used by Roiter [Ro] in his proof of the ﬁrst Brauer-Thrall conjecture. Roiter’s proof of this conjecture marks the beginning of the new representation theory of ﬁnite dimensional algebras. Let us recall the precise statement of both Brauer-Thrall conjectures.

BTh 1. Bounded representation type (that means a(d) = 0 for large d) implies ﬁnite representation type. (Under the assumption that a(1) < ∞.)

If a(d) < ∞ for all d, then bounded representation type implies ﬁnite representation type, trivially. Thus we can reformulate BTh 1 as follows:

BTh 1′ If a(d) = ∞ for some d, then a(d) = 0 for inﬁnitely many d.

BTh 2. Unbounded representation type implies strongly unbounded representation type. (Under the assumption that we deal with ﬁnite dimensional algebras over an inﬁnite ﬁeld, or more generally, with inﬁnite (and connected) artin algebras.)

The assumption is quite essential, since there are plenty of ﬁnite length categories which are of unbounded type, but not of strongly unbounded type. Example 1 above is a typical such example.

BTh 1 is true for R a semiprimary ring: for ﬁnite dimensional algebras this was shown by Roiter, for left artinian rings by Auslander. Note: Bounded representation type means that every simple module has a relative projective cover and a relative injective envelope in the category of ﬁnite length modules; thus a semiprimary ring of bounded representation type is necessarily left artinian.

Remark*. One should be aware of the V -rings as constructed by Cozzens: here all simple modules are injective, thus also relative projective in the category of ﬁnite length modules. These rings satisfy a(d) = 0 for d ≥ 2, however they are not semiprimary. Note that there are such examples with a(1) being ﬁnite, as well as examples with a(1) being inﬁnite.

FOUNDATION, USING THE GABRIEL-ROITER MEASURE

5

0.3. Smalø [S] has shown the following:

Theorem. Let R be an artin algebra with a(d) = ∞ for some d, then a(d′) = ∞ for some d′ > d.

We see that in order to show that an artin algebra is of strongly unbounded representation type, it is suﬃcient to show that a(d) = ∞ for some d.

Thus we can reformulate BTh 2 as follows:

BTh 2′. Let R be an inﬁnite artin algebra. If a(d) = 0 for inﬁnitely many d, then a(d) = ∞ for some d.

We therefore see: The two assertions BTh 1′ and BTh 2′ are inverse to each other. They claim that there is a strong interrelation between large indecomposables on the one hand, and families of indecomposables of the same length on the other hand.

The investigations presented below (and those in [R5]) yield a lot of insight into BTh 1, but there is not yet any corresponding result concerning BTh 2. But it seems that the Gabriel-Roiter measure should also be helpful in dealing with BTh 2.

I. General Results.

1. The Gabriel-Roiter measure

In the ﬁrst three sections of the paper, all the modules considered are ﬁnite length modules.

1.1. The deﬁnition. We deﬁne the Gabriel-Roiter measure µ(M ) for modules

M of ﬁnite length by induction on the length. It will be a rational number. For the

zero module 0, let µ(0) = 0. Given a module M of length n > 0, we may assume by

induction that µ(M ′) is already deﬁned for any proper submodule M ′ of M . Let

2−n µ(M ) = max µ(M ′) + 0

in case M is

indecomposable, .

decomposable,

Here, the maximum is taken over all proper submodules M ′ of M ; in order to see

that this maximum exists, we have to observe inductively: If M is of length n ≥ 1, there is a set of natural numbers I(M ) ⊆ [1, n] such that µ(M ) = i∈I(M) 2−i. (The set I(M ) will be analyzed in more detail below.)

1.2. Here are some elementary properties:

Property 1. For any non-zero module M , there is an indecomposable submodule M ′ of M with µ(M ′) = µ(M ).

Thus, when taking the maximum µ(M ′) in the deﬁnition of µ(M ), it is suﬃcient to consider only proper submodules M ′ of M which are indecomposable.

Property 2. Let Y be a module and X ⊆ Y a submodule. Then µ(X) ≤ µ(Y ). If Y is indecomposable and X is a proper submodule of Y , then µ(X) < µ(Y ).

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CLAUS MICHAEL RINGEL

Property 3. For any module M of length n ≥ 1, we have 1 ≤ µ(M ) ≤ n 2−i = 2n − 1 < 1. 2 i=1 2n

The lower bound is clear, since M contains a simple submodule S and µ(S) = 2−1. The upper bound follows from the fact that there is the subset I(M ) ⊆ [1, n] with µ(M ) = i∈I(M) 2−i.

1.3. Some examples: • If M is a local module of length n, then µ(M ) = µ(rad M ) + 2−n.

• If M is a simple module, then µ(M ) = 21 .

• If M is of length two, then µ(M ) is equal to:

1

2

1 2

+

1 4

=

3 4

if M is decomposable if M is indecomposable,

• If M is of length three, there are already four possibilities: µ(M ) may be one

of the following numbers:

1

2

1 2

+

1 8

=

5 8

1 2

+

1 4

=

3 4

1 2

+

1 4

+

1 8

=

7 8

if M is semisimple, if M is indecomposable with socle of length 2, if M has an indecomposable direct summand of length 2, if M has simple socle.

• In general, if M is of length n ≥ 1, there are 2n−1 possibilities for µ(M ), namely I(M ) may be an arbitrary subset of [1, n] containing 1. Proof, by induction, that all these possibilities actually do occur (for suitable ﬁnite dimensional algebras): The assertion is clear for n = 1. Let n ≥ 2. Let J ⊆ [1, n] be an arbitrary subset containing 1. By induction there is a ﬁnite-dimensional algebra R and an R-module N of length n − 1, such that I(N ) = J ∩ [1, n − 1]. If n ∈/ J, let S be a simple module. Then I(N ⊕ S) = J, see property 5 below.

If n ∈ J, let R′ = R0 Nk , this is again a ﬁnite-dimensional k-algebra called the one-point-extension. There is an indecomposable projective R′-module P with radical equal to N , and I(P ) = I(N ) ∪ {n} = J.

