Tame and wild subspace problems
Abstract
Assume that B is a finite-dimensional algebra over an algebraically closed field k, Bd=Speck[Bd] is the affine algebraic scheme whose R-points are the B⊗kk[Bd]-module structures on Rd, and Md is a canonical B⊗kk[Bd]-module supported by k[Bd]d. Further, say that an affine subscheme Ν of Bd isclass true if the functor Fgn∶X→Md⊗k[B]X induces an injection between the sets of isomorphism classes of indecomposable finite-dimensional modules over k[Ν] and B. If Bd contains a class-true plane for some d, then the schemes Be contain class-true subschemes of arbitrary dimensions. Otherwise, each Bd contains a finite number of classtrue puncture straight lines L(d,i) such that for eachn, almost each indecomposable B-module of dimensionn is isomorphic to some FL(d,i)(X); furthermore, FL(d,i)(X) is not isomorphic to FL(l,j)(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.
Published
25.03.1993
How to Cite
GabrielР., L. A. Nazarova, A. V. Roiter, V. V. Sergeychuk, and D. Vossieck. “Tame and Wild Subspace Problems”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 45, no. 3, Mar. 1993, pp. 313–352, https://umj.imath.kiev.ua/index.php/umj/article/view/5815.
Issue
Section
Research articles