The standard form of a representation-finite algebra

The importance of covering techniques for the investigation of the representations of finite-dimensional algebras is well-known since [RI 1] and [BG]. When the algebra A admits only finitely many isomorphism classes of indecomposable representations, the definition of the fundamental group 0 of A given in [BG] requires the knowledge of all indecomposables. In the sequel, we describe <1>, the universal cover and the standard form of-<4 directly in terms of A. A very convenient way to determine whether a given algebra A is representation-finite (and to compute its representations) consists in finding a simply connected cover of a suitable degeneration of A ([BG], [G2], [GO]). In theorem 3.1. we describe the standard form A of A, which is the best possible degeneration, and we determine the universal Galois covering of A in terms of A (Theorem 1.5 and Remark 3.3 a). Our results heavily rest on the existence of a multiplicative basis in A, i. e. of a basis such that the product of two basis vectors is again a basis vector or else is zero. The existence of such bases in representation-finite algebras was first stated by KUPISCH in the symmetric case [K] (see also [KS]). It was then proved by BONGARTZ for algebras whose quiver contains no oriented cycle [B 1]. BONGARTZ' proof is easily extended to simply connected locally representation-finite "algebras" [BRL], a case to which our existence proof (*) Texie recu Ie 20 avril 1982, legerement revise Ie 10 decembre 1982. 0. BRETSCHER and P. GABRIEL, Mathematisches Institut Universitat Zurich, Ramistrasse 74, CH 8001 Zurich. Suisse. BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE 0037-9484/1983/21 /$ 5.00 © Gauthier-Villars 22 0. BRETSCHER AND P. GABRIEL can be reduced. Our own researches were started by ROITER 's publication [RO], in which the existence of a multiplicative basis for general representation-finite algebras is stated. Clearly, ROITER 's work lies on a higher level of difficulty, and we have not been in a position to check all the points in his developments. Paraphrasing Picard's comment on Poincare 's Duality Theorem (Picard et Simart, Theorie des Fonctions Algebriques, tome I, chap. II, 1897), we might say: « ROITER a donne une demonstration generate de Fexistence d'une base multiplicative. Sa demonstration repose sur des considerations entierement differentes, mais peut-etre plusieurs points auraient-ils besoin d'etre completes. Aussi, avons-nous suivi une autre voie, mais avons du nous limiter au cas des algebres standard. » Well-read mathematicians will easily interpret our proofs as statements on singular homology of triangulated topological spaces associated with the considered algebras. In fact, our intuition is geometrical, and so were our original proofs. The translation from geometry to algebra has been worked out during the spring vacation 1982 in order to take care of pure algebraists. For them we replaced short references to algebraic topology (to [ML], IV, 11.5 or [GZ], Ap. 11,3.6 for instance) by longer elementary variations on the snake-lemma. We hope that these variations will not be dismissed as mere exercises for a first-year course in homological algebra, since they come from, rest on and elucidate a mass of examples in representation-theory ([NR], [BGP], [L], [SZ], [BR], [BG], [BRL], [B2],[CG]). The method used here was presented in June 1981 in Oberwolfach, where corollary 2.2, 2.6, 2.11 and 3.3 b were stated. The remaining results were presented in november 1981 in a lecture at the university of Trondheim. Since then, we received from MARTINEZ and DE LA PE^IA the proof [MP1] of an older conjecture of ours which simplifies the demonstration of Theorem 1.5. Nevertheless, we maintained the first proof in paragraph 2 because of its own virtues (2.6-2.8). As they informed us at the beginning of March, MARTINEZ and DE LA PE^A also noticed that our Oberwolfach-proof for the existence of a multiplicative basis in schurian algebras extends to standard algebras [MP2]. Our investigations commenced with BONGARTZ and ROITER as a joint discussion which diverged too rapidly. We like to express our thanks to both of them. The notations are those introduced in [G I], [BG] and [G 2]. In particular, *• always denotes an algebraically closed field. TOME i l l 1983 N° l STANDARD FORM OF A REPRESENTATION-FINITE ALGEBRA 23 1. The group of constraints of a representation-finite algebra 1.1. We first fix the notations: Let A be a locally bounded ^--category ([BG], 2.1). By A (a, b) we denote the space ofmorphisms from a to b in A, by ^^(a,b)==A(a,b)^^A{a,b)^^A(a,b)... the radical series of A (a, b) considered as a bimodule over A (b, b) and A (a, a). We say that a morphism [ieA(a,b) has level ne^ if He^"A(fl, ^)\^"'" A(o, b)', the zero-morphism has level oo by definition. In the sequel, we assume that A is locally representationfinite ([BG], 2.2). We denote by 7==indA the category formed by chosen representatives of the isoclasses of indecomposable finitedimensional A-modules, and we agree that o*=A(?,fl) is chosen as a representative for each aeA. Given two objects a.be\. we set g(a, b,n)=sup{p : ^ I ^ a * , &*)=^"A (o,fc)*} for each w 6 ^ u { o c } , where J=>^J=>^ 2 1 . . . is the radical series of J([BG], 2.1). Wesaythata morphism [IE A (a, b) of level n has grade gW^g (a, b, n) in A. In case ^(n)=^< °o. this implies the existence of irreducible morphisms ([AR], § 1; [Gl],1.6) * pl p2 ^ i.* cr -^ Wi -»• w ^ . . . w^_ i --»• b* of J such that p* ̂ ... ̂ p^ e ̂ n + A A (a, b)* (remember that ^ +1 A (a, A) has codimension ^1 in ^A(fl, A); see [G2], 2.4). v In the following lemma, we denote by m -* n the arrow of r\ (= the Auslander-Reiten quiver of A) which is associated with an irreducible morphism v : m -»n. LEMMA. — With the above notations, the homotopy-cluss ([BG], 1 .2 ) S(a, 6, n) of the path * pl p; ^ i * cr -»Wi -> w ^ . . .w^-i -* o* ^/ r^ depends only on a, b and n. Proof. Let 71 : I\ -»• r\ be a universal covering and F : A (F^) -»/ a wellbehaved functor ([RI 2], 2.5;[BG], 3.1). Choose a sequence of morphisms BULLETIN DE LA SOCIETE MATHEMAT1QUE DE FRANCE 24 0. BRETSCHER AND P. GABRIEL of fc(!\) such that Fv,-n,e^/ for each L We then get F ( V g . . .Vi)-^.. .pie^'"/. Since we have