1.4. Gabriel-Roiter submodules. If M is an indecomposable module, and M ′ is an indecomposable submodule of M with µ(M ′) maximal, then we call M ′ a Gabriel-Roiter submodule of M (and the embedding M ′ ⊂ M a Gabriel-Roiter

inclusion). We may reformulate this as follows: If X ⊂ Y is an inclusion of in-

decomposable modules, then X is a Gabriel-Roiter submodule of Y if and only if µ(X) = µ(Y ) − 2−|Y |.

Property

4.

If

M

is

indecomposable

and

|M | = n,

then

µ(M ) =

a 2n

with

a

an odd natural number, such that 2n−1 ≤ a < 2n. In particular, the Gabriel-Roiter

measure of an indecomposable module M determines the length of M .

Proof, by induction on the length of M . If M is simple, then µ(M ) = 21 . Otherwise, take a Gabriel-Roiter submodule M ′ of M . Let n′ = |M ′|, thus n′ < n.

By induction µ(M ′) =

a′ 2n′

with a′ odd.

Thus µ(M ) = µ(M ′) + 2−n =

a 2n

with

FOUNDATION, USING THE GABRIEL-ROITER MEASURE

7

a = a′2n−n′ + 1. Since n − n′ ≥ 1, it follows that a is odd. The bounds on a follow from the bounds on µ(M ) mentioned in Property 3.

Corollary. Let X, Y be modules with µ(X) = µ(Y ). If X is indecomposable, then |X| ≤ |Y |.

Proof: According to Property 1, there is an indecomposable submodule Y ′ of Y with µ(Y ′) = µ(Y ). Now X, Y ′ are indecomposable modules with same GabrielRoiter measure, thus they have the same length. Therefore |X| = |Y ′| ≤ |Y |.

1.5. Gabriel-Roiter ﬁltrations. A chain

(∗)

X1 ⊂ X2 ⊂ · · · ⊂ Xt

will be called a Gabriel-Roiter ﬁltration (of Xt), provided X1 is simple and all the inclusions Xi−1 ⊂ Xi are Gabriel-Roiter inclusions, for 2 ≤ i ≤ t. Note that all the modules Xi involved in a Gabriel-Roiter ﬁltration are indecomposable.

1.6. There is a another possibility for introducing and calculating the Gabriel-

Roiter measure of a module M of length n, concentrating on the set I(M ) ⊆ [1, n] such that µ(M ) = i∈I(M) 2−i.

This approach deals with the set U of all possible chains

U• = (U1 ⊂ U2 ⊂ · · · ⊂ Ut)

of indecomposable submodules Ui of M . Given such a chain U•, we consider the set |U•| = {|Ui| | 1 ≤ i ≤ t} of the length of these submodules (or ﬁnally the corresponding rational number s∈|U•| 2−s). Thus we deal with ﬁnite sets of natural numbers, let us denote by Pf (N1) the set of ﬁnite subsets of N1. We want to consider this set as a totally ordered set. We introduce the following order relation:

Let I = J be elements of Pf (N1). Set I < J provided the smallest element in the symmetric diﬀerence (I \ J) ∪ (J \ I) belongs to J (since we assume that I = J, the

set (I \ J) ∪ (J \ I) cannot be empty, thus there is a smallest element). It is easy

to see that this relation is transitive, and of course it is anti-symmetric. It follows

that Pf (N1) with this ordering is totally ordered. For any module M , let I(M ) be the maximum of the sets |U•| in the totally ordered set (Pf (N1), ≤), where U• belongs to U. Note that if M is of length n, then all the sets |U•| are subsets of the set [1, n] = {1, 2, ..., n}, thus there are only ﬁnitely many possible |U•|.

The relationship between I(M ) and µ(M ) is established via the equality:

µ(M ) =

2−i.

i∈I (M )

Note that if I, J belong to Pf (N1), then

I < J ⇐⇒ 2−i < 2−j.

i∈I

j∈J

This shows that the order introduced on Pf (N1) and the usual ordering of rational numbers are compatible.

If U• = (U1 ⊂ U2 ⊂ · · · ⊂ Ut)

belongs to U, then I(Ut) = |U•| is and only if U• is a Gabriel-Roiter ﬁltration.

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CLAUS MICHAEL RINGEL

In future, when referring to the Gabriel-Roiter measure of module M , we will deal either with µ(M ) or with I(M ), whatever is more suitable.

When Gabriel introduced the (Gabriel-)Roiter measure, he used this second

approach of dealing with I(M ) as an element in the totally ordered set (Pf (N1), ≤).

We should stress that the set I(M ) may be considered to be more intrinsic than the

rational number µ(M ). After all, the reference to the base number 2 used in the

deﬁnition of µ(M ) is really arbitrary, and 2 could be replaced by any other prime

number. On the other hand, to deal with an invariant which takes values in the

well-known set of rational numbers seems to be quite satisfactory for psychological

reasons – in contrast to an invariant which takes values in the rather strange totally

ordered set (Pf (N1), ≤) (by the way, it is more the totally ordering which seems

to be horrifying than the set itself). Of course, it is well-known (and easy to see)

that any countable totally ordered set can be realized as a subset of the totally

ordered set Q, however the embedding used here (mapping I to to be really manageable.

i∈I 2−i) seems

1.7. Main property (Gabriel). Let X, Y1, . . . , Yn be indecomposable modules, and let u : X → i Yi be a monomorphism. Then (a) µ(X) ≤ maxi µ(Yi). (b) If µ(X) = maxi µ(Yi), then u is a split monomorphism.

For the proof we refer to [R5], or see [G].

Property 5. If X, X′ are modules, then

µ(X ⊕ X′) = max(µ(X), µ(X′)).

For the proof we ﬁrst show: If X1, . . . , Xn are indecomposable modules, then µ( i Xi) = maxi µ(Xi). The inequality ≥ follows directly from the deﬁnition. The inequality ≤ follows from Gabriel’s main property (a) and the deﬁnition. Now, if X, X′ are modules, write them as direct sums of indecomposables and apply the ﬁrst assertion.

Corollary. If X is an indecomposable submodule of M with µ(X) = µ(M ), then X is a direct summand. In particular, if M = 0, then M has an indecomposable direct summand X with µ(X) = µ(M ).

Proof: Write M = Mi with indecomposable modules Mi. Property 5 asserts that µ(M ) = max µ(Mi). Gabriel’s main property (b) shows that the embedding X → M = Mi splits. The second assertion follows from property 1.

