Abstract
Assume thatB is a finite-dimensional algebra over an algebraically closed fieldk, B d =Spec k[(B d ] is the affine algebraic scheme whoseR-points are theB ⊗k k[Bd]-module structures onR d, and Md is a canonical B⊗k k[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] andB. If Bd contains a class-true plane for somed, then the schemes Be contain class-true subschemes of arbitrary dimensions. Otherwise, each Bd contains a finite number of classtrue puncture straight linesL(d, i) such that for eachn, almost each indecomposableB-module of dimensionn is isomorphic to someF L(d, i) (X); furthermore,F L(d, i) (X) is not isomorphic toF L(l, j) (Y) if(d, i) ≠ (l, j) andX ≠ 0. The proof uses a reduction to subspace problems, for which an inductive algorithm permits us to prove corresponding statements.
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References
I. M. Gel'fand and V. A. Ponomarev, “Remarks on the classification of a pair of commuting linear transformations in finite-dimensional spaces,”Funkts. Anal. Prilozh.,3, 81–82 (1969).
P. Donovan and M. R. Freislich, “Some evidence for an extension of the Brauer-Thrall conjecture,”Sonderforschungsbereich Theor. Math.,40, 24–26 (1972).
Yu. A. Drozd, “Tame and wild matrix problems,”Lect. Notes Math.,832, 242–258 (1980).
W. W. Grawley-Boevey, “On tame algebras and bocses,”Proc. London Math. Soc. III., Ser. 56, 451–483 (1988).
W. W. Grawley-Boevey, “Tame algebras and generic modules,” Ibid.,Ser. 63, 241–265 (1991).
A. V. Roiter, “Matrix problems and representations of bosses,”Lect. Notes Math.,831, 288–324 (1980).
L. A. Nazarova, A. V. Roiter, and P. Gabriel, “Representations idecomposables: un algorithme,”C. R. Acad. Sci. Paris,307, ser. 1, 701–706 (1988).
M. Auslander and S.O. Smalo, “Almost split sequences in subcategories,”J. Algebra,69, 426–454 (1981); Addendum,71,592–594 (1981).
P. Gabriel and A. V. Roiter, “Representations of finite-dimensional algebras,” in:Encyclopaedia of Math. Sci., Vol. 73., Algebra VIII, Springer-Verlag (1992).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 313–352, March, 1993.
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Gabriel, P., Nazarova, L.A., Roiter, A.V. et al. Tame and wild subspace problems. Ukr Math J 45, 335–372 (1993). https://doi.org/10.1007/BF01061008
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DOI: https://doi.org/10.1007/BF01061008