Tame and wild subspace problems - Gabriel Р.; Nazarova, L. A.; Roiter, A. V.; Sergeychuk, V. V.; Vossieck, D. (Ukrains’kyi Matematychnyi Zhurnal, 1993)
Assume that B is a finite-dimensional algebra over an algebraically closed field k, B_d = Spec k[B_d] is the affine algebraic scheme whose R-points are the B ⊗_k k[B_d]-module structures on R^d, and M_d is a canonical B ⊗_k k[B_d]-module supported by k[Bd^]d. Further, say that an affine subscheme Ν of B_d isclass true if the functor F_gn ∶ X → M_d ⊗_k[B] X induces an injection between the sets of isomorphism classes of indecomposable finite-dimensional modules over k[Ν] and B. If B_d contains a class-true plane for some d, then the schemes B_e contain class-true subschemes of arbitrary dimensions. Otherwise, each B_d 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 F_L(d, i) (X); furthermore, F_L(d, i) (X) is not isomorphic to F_L(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.