1.8. Before we proceed, let us insert here several characterizations of modules with simple socle. Of special interest seems to be condition (5) and the equivalence of this condition with the other ones is again an immediate consequence of Gabriel’s main properties.

Lemma. Let M be a module of length n. Then the following conditions are equivalent: (1) The socle of M is simple. (2) Any non-zero submodule of M is indecomposable. (3) There exists a composition series of M with all terms indecomposable. (4) I(M ) = [1, n].

FOUNDATION, USING THE GABRIEL-ROITER MEASURE

9

(5) µ(M ′) < µ(M ), for any proper factor module M ′ of M .

Modules with these properties are often called uniform modules.

Proof: The implications (1) =⇒ (2) =⇒ (3) are obvious. If there exists a composition series U• of M with all terms indecomposable, then clearly |U•| = [1, n] is the maximal possibility, thus U• is a Gabriel-Roiter ﬁltration and I(M ) = [1, n]. This shows that (3) implies (4). If we assume (4), and M ′ is a proper factor module, say of length n′ < n, then I(M ′) ⊆ [1, n′] ⊂ [1, n] = I(M ), thus also I(M ′) < I(M ). It remains to show that (5) implies (1). Assume M has two diﬀerent simple submodules S and S′. Then we can form the factor modules M/S and M/S′ and the canonical maps give rise to an embedding M → M/S ⊕ M/S′. Main property (a) yields I(M ) ≤ max(I(M/S), I(M/S′)), but by assumption both I(M/S) < I(M ) and I(M/S′) < I(M ), thus also max(I(M/S), I(M/S′)) < I(M ), a contradiction.

We may reformulate the essential implication (5) =⇒ (1) as follows: If M is a module, then either the socle of M is simple, or else there is an indecomposable factor module M ′ of M with µ(M ) ≤ µ(M ′). Indeed we have:

Property 6. If M is indecomposable, then either M has simple socle, or else there is a factor module M ′ of M with simple socle such that µ(M ) < µ(M ′).

Proof. If the socle of M is not simple, then write soc M = Si with Si simple. For any i, choose a submodule Ui of M with Si ∩ Ui = 0 and such that Ui is maximal with this property. Then M/Ui has simple socle and the canonical maps M → M/Ui combine to an embedding M → i M/Ui. Gabriel’s main property (a) asserts that µ(M ) ≤ max µ(M/Ui). We even must have µ(M ) < max µ(M/Ui), since otherwise M would be isomorphic to M/Ui for some i by property (b). Thus there is M ′ = M/Ui with µ(M ) < µ(M/Ui).

1.9. We consider now maps f : X → Y, where X, Y are indecomposable modules with µ(X) > µ(Y ).

Property 7. For any indecomposable module M , let M ′ be the intersection of the kernels of all maps M → Z with µ(Z) < µ(M ). Then

• µ(M/M ′) < µ(M ), and • if U is a submodule of M , such that µ(M/U ) < µ(M ), then M ′ ⊆ U.

This means that M/M ′ is the largest factor module of M with Gabriel-Roiter measure smaller than the µ(M ). Of course, M ′ = 0, since µ(M/M ′) < µ(M ).

Proof: Let M ′ be the intersection of the kernels of all maps f : M → Z with µ(Z) < µ(M ). There are ﬁnitely many maps fi : M → Zi, say 1 ≤ i ≤ t, such that µ(Zi) < µ(M ), and such that the intersection of the kernels of these maps fi is equal to M ′ (since M is of ﬁnite length, thus artinian). So (f1, . . . , ft) : M/M ′ →

Zi is injective. But this implies, by Gabriel’s main property, that µ(M/M ′) ≤ max(µ(Zi)) < µ(M ). If M ′ = 0, then we get µ(M ) < µ(M ), a contradiction. Thus M ′ = 0. Of course, for any f : M → X with µ(X) < µ(M ), the kernel of f contains M ′. In particular, if M ′′ is a submodule of M such that µ(M/M ′′) < µ(M ), then M ′ ⊆ M ′′.

Note that property 7 conversely implies Gabriel’s main property (a): Namely, assume that indecomposable modules X, Y1, . . . , Yn and a monomorphism u : X →

10

CLAUS MICHAEL RINGEL

Yi are given. Denote by ui : X → Yi the composition of u with the corresponding projections. If µ(X) > µ(Yi), then property 7 asserts that X′ ⊆ Ker(ui). But since u is a monomorphism, 0 = X′ cannot be contained in Ker(ui) for all i, thus

µ(X) ≤ µ(Yi) for some i.

1.10. Deﬁnition. For every rational number γ, we denote by A(γ) the class of all indecomposable modules (of ﬁnite length) with Gabriel-Roiter measure γ. Similarly, let us denote by A( ≤ γ) the class of all indecomposable modules (of ﬁnite length) with Gabriel-Roiter measure ≤ γ (here we may start even with a real number γ). We say that γ is a Gabriel-Roiter measure (for Λ, in case there could be a doubt) provided A(γ) is not empty.

There

always

is

a

smallest

Gabriel-Roiter

measure

I1

=

1 2

(provided

Λ

is

non-

zero) and A(I1) is the class of all simple modules. This holds for any ring R. In

our case of an artin algebra Λ, there also is a largest Gabriel-Roiter measure (again

provided Λ is non-zero), namely I1 = [1, q], where q is the maximal length of an

indecomposable injective module. (For a general ring R, there may exist arbitrary

large ﬁnite length modules with simple socle, as in the case of a discrete valuation

ring; then there is no largest Gabriel-Roiter measure.)

We will use the Gabriel-Roiter measure µ in order to visualize the category

mod Λ.

A( ≤ γ)

.......................................................................

.

.

.

.

. . .

....................................................................................................................................................................................................................................

. .

..........................................................................................................................................................................................................................................................................

. .

mod Λ .

............................................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................

A(γ ) .......................................................................................................................................................................................................................................................................................................................

.....

....................................................................................................................................................................................................................................................................................................................

......

.

µ .

...........................................................................................................................................................................................................................................................................................

.....

. . .

.......................................................................................................................................................................................................................................

.... ......

.

