Vossieck: Tame and wild subspace problems

Assume that B is a finite-dimensional algebra over an algebraically closed field k, B a = Speck [B a] is the affme algebraic scheme whose R-points are the B | k[Ba]-m ~ structures on R a, and Mg is a canonical B | k[Bg]-module supported by k [Bg] d. Further, say that an affme subscheme V of Bg is class true if the functor F~: X ~ M ~ | X induces an injection between the sets of isomorph-ism classes of indecomposable finite-dimensional modules over k [q ~ and B. If B d contains aclass-true plane for some d, then the schemes B, contain class-true subschemes of arbitrary dimensions. Otherwise, ach Bg contains a finite number of classtme puncture straight lines L(d, i) such that for each n, almost each indecomposable B-module of dimension n is isomorphic to some Fz~d. ~)(X); furthermore, F aa,, ~ (X) is not isomorphic to F z(t,~} (Y) if ( d, i) ~ (l, j) and X ~ 0. The proof uses a reduction to subspace problems, for which an inductive algorithm permits us to prove corresponding statements. 1. Notation, Terminology, Objective. Throughout the paper, k denotes an algebraically closed field. By A we denote a k-category, i.e., a category whose morphism sets A(X, Y) are endowed with vector space structures over k such that the composition maps are bilinear. Furthermore, we suppose that A is an. aggregat