Tame and wild subspace problems

• Datum: 1993
• Hauptverfasser: Gabriel, Р., Nazarova, L. A., Roiter, A. V., Sergeychuk, V. V., Vossieck, D., Габріель, Р., Назарова, Л. А., Ройтер, А. В., Сергійчук, В. В., Воссик, Д.
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 1993
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/5815
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

Assume that $B$ is a finite-dimensional algebra over an algebraically closed field $k$, $B_d = \text{Spec} k[B_d]$ 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.