...................................................................................................

...

.

.

.

.

.

.

Q ........................................................................................................................................................................................................................................................................................

12 γ.. 1

What really matters is the fact that all the subcategories add A( ≤ γ) are closed

under submodules (or, equivalently, that the categories A( ≤ γ) are closed under co-

generation). By the way, there are a lot of investigations dealing with subcategories

which are closed under submodules — but often one requires in addition that the

subcategory is also closed under extensions. In our setting, this latter assumption is satisﬁed only in trival cases: Namely, as soon as γ ≥ 12 , the subcategory add A( ≤ γ) contains all the simple objects. Thus, if we require that such a subcategory is closed

under extension, we will just obtain all the ﬁnite length modules!

The set of Gabriel-Roiter measures which occur for a given Λ gives a lot of information about Λ (or better, its Morita equivalence class). But note that diﬀerent types of rings may yield the same Gabriel-Roiter measures. For example, there are four diﬀerent quivers of type A4, they are distinguished by the number of sinks and sources. The linear orientation leads to just one sink and just one source, the corresponding path algebra is serial, and the Gabriel-Roiter measures are

{1} < {1, 2} < {1, 2, 3} < {1, 2, 3, 4}.

The same Gabriel-Roiter measures occur for the second orientation with precisely one sink (and two sources). For the remaining two orientations (with two sinks,

Foundation of the Representation Theory of Artin Algebras,

Using the Gabriel-Roiter Measure

Claus Michael Ringel

Abstract. These notes are devoted to a single invariant, the Gabriel-Roiter measure of ﬁnite length modules: this invariant was introduced by Gabriel (under the name ‘Roiter measure’) in 1972 in order to give a combinatorial interpretation of the induction scheme used by Roiter in his 1968 proof of the ﬁrst Brauer-Thrall conjecture. It is strange that this invariant (and Roiter’s proof itself) was forgotten in the meantime. One explanation may be that both Roiter and Gabriel pretend that their considerations are restricted to algebras of bounded representation type which are shown to be of ﬁnite representation type, thus restricted to algebras of ﬁnite representation type. But, as we are going to show, this invariant is of special interest when dealing with algebras of inﬁnite representation type! And there may be a second explanation: in the early seventies, it was possible to calculate this invariant only for few examples, whereas nowadays there is a wealth of methods available. Looking at such examples, we are convinced that the Gabriel-Roiter measure has to be considered as a very important invariant and that it can be used to lay the foundation of the representation theory of artin algebras.

The notes are based on lectures given at Hirosaki (2003), Queretaro (2004), Hangzhou and Beijing (2005) looking at some basic questions in the representation theory of artin algebras. The main emphasis in the Hirosaki lectures was to develop a direct approach to the representation theory of artin algebras, using the GabrielRoiter measure, independent of Auslander-Reiten theory. Indeed, the GabrielRoiter measure should be considered as being elementary: only individual modules are considered (as abelian groups with a set of prescribed endomorphisms), whereas the notions of Auslander-Reiten theory relate from the beginning the given module to the whole module category. In the later lecture series (at Queretaro, Hangzhou and Beijing), some useful connections between the Gabriel-Roiter measure and Auslander-Reiten theory have been included. In the present notes we do not hesitate to mingle the diﬀerent approaches whenever it seems suitable.

2000 Mathematics Subject Classiﬁcation. Primary: 16G10, 16G20, 16G60, 16G70 Secondary: 16D10, 16D70..

Key words and phrases. Brauer-Thrall conjectures. Artin algebras. Finite dimensional associative algebras. Quivers and their representations. Representation type. Large indecomposable modules. Direct sums of ﬁnitely generated modules. Relative Σ-injectivity..

c XXXX American Mathematical Society

1

2

CLAUS MICHAEL RINGEL

The presentation centers on some old topics in representation theory, and we hope to convince the reader of the usefulness of the Gabriel-Roiter measure when dealing with results of the following kind:

• The ﬁrst Brauer-Thrall conjecture established by Roiter and Auslander, which asserts that inﬁnite representation type implies unbounded representation type.

• Any module for an artin algebra of ﬁnite representation type is a direct sum of ﬁnitely generated modules.

• Auslander’s theorem asserting that artin algebras of inﬁnite representation type have indecomposable modules which are not ﬁnitely generated.

Of course, it is not surprising that the Gabriel-Roiter measure is useful when dealing with the Brauer-Thrall conjectures, since it was introduced in this context. But it seems that the relationship to the other two topics was not realized before.

This report has three parts. The ﬁrst part develops general properties of the Gabriel-Roiter measure, the second part deals with what we call the “take-oﬀ part” of the module category of an artin algebra, the third one with the “landing part”. With respect to the Gabriel-Roiter measure, the category mod Λ for any artin algebra Λ is divided into three diﬀerent subcategories: the take-oﬀ part, the central part and the landing part. Of highest concern should be the central part — however, at present, there is not yet a single result available in print concerning the central part.

We do not touch at all the dual construction, the Gabriel-Roiter “comeasure”. Concerning the Gabriel-Roiter comeasure, and the interrelation between the measure and the comeasure, we propose to look at [R5]: the rhombic picture seems to provide a fascinating description of the module category, but also here we have to wait for future research in order to get a full understanding of what is going on.

The core of the notes is self-contained, but for some of the proofs we refer to [R5]. There are several remarks and examples (usually marked by a star: Example*, Remark*) which involve notions which are unexplained. They may be helpful for some (or most) of the readers, but can be skipped in a ﬁrst reading.

Notation: N1 = {1, 2, . . . } denotes the positive natural numbers. Given two integers a ≤ b, let [a, b] be the set of integers z with a ≤ z ≤ b.

0. Preliminaries

Let R be an arbitrary ring, we consider left R-modules and call them just modules. We will assume some basic results from module theory and homological algebra. All our consideration are related to ﬁnite length modules: we only deal with modules which are unions of modules of ﬁnite length. Recall that ﬁnite length modules are modules which are both artinian and noetherian; they have a ﬁnite chain of submodules which cannot be reﬁned (such a chain is called a composition series). The Jordan-Ho¨lder theorem asserts that the length of such a chain depends only on the module, it is called the length of the module. We denote the length of a module M by |M |. A module M is said to be indecomposable, provided it is non-zero and cannot be written as a direct sum M = M1 ⊕ M2 of two non-zero modules (a direct sum decomposition M = M1 ⊕ M2 is given by two submodules M1, M2 of M such that M1 + M2 = M and M1 ∩ M2 = 0.) It is obvious that any ﬁnite length module is a ﬁnite direct sum of indecomposable modules (and they are again of ﬁnite length). The Fitting Lemma tells us that the endomorphism ring of

FOUNDATION, USING THE GABRIEL-ROITER MEASURE

3

an indecomposable module of ﬁnite length is a local ring (the set of non-invertible endomorphisms forms an ideal), and this implies the Krull-Remak-Schmidt theorem: A decomposition of a ﬁnite length module into indecomposable modules is unique up to isomorphism.

0.1. Questions. Given a natural number d, are there indecomposable modules of length d? And how many such modules are there? In particular: are there inﬁnitely many isomorphism classes of indecomposable modules of length d? Let a(d) be the number of isomorphism classes of indecomposable modules of length d (recall: we mean left R-modules, the R being ﬁxed; of course we could write aR(d) in order to specify the ring). Thus, we consider the questions:

• When is a(d) = 0 ? • When is a(d) = ∞?

Example* 1. Let R be a discrete valuation ring, for example R = k[[T ]], the

power series ring in one variable, or R = Z(p) the ring of p-adic integers. The maximal ideal of R is a principal ideal, say generated by the element π. The module R/Rπd is indecomposable and of length d, and is, up to isomorphism the

only indecomposable module of length d. Thus we see that a(d) = 1 for any d ∈ N1.

Example 2. Let k be a ﬁeld and let R = Tn(k) be the ring of upper triangular

(n × n)-matrices with coeﬃcients in k. Then a(d) = n − d + 1, for 1 ≤ d ≤ n and 0

for n < d, thus altogether we obtain

n+1 2

isomorphism classes. Using “quivers” (as

we will often do), we can interprete R = Tn(k) as the path algebra of the linearly

ordered quiver of type An, with vertices 1 → 2 → · · · → n, and the indecomposables

of length d correspond bijectively to the intervals [i, i + d − 1] with 1 ≤ i ≤ n − d.

Remark*. More generally, Gabriel’s theorem asserts that the following quivers are of ﬁnite representation type (this means: d a(d) < ∞) and that the number of indecomposables is as follows (and in particular independent of the orientation):

....

type A D E E E .... ..

n

n

6

7

8

....

..............................................................................................................................................................................................................................................................................................................................

....

.... ..

n+1

a(d) n(n − 1) 36 63 120 d

.... ...

2

.

The types ∆ which occur here are just the simply laced Dynkin diagrams (which occur in Lie theory, but also in many other parts of mathematics); and the numbers below are known as the number of positive roots of ∆: in fact, there is a natural bijection between the indecomposable representations and the positive roots.

Example 3. The Kronecker quiver: it has two vertices x, y and two arrows, say going from x to y. Then one knows that a(n) = 2 for n odd, and a(n) ≥ 3 for n even. For n even, the number a(n) depends on the base ﬁeld k. In case we deal with representations over an inﬁnite ﬁeld k, then a(n) = ∞ for n even. (Here we have an example of what has been called strongly unbounded representation type: there are inﬁnitely many natural numbers d1 < d2 < . . . such that a(di) = ∞ for all i.)

The main problem of representation theory is to ﬁnd invariants for modules and to describe the isomorphism classes of all the indecomposable modules for which such an invariant takes a ﬁxed value. A typical such invariant is the length of a

4

CLAUS MICHAEL RINGEL

module: the simple modules are those of length 1 (and there is just a ﬁnite number of such modules), the information concerning the indecomposable modules of length 2 is stored in the quiver (in case we deal with a ﬁnite dimensional algebra over some algebraically closed ﬁeld) or the “species” of Λ. Given any invariant γ, as a ﬁrst question one may look for values of ﬁnite type: these are those values v such that there are only ﬁnitely many isomorphisms classes of indecomposable modules M with γ(M ) = v. The invariant to be discussed here is the Gabriel-Roiter measure.

In most parts of these lectures we will assume that R is an artin algebra. This means that the center z(R) of R is artinian and that R is a ﬁnitely generated z(R)-module. Typical examples are the ﬁnite-dimensional k-algebras, where k is a ﬁeld, but also all ﬁnite rings. Some elementary properties of the Gabriel-Roiter measure hold for ﬁnite length modules over an arbitrary ring R, or, more generally, for arbitrary ﬁnite length categories (a ﬁnite length category is an abelian category such than any object is both artinian and noetherian, thus has a composition series; for any ring R, the category of all ﬁnite length R-modules is of course a length category).

0.2. The Brauer-Thrall conjectures. As we have mentioned the GabrielRoiter measure was introduced by Gabriel in [G] in order to clarify the intricate induction scheme used by Roiter [Ro] in his proof of the ﬁrst Brauer-Thrall conjecture. Roiter’s proof of this conjecture marks the beginning of the new representation theory of ﬁnite dimensional algebras. Let us recall the precise statement of both Brauer-Thrall conjectures.

BTh 1. Bounded representation type (that means a(d) = 0 for large d) implies ﬁnite representation type. (Under the assumption that a(1) < ∞.)

If a(d) < ∞ for all d, then bounded representation type implies ﬁnite representation type, trivially. Thus we can reformulate BTh 1 as follows:

BTh 1′ If a(d) = ∞ for some d, then a(d) = 0 for inﬁnitely many d.

BTh 2. Unbounded representation type implies strongly unbounded representation type. (Under the assumption that we deal with ﬁnite dimensional algebras over an inﬁnite ﬁeld, or more generally, with inﬁnite (and connected) artin algebras.)

The assumption is quite essential, since there are plenty of ﬁnite length categories which are of unbounded type, but not of strongly unbounded type. Example 1 above is a typical such example.

BTh 1 is true for R a semiprimary ring: for ﬁnite dimensional algebras this was shown by Roiter, for left artinian rings by Auslander. Note: Bounded representation type means that every simple module has a relative projective cover and a relative injective envelope in the category of ﬁnite length modules; thus a semiprimary ring of bounded representation type is necessarily left artinian.

Remark*. One should be aware of the V -rings as constructed by Cozzens: here all simple modules are injective, thus also relative projective in the category of ﬁnite length modules. These rings satisfy a(d) = 0 for d ≥ 2, however they are not semiprimary. Note that there are such examples with a(1) being ﬁnite, as well as examples with a(1) being inﬁnite.

FOUNDATION, USING THE GABRIEL-ROITER MEASURE

5

0.3. Smalø [S] has shown the following:

Theorem. Let R be an artin algebra with a(d) = ∞ for some d, then a(d′) = ∞ for some d′ > d.

We see that in order to show that an artin algebra is of strongly unbounded representation type, it is suﬃcient to show that a(d) = ∞ for some d.

Thus we can reformulate BTh 2 as follows:

BTh 2′. Let R be an inﬁnite artin algebra. If a(d) = 0 for inﬁnitely many d, then a(d) = ∞ for some d.

We therefore see: The two assertions BTh 1′ and BTh 2′ are inverse to each other. They claim that there is a strong interrelation between large indecomposables on the one hand, and families of indecomposables of the same length on the other hand.

The investigations presented below (and those in [R5]) yield a lot of insight into BTh 1, but there is not yet any corresponding result concerning BTh 2. But it seems that the Gabriel-Roiter measure should also be helpful in dealing with BTh 2.

I. General Results.

1. The Gabriel-Roiter measure

In the ﬁrst three sections of the paper, all the modules considered are ﬁnite length modules.

1.1. The deﬁnition. We deﬁne the Gabriel-Roiter measure µ(M ) for modules

M of ﬁnite length by induction on the length. It will be a rational number. For the

zero module 0, let µ(0) = 0. Given a module M of length n > 0, we may assume by

induction that µ(M ′) is already deﬁned for any proper submodule M ′ of M . Let

2−n µ(M ) = max µ(M ′) + 0

in case M is

indecomposable, .

decomposable,

Here, the maximum is taken over all proper submodules M ′ of M ; in order to see

that this maximum exists, we have to observe inductively: If M is of length n ≥ 1, there is a set of natural numbers I(M ) ⊆ [1, n] such that µ(M ) = i∈I(M) 2−i. (The set I(M ) will be analyzed in more detail below.)

1.2. Here are some elementary properties:

Property 1. For any non-zero module M , there is an indecomposable submodule M ′ of M with µ(M ′) = µ(M ).

Thus, when taking the maximum µ(M ′) in the deﬁnition of µ(M ), it is suﬃcient to consider only proper submodules M ′ of M which are indecomposable.

Property 2. Let Y be a module and X ⊆ Y a submodule. Then µ(X) ≤ µ(Y ). If Y is indecomposable and X is a proper submodule of Y , then µ(X) < µ(Y ).

6

CLAUS MICHAEL RINGEL

Property 3. For any module M of length n ≥ 1, we have 1 ≤ µ(M ) ≤ n 2−i = 2n − 1 < 1. 2 i=1 2n

The lower bound is clear, since M contains a simple submodule S and µ(S) = 2−1. The upper bound follows from the fact that there is the subset I(M ) ⊆ [1, n] with µ(M ) = i∈I(M) 2−i.

1.3. Some examples: • If M is a local module of length n, then µ(M ) = µ(rad M ) + 2−n.

• If M is a simple module, then µ(M ) = 21 .

• If M is of length two, then µ(M ) is equal to:

1

2

1 2

+

1 4

=

3 4

if M is decomposable if M is indecomposable,

• If M is of length three, there are already four possibilities: µ(M ) may be one

of the following numbers:

1

2

1 2

+

1 8

=

5 8

1 2

+

1 4

=

3 4

1 2

+

1 4

+

1 8

=

7 8

if M is semisimple, if M is indecomposable with socle of length 2, if M has an indecomposable direct summand of length 2, if M has simple socle.

• In general, if M is of length n ≥ 1, there are 2n−1 possibilities for µ(M ), namely I(M ) may be an arbitrary subset of [1, n] containing 1. Proof, by induction, that all these possibilities actually do occur (for suitable ﬁnite dimensional algebras): The assertion is clear for n = 1. Let n ≥ 2. Let J ⊆ [1, n] be an arbitrary subset containing 1. By induction there is a ﬁnite-dimensional algebra R and an R-module N of length n − 1, such that I(N ) = J ∩ [1, n − 1]. If n ∈/ J, let S be a simple module. Then I(N ⊕ S) = J, see property 5 below.

If n ∈ J, let R′ = R0 Nk , this is again a ﬁnite-dimensional k-algebra called the one-point-extension. There is an indecomposable projective R′-module P with radical equal to N , and I(P ) = I(N ) ∪ {n} = J.

1.4. Gabriel-Roiter submodules. If M is an indecomposable module, and M ′ is an indecomposable submodule of M with µ(M ′) maximal, then we call M ′ a Gabriel-Roiter submodule of M (and the embedding M ′ ⊂ M a Gabriel-Roiter

inclusion). We may reformulate this as follows: If X ⊂ Y is an inclusion of in-

decomposable modules, then X is a Gabriel-Roiter submodule of Y if and only if µ(X) = µ(Y ) − 2−|Y |.

Property

4.

If

M

is

indecomposable

and

|M | = n,

then

µ(M ) =

a 2n

with

a

an odd natural number, such that 2n−1 ≤ a < 2n. In particular, the Gabriel-Roiter

measure of an indecomposable module M determines the length of M .

Proof, by induction on the length of M . If M is simple, then µ(M ) = 21 . Otherwise, take a Gabriel-Roiter submodule M ′ of M . Let n′ = |M ′|, thus n′ < n.

By induction µ(M ′) =

a′ 2n′

with a′ odd.

Thus µ(M ) = µ(M ′) + 2−n =

a 2n

with

FOUNDATION, USING THE GABRIEL-ROITER MEASURE

7

a = a′2n−n′ + 1. Since n − n′ ≥ 1, it follows that a is odd. The bounds on a follow from the bounds on µ(M ) mentioned in Property 3.

Corollary. Let X, Y be modules with µ(X) = µ(Y ). If X is indecomposable, then |X| ≤ |Y |.

Proof: According to Property 1, there is an indecomposable submodule Y ′ of Y with µ(Y ′) = µ(Y ). Now X, Y ′ are indecomposable modules with same GabrielRoiter measure, thus they have the same length. Therefore |X| = |Y ′| ≤ |Y |.

1.5. Gabriel-Roiter ﬁltrations. A chain

(∗)

X1 ⊂ X2 ⊂ · · · ⊂ Xt

will be called a Gabriel-Roiter ﬁltration (of Xt), provided X1 is simple and all the inclusions Xi−1 ⊂ Xi are Gabriel-Roiter inclusions, for 2 ≤ i ≤ t. Note that all the modules Xi involved in a Gabriel-Roiter ﬁltration are indecomposable.

1.6. There is a another possibility for introducing and calculating the Gabriel-

Roiter measure of a module M of length n, concentrating on the set I(M ) ⊆ [1, n] such that µ(M ) = i∈I(M) 2−i.

This approach deals with the set U of all possible chains

U• = (U1 ⊂ U2 ⊂ · · · ⊂ Ut)

of indecomposable submodules Ui of M . Given such a chain U•, we consider the set |U•| = {|Ui| | 1 ≤ i ≤ t} of the length of these submodules (or ﬁnally the corresponding rational number s∈|U•| 2−s). Thus we deal with ﬁnite sets of natural numbers, let us denote by Pf (N1) the set of ﬁnite subsets of N1. We want to consider this set as a totally ordered set. We introduce the following order relation:

Let I = J be elements of Pf (N1). Set I < J provided the smallest element in the symmetric diﬀerence (I \ J) ∪ (J \ I) belongs to J (since we assume that I = J, the

set (I \ J) ∪ (J \ I) cannot be empty, thus there is a smallest element). It is easy

to see that this relation is transitive, and of course it is anti-symmetric. It follows

that Pf (N1) with this ordering is totally ordered. For any module M , let I(M ) be the maximum of the sets |U•| in the totally ordered set (Pf (N1), ≤), where U• belongs to U. Note that if M is of length n, then all the sets |U•| are subsets of the set [1, n] = {1, 2, ..., n}, thus there are only ﬁnitely many possible |U•|.

The relationship between I(M ) and µ(M ) is established via the equality:

µ(M ) =

2−i.

i∈I (M )

Note that if I, J belong to Pf (N1), then

I < J ⇐⇒ 2−i < 2−j.

i∈I

j∈J

This shows that the order introduced on Pf (N1) and the usual ordering of rational numbers are compatible.

If U• = (U1 ⊂ U2 ⊂ · · · ⊂ Ut)

belongs to U, then I(Ut) = |U•| is and only if U• is a Gabriel-Roiter ﬁltration.

8

CLAUS MICHAEL RINGEL

In future, when referring to the Gabriel-Roiter measure of module M , we will deal either with µ(M ) or with I(M ), whatever is more suitable.

When Gabriel introduced the (Gabriel-)Roiter measure, he used this second

approach of dealing with I(M ) as an element in the totally ordered set (Pf (N1), ≤).

We should stress that the set I(M ) may be considered to be more intrinsic than the

rational number µ(M ). After all, the reference to the base number 2 used in the

deﬁnition of µ(M ) is really arbitrary, and 2 could be replaced by any other prime

number. On the other hand, to deal with an invariant which takes values in the

well-known set of rational numbers seems to be quite satisfactory for psychological

reasons – in contrast to an invariant which takes values in the rather strange totally

ordered set (Pf (N1), ≤) (by the way, it is more the totally ordering which seems

to be horrifying than the set itself). Of course, it is well-known (and easy to see)

that any countable totally ordered set can be realized as a subset of the totally

ordered set Q, however the embedding used here (mapping I to to be really manageable.

i∈I 2−i) seems

1.7. Main property (Gabriel). Let X, Y1, . . . , Yn be indecomposable modules, and let u : X → i Yi be a monomorphism. Then (a) µ(X) ≤ maxi µ(Yi). (b) If µ(X) = maxi µ(Yi), then u is a split monomorphism.

For the proof we refer to [R5], or see [G].

Property 5. If X, X′ are modules, then

µ(X ⊕ X′) = max(µ(X), µ(X′)).

For the proof we ﬁrst show: If X1, . . . , Xn are indecomposable modules, then µ( i Xi) = maxi µ(Xi). The inequality ≥ follows directly from the deﬁnition. The inequality ≤ follows from Gabriel’s main property (a) and the deﬁnition. Now, if X, X′ are modules, write them as direct sums of indecomposables and apply the ﬁrst assertion.

Corollary. If X is an indecomposable submodule of M with µ(X) = µ(M ), then X is a direct summand. In particular, if M = 0, then M has an indecomposable direct summand X with µ(X) = µ(M ).

Proof: Write M = Mi with indecomposable modules Mi. Property 5 asserts that µ(M ) = max µ(Mi). Gabriel’s main property (b) shows that the embedding X → M = Mi splits. The second assertion follows from property 1.

1.8. Before we proceed, let us insert here several characterizations of modules with simple socle. Of special interest seems to be condition (5) and the equivalence of this condition with the other ones is again an immediate consequence of Gabriel’s main properties.

Lemma. Let M be a module of length n. Then the following conditions are equivalent: (1) The socle of M is simple. (2) Any non-zero submodule of M is indecomposable. (3) There exists a composition series of M with all terms indecomposable. (4) I(M ) = [1, n].

FOUNDATION, USING THE GABRIEL-ROITER MEASURE

9

(5) µ(M ′) < µ(M ), for any proper factor module M ′ of M .

Modules with these properties are often called uniform modules.

Proof: The implications (1) =⇒ (2) =⇒ (3) are obvious. If there exists a composition series U• of M with all terms indecomposable, then clearly |U•| = [1, n] is the maximal possibility, thus U• is a Gabriel-Roiter ﬁltration and I(M ) = [1, n]. This shows that (3) implies (4). If we assume (4), and M ′ is a proper factor module, say of length n′ < n, then I(M ′) ⊆ [1, n′] ⊂ [1, n] = I(M ), thus also I(M ′) < I(M ). It remains to show that (5) implies (1). Assume M has two diﬀerent simple submodules S and S′. Then we can form the factor modules M/S and M/S′ and the canonical maps give rise to an embedding M → M/S ⊕ M/S′. Main property (a) yields I(M ) ≤ max(I(M/S), I(M/S′)), but by assumption both I(M/S) < I(M ) and I(M/S′) < I(M ), thus also max(I(M/S), I(M/S′)) < I(M ), a contradiction.

