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

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

Representations of nodal algebras of type A

We define nodal finite dimensional algebras and describe their structure over an algebraically closed field. For a special class of such algebras (type A) we find a criterion of tameness.Comment: 19 pages