Tame and wild subspace problems

Gabriel, Patrick; Назарова, Л. А.; Roiter, A. V.; Sergeichuk, Vladimir V.; Vossieck, Dieter

doi:10.1007/bf01061008

Cited by 44 publications

(42 citation statements)

References 6 publications

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“…In these cases C is tame; see [7,9] for a precise definition. In all other cases C is wild, and a classification of the indecomposable modules is regarded to be impossible.…”

Section: Remarks On Previous Workmentioning

confidence: 98%

Module Varieties over Canonical Algebras

Barot¹,

Schröer²

2001

Journal of Algebra

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The main purpose of this paper is the study of module varieties over the class of canonical algebras, providing a rich source of examples of varieties with interesting properties. Our main tool is a stratification of module varieties, which was recently introduced by Richmond. This stratification does not require a precise knowledge of the module category. If it is finite, then it provides a method to classify irreducible components. We determine the canonical algebras for which this stratification is finite. In this case, we describe the algorithm for calculating the dimension of the variety and the number of irreducible components of maximal dimension. For an infinite family of examples we give easy combinatorial criteria for irreducibility, Cohen-Macaulay and normality.  2001 Elsevier Science

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“…In these cases C is tame; see [7,9] for a precise definition. In all other cases C is wild, and a classification of the indecomposable modules is regarded to be impossible.…”

Section: Remarks On Previous Workmentioning

confidence: 98%

Module Varieties over Canonical Algebras

Barot¹,

Schröer²

2001

Journal of Algebra

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“…On the other hand, it was proved in ([1], 4.2, 4.3) and ( [3], 9.1, 9.4) that the category mod A of representations of an arbitrary finite-dimensional algebra A over k coincides with the category of representations of a certain vectroid V in the following sense: There exists an injective indecomposable A -module P such that the category of all A -modules that do not contain P as a direct summand is epivalent to Rep V. In this case, all these submodules are cyclic, X i m x V(X,X), and ml X .... x = , mdimX is a triangular basis of X e V (see Lemma 1 ).…”

Section: A Triple ( Uf X) Consisting Of the Spaces U ~ Modk And X~ mentioning

confidence: 99%

Elementary and multielementary representations of vectroids

et al. 1995

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“…Lemma 4 implies the following assertion: (1,1), (2,2), and (3,3) and is equal to implies that q0 is aproduct of basis morphisms; hence, M(q0) = eii + ejj and char(k) = 2.…”

Section: Existence Of a Multiplicative Basis For A Finitely Spaced Momentioning

confidence: