Degrees of irreducible morphisms and finite-representation type

We study the degree of irreducible morphisms in any Auslander-Reiten component of a finite dimensional algebra over an algebraically closed field. We give a characterization for an irreducible morphism to have finite left (or right) degree. This is used to prove our main theorem: An algebra is of finite representation type if and only if for every indecomposable projective the inclusion of the radical in the projective has finite right degree, which is equivalent to require that for every indecomposable injective the epimorphism from the injective to its quotient by its socle has finite left degree. We also apply the techniques that we develop: We study when the non-zero composite of a path of n irreducible morphisms between indecomposable modules lies in the n+ 1-th power of the radical; and we study the same problem for sums of such paths when they are sectional, thus proving a generalisation of a pioneer result of Igusa and Todorov on the composite of a sectional path. Introduction Let A be an artin algebra over an artin commutative ring k. The representation theory of A deals with the study of the category modA of (right) A-modules of finite type. One of the most powerful tools in this study is the Auslander-Reiten theory, based on irreducible morphisms and almost split sequences (see [1]). Although irreducible morphisms have permitted important advances in representation theory, some of their basic properties still remain mysterious to us. An important example is the composition of two irreducible morphisms: It obviously lies in rad (where rad is the l-th power of the radical ideal rad of modA) but it may lie in rad, rad or even be the zero morphism. Of course, the situation still makes sense with the composite of arbitrary many irreducible morphisms. A first, but partial, treatment of this situation was given by Igusa and Todorov ([10]) with the following result: “If X0 f1 −→ X1 → · · · → Xn−1 fn −−→ Xn is a sectional path of irreducible morphisms between indecomposable modules, then the composite fn · · · f1 lies in rad (X0,Xn) and not in rad (X0,Xn), in particular, it is non-zero.” In [11], Liu introduced the left and right degrees of an irreducible morphism f : X → Y as follows: The left degree dl(f) of f is the least integer m > 1 such that there exists Z ∈ modA and g ∈ rad(Z,X)\rad(Z,X) satisfying fg ∈ rad(Z, Y ). If no such an integer m exists, then dl(f) = ∞. The right degree is defined dually. This notion was introduced to study the composition of irreducible morphisms. In particular, Liu extended the above study of Igusa and Todorov to presectional paths. Later it was used to determine the possible shapes of the Auslander-Reiten components of A (see [11, 12]). More recently, the composite of irreducible morphisms was studied in [5], [6], [7] and [9]. The work made in the first three of these papers is based on the notion of degree of irreducible morphisms. The definition of the degree raises the following problem: Determine when dl(f) = ∞ or dr(f) = ∞. Consider an irreducible morphism f : X → Y with X indecomposable. Then, the following conditions have been related in the recent literature: (1)dl(f) = n < ∞, (2)Ker(f) lies in the Auslander-Reiten component containing X. Indeed, these two conditions were proved to be equivalent if the Auslander-Reiten component containing X is convex, generalized standard and with length ([8], actually this equivalence The first and third authors acknowledge partial support of CONICET, Argentina. The third author is a researcher of CONICET. The third author acknowledges financial support from the Department of Mathematics of FCEyN, Universidad Nacional de Mar del Plata and also from the PICS 3410 of CNRS, France. ha l-0 04 31 29 5, v er si on 1 11 N ov 2 00 9 Page 2 of 20 CLAUDIA CHAIO, PATRICK LE MEUR, SONIA TREPODE still holds true if one removes the convex hypothesis) and when the Auslander-Reiten quiver is standard ([4]). In this text, we shall see that such results are key-steps to show that the degree of irreducible morphisms is a useful notion to determine the representation type of A. Indeed, we recall the following well-known conjecture appeared first in [12] and related to the Brauer-Thrall conjectures: ”If the Auslander-Reiten quiver of A is connected, then A is of finite representation type.” This conjecture is related to the degree of irreducible morphisms as follows: In the above situation of assertions (1) and (2), the existence of f such that dl(f) = ∞ is related to the existence of at least two Auslander-Reiten components. Actually, it was proved in [8, Thm. 3.11] that if A is of finite representation type, then every irreducible morphism between indecomposables either has finite right degree or has finite left degree. Conversely, one can wonder if the converse holds true. In this text, we prove the following main theorem where we assume that k is an algebraically closed field. Theorem A. Let A be a connected finite dimensional k-algebra over an algebraically closed field. The following conditions are equivalent: (a) A is of finite representation type. (b) For every indecomposable projective A-module P , the inclusion rad(P ) →֒ P has finite right degree. (c) For every indecomposable injective A-module I , the quotient I → I/soc(I) has finite left degree. (d) For every irreducible epimorphism f : X → Y with X or Y indecomposable, the left degree of f is finite. (e) For every irreducible monomorphism f : X → Y with X or Y indecomposable, the right degree of f is finite. Hence, going back to the above conjecture, if one knows that the Auslander-Reiten quiver of A is connected, by (b) and (c) it suffices to study the degree of finitely many irreducible morphisms in order to prove that A is of finite representation type. Our proof of the above theorem only uses considerations on degrees and their interaction with coverings of translation quivers. In particular it uses no advanced characterization of finite representation type (such as the Brauer-Thrall conjectures or multiplicative bases, for example). The theorem shows that the degrees of irreducible morphisms are somehow related to the representation type of A. Note also that our characterization is expressed in terms of the knowledge of the degree of finitely many irreducible morphisms. In order to prove the theorem we investigate the degree of irreducible morphisms and more particularly assertions (1) and (2) above. Assuming that k is an algebraically closed field and given f : X → Y an irreducible epimorphism with X indecomposable, we prove that the assertion (1) is equivalent to (3) below and implies (2), with no assumption on the Auslander-Reiten component Γ containing X: (3)There exists Z ∈ Γ and h ∈ rad(Z,X)\rad(Z,X) such that fh = 0. Therefore, the existence of an irreducible monomorphism (or epimorphism) with infinite left (or right) degree indicates that there are more than one component in the Auslander-Reiten quiver (at least when Γ is generalized standard). We also prove that (2) implies (1) (and therefore implies (3)) under the additional assumption that Γ is generalized standard. The equivalence between (1) and (3) and the fact that it works for any Auslander-Reiten component are the chore facts in the proof of the theorem. For this purpose we use the covering techniques introduced in [13]. Indeed, these techniques allow one to reduce the study of the degree of irreducible morphisms in a component to the study of the degree of irreducible morphisms in a suitable covering called the generic covering. Among other things, the generic covering is a translation quiver with length. As was proved in [8] such a condition is particularly useful in the study of the degree of an irreducible morphism. The text is therefore organized as follows. In the first section we recall some needed definitions. In the second section we extend to any Auslander-Reiten component the pioneer result [13, 2.2, 2.3] on covering techniques which, in its original form, only works for the Auslander-Reiten quiver of representation-finite algebras. The results of this section are used in the third one to prove the various implications between assertions (1), (2) and (3) in 3.1 and 3.5. As explained above, these results have been studied previously and they were proved under additional assumptions. In particular, the corresponding corollaries proved at that time can be generalized accordingly. In the fourth section we prove our main Theorem A using the previous results. The proof of our main results are based on the covering techniques developed in the second section. In the ha l-0 04 31 29 5, v er si on 1 11 N ov 2 00 9 DEGREE AND REPRESENTATION TYPE Page 3 of 20 last section, we use these to study when the non-zero composite of n irreducible morphisms lies in the n+ 1-th power of the radical and we extend the cited-above result ([10]) of Igusa and Todorov on the composite of a sectional paths to sums of composites of sectional paths.