The existence of bounded infinite Tr-orbits

We construct an indecomposable module over a symmetric algebra whose DTr-orbit is infinite and bounded. This yields a counterexample to a conjecture which states that the number of modules in an Auslander-Reiten component having the same length is finite. Let A be an Artin algebra, W a connected component of the AuslanderReiten quiver of A, and DTr the Auslander-Reiten translation [1]. In [8], Ringel asked whether the number of modules having the same length in W is always finite. This is the case when A is a hereditary algebra [2, 10] or a tame algebra [4]. For an arbitrary algebra A, the question has an affirmative answer if W has at most finitely many nonperiodic DTr-orbits [3, 6] or is a regular component of the form ZA with A one of Al, Boo, COO, or Do [7]. The aim of this paper is to show that the above problem has no affirmative answer in general. We shall construct a local symmetric algebra whose Auslander-Reiten quiver contains a bounded infinite DTr-orbit. Our example will be a modification of that given by the second author in a different context [9]. Let K be a field which contains an element p of infinite multiplicative order. Let R be the polynomial ring over K in noncommutating variables X and Y modulo the ideal generated by X21 y2, and YX pXY. Then R is a local Frobenius algebra over K with radical J(R) = xR + yR, J(R)2 = Soc(R) = xyR, and J(R)3 = 0, where x, y denote the residue classes of X, Y, respectively. Let DR = HomK(R, K) be the dual of R with the following RR-bimodule structure: given r', r" E R and f E DR, r'fr" is the K-linear map which sends r E R to f(r"rr'). Let T be the trivial extension algebra of R by DR which is the K-vector space T = R e DR with multiplication given by (r, f)(r', f') = (rr', rf'+fr') for r, r' E R and f, f' E DR. Then T is a local symmetric K-algebra with radical J(T) = {(r, f)Ir E J(R), f E DR} Received by the editors March 25, 1993. 1991 Mathematics Subject Classification. Primary 16G10, 16G70.