We may reformulate the essential implication (5) =⇒ (1) as follows: If M is a module, then either the socle of M is simple, or else there is an indecomposable factor module M ′ of M with µ(M ) ≤ µ(M ′). Indeed we have:

Property 6. If M is indecomposable, then either M has simple socle, or else there is a factor module M ′ of M with simple socle such that µ(M ) < µ(M ′).

Proof. If the socle of M is not simple, then write soc M = Si with Si simple. For any i, choose a submodule Ui of M with Si ∩ Ui = 0 and such that Ui is maximal with this property. Then M/Ui has simple socle and the canonical maps M → M/Ui combine to an embedding M → i M/Ui. Gabriel’s main property (a) asserts that µ(M ) ≤ max µ(M/Ui). We even must have µ(M ) < max µ(M/Ui), since otherwise M would be isomorphic to M/Ui for some i by property (b). Thus there is M ′ = M/Ui with µ(M ) < µ(M/Ui).

1.9. We consider now maps f : X → Y, where X, Y are indecomposable modules with µ(X) > µ(Y ).

Property 7. For any indecomposable module M , let M ′ be the intersection of the kernels of all maps M → Z with µ(Z) < µ(M ). Then

• µ(M/M ′) < µ(M ), and • if U is a submodule of M , such that µ(M/U ) < µ(M ), then M ′ ⊆ U.

This means that M/M ′ is the largest factor module of M with Gabriel-Roiter measure smaller than the µ(M ). Of course, M ′ = 0, since µ(M/M ′) < µ(M ).

Proof: Let M ′ be the intersection of the kernels of all maps f : M → Z with µ(Z) < µ(M ). There are ﬁnitely many maps fi : M → Zi, say 1 ≤ i ≤ t, such that µ(Zi) < µ(M ), and such that the intersection of the kernels of these maps fi is equal to M ′ (since M is of ﬁnite length, thus artinian). So (f1, . . . , ft) : M/M ′ →

Zi is injective. But this implies, by Gabriel’s main property, that µ(M/M ′) ≤ max(µ(Zi)) < µ(M ). If M ′ = 0, then we get µ(M ) < µ(M ), a contradiction. Thus M ′ = 0. Of course, for any f : M → X with µ(X) < µ(M ), the kernel of f contains M ′. In particular, if M ′′ is a submodule of M such that µ(M/M ′′) < µ(M ), then M ′ ⊆ M ′′.

Note that property 7 conversely implies Gabriel’s main property (a): Namely, assume that indecomposable modules X, Y1, . . . , Yn and a monomorphism u : X →

10

CLAUS MICHAEL RINGEL

Yi are given. Denote by ui : X → Yi the composition of u with the corresponding projections. If µ(X) > µ(Yi), then property 7 asserts that X′ ⊆ Ker(ui). But since u is a monomorphism, 0 = X′ cannot be contained in Ker(ui) for all i, thus

µ(X) ≤ µ(Yi) for some i.

1.10. Deﬁnition. For every rational number γ, we denote by A(γ) the class of all indecomposable modules (of ﬁnite length) with Gabriel-Roiter measure γ. Similarly, let us denote by A( ≤ γ) the class of all indecomposable modules (of ﬁnite length) with Gabriel-Roiter measure ≤ γ (here we may start even with a real number γ). We say that γ is a Gabriel-Roiter measure (for Λ, in case there could be a doubt) provided A(γ) is not empty.

There

always

is

a

smallest

Gabriel-Roiter

measure

I1

=

1 2

(provided

Λ

is

non-

zero) and A(I1) is the class of all simple modules. This holds for any ring R. In

our case of an artin algebra Λ, there also is a largest Gabriel-Roiter measure (again

provided Λ is non-zero), namely I1 = [1, q], where q is the maximal length of an

indecomposable injective module. (For a general ring R, there may exist arbitrary

large ﬁnite length modules with simple socle, as in the case of a discrete valuation

ring; then there is no largest Gabriel-Roiter measure.)

We will use the Gabriel-Roiter measure µ in order to visualize the category

mod Λ.

A( ≤ γ)

.......................................................................

.

.

.

.

. . .

....................................................................................................................................................................................................................................

. .

..........................................................................................................................................................................................................................................................................

. .

mod Λ .

............................................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................

A(γ ) .......................................................................................................................................................................................................................................................................................................................

.....

....................................................................................................................................................................................................................................................................................................................

......

.

µ .

...........................................................................................................................................................................................................................................................................................

.....

. . .

.......................................................................................................................................................................................................................................

.... ......

.

...................................................................................................

...

.

.

.

.

.

.

Q ........................................................................................................................................................................................................................................................................................

12 γ.. 1

What really matters is the fact that all the subcategories add A( ≤ γ) are closed

under submodules (or, equivalently, that the categories A( ≤ γ) are closed under co-

generation). By the way, there are a lot of investigations dealing with subcategories

which are closed under submodules — but often one requires in addition that the

subcategory is also closed under extensions. In our setting, this latter assumption is satisﬁed only in trival cases: Namely, as soon as γ ≥ 12 , the subcategory add A( ≤ γ) contains all the simple objects. Thus, if we require that such a subcategory is closed

under extension, we will just obtain all the ﬁnite length modules!

The set of Gabriel-Roiter measures which occur for a given Λ gives a lot of information about Λ (or better, its Morita equivalence class). But note that diﬀerent types of rings may yield the same Gabriel-Roiter measures. For example, there are four diﬀerent quivers of type A4, they are distinguished by the number of sinks and sources. The linear orientation leads to just one sink and just one source, the corresponding path algebra is serial, and the Gabriel-Roiter measures are

{1} < {1, 2} < {1, 2, 3} < {1, 2, 3, 4}.

The same Gabriel-Roiter measures occur for the second orientation with precisely one sink (and two sources). For the remaining two orientations (with two sinks,

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