On K_0 of locally finte categories

We calculate the Grothendieck group $K_0(\cal A)$, where $\cal A$ is an additive category, locally finite over a Dedekind ring and satisfying some additional conditions. The main examples are categories of modules over finite algebras and the stable homotopy category $\mathsf{SW}$ of finite CW-complexes. We show that this group is a direct sum of a free group arising from localizations of the category $\cal A$ and a group analogous to the groups of ideal classes of maximal orders. As a corollary, we obtain a new simple proof of the Freyd's theorem describing the group $K_0(\mathsf{SW})$.Comment: 21 page

Derived tame and derived wild algebras

We prove that every finite dimensional algebra over an algebraically closed field is either derived tame or derived wild. We also prove that any deformation of a derived tame algebra is derived tame.Comment: 15 page

Dimensions of finite type for representations of partially ordered sets

We consider the dimensions of finite type of representations of a partially ordered set, i.e. such that there is only finitely many isomorphism classes of representations of this dimension. We give a criterion for a dimension to be of finite type. We also characterize those dimensions of finite type, for which there is an indecomposable representation of this dimension, and show that there can be at most one indecomposable representation of any dimension of finite type. Moreover, if such a representation exists, it only has scalar endomorphisms.Comment: 14 page

Generators and relations for wreath products

Generators and defining relations for wreath products of groups are given. Under some condition (conormality of the generators) they are minimal. In particular, it is just the case for the Sylow subgroups of the symmetric groups.Comment: 4 page

Semicontinuity for representations of one-dimensional Cohen-Macaulay rings

We show that the number of parameters for CM-modules of prescribed rank is semi-continuous in families of CM rings of Krull dimension 1. This transfers a result of Knoerrer from the commutative to the not necessarily commutative case. For this purpose we introduce the notion of ``dense subrings'' which seems rather technical but, nevertheless, useful. It enables the construction of ``almost versal'' families of modules for a given algebra and the definition of the ``number of parameters''. The semi--continuity implies, in particular, that the set of so-called ``wild algebras'' in any family is a countable union of closed subsets. A very exciting problem is whether it is actually closed, hence whether the set of tame algebras is open. However, together with the results of a former paper of the authors the semi-continuity implies that tame is indeed an open property for curve singularities (commutative CM rings). An analogous procedure leads to the semicontinuity of the number of parameters in other cases, like representations of finite dimensional algebras or finite dimensional bimodules.Comment: LaTeX2

One branch curve singularities with at most 2-parameter families of ideals

A criterion is given in order that the ideals of a one branch curve singularity form at most 2-parameter families. Namely, we present a list of plane curve singularities from the Arnold's classification which are the smallest among all one branch singularities having at most 2-parameter families of ideals.Comment: 10 page