Piotr Dowbor

A characterization of hereditary rings of finite representation type

In this note we announce a characterization of hereditary rings R of finite representation type in terms of a sequence of partial Coxeter functors associated toR. The methods we use are theory of almost split sequences [1], partial Coxeter functors [2], [3], [5], representations of species [7], and some ideas of Ringel [8, §6]. We recall from [7] that a species M = (F(, ^fy)l>/e/ is a finite set of division rings Ft and F. F;. bimodules jMj. Throughout we will suppose M is a species such that jMj -=h 0 implies jMt = 0 for i ¥= /. M is called finite dimensional if the dimensions dif = dimQM,.)̂ ., dfi = dimF.(^My) are finite for i =£ ƒ. We denote by /i(M) the category of all finite dimensional right representations of M (see [6], [8] ). From M we derive an oriented valued graph (r, d, Ü) (not necessarily symmetrizable) with valued edges

A Comparison of Various Types of Degenerations for Algebras

Three different notions of geometric degeneration for finite dimensional algebras are compared. It is proved that for algebras A0 and A1 of same dimension, if A0 is a CB-degeneration of A1, then A0 is a degeneration of A1 in the classical sense; moreover, if additionally A0 and A1 are basic, then A0 is a CB-degeneration of A1 if and only if A0 is a rigid degeneration of A1. It is also shown that CB-degenerations along affine lines in the different dimension case do not differ so much from those in the equal dimension case.

Galois covering technique and tame non-simply connected posets of polynomial growth

Abstract A combinatorial criterion for polynomial growth of partially ordered sets which are not simply connected is given. It is obtained by use of Galois covering techniques applied to poset representations. For this purpose a relative version of the basic Galois covering theory of k -categories is developed in the paper. The remaining part of the proof of the main result is based on standard methods like “peak reductions” and Splitting Lemma used in the theory of multipeak posets.

On equivalences of bernstein-gelfand-gelfand, beilinson and happel

Bernstein-Gelfand-Gelfand showed that the derived category of coherent sheaves on the projective n-space Db(cohPn) is equivalent to the stable ategory of the category of Z-graded finite-dimensional modules over the exterior algebra. At the same time Beilinson gave a description of Db(cohPn) which can be interpreted in terms of tilting theory. We compare these two characterizations using Happels description of the derived category of modules over a finite-dimensional algebra of fnite global dimension.