Über einen Zusammenhang zwischen darstellungsendlichen Algebren und geordneten Mengen. (On a connection between representation finite algebras and ordered sets) (Q1100539)

Let A be a finite dimensional associative algebra with unity over an algebraically closed field K with characteristic \(\neq 2\), for convenience. If A is representation-finite, i.e. A has a finite number of indecomposable matrix representations, then \textit{R. Bautista}, \textit{P. Gabriel}, \textit{A. Rojter} and \textit{L. Salmerón} [Invent. Math. 81, 217- 285 (1985; Zbl 0575.16012)] showed that up to Morita equivalence, A is uniquely determined by its ray-category P(A). This category P(A) is derived from the lattice-ordered semigroup of ideals of A which is finite since representation-finite algebras A are arithmetical in the sense of \textit{L. Fuchs} [Comment. Math. Helv. 23, 334-341 (1949; Zbl 0040.301)]. In passing to the universal cover of P(A), the representation-finiteness of A can be reduced to local representation-finiteness of a simply connected ray category P with finite chains and intervals. This, however, is the case iff P does not contain a convex subcategory isomorphic to a member of the Bongartz-Happel-Vossieck list [cf. Bautista et. al., loc. cit.]. The author achieves a remarkable reduction of this criterion to its (in our opinion) natural form, replacing the BHV-list by \textit{M. M. Kleiner}'s list of five critical posets [J. Sov. Math. 23, 607-615 (1975); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 28, 32-41 (1972; Zbl 0345.06001)]. He constructs to each object s in P some finite poset consisting of certain special convex subcategories of P such that P is locally representation finite if and only if all these posets are representation-finite. One half of his proof depends on the theory of generalized fiber sums developed by W. Müller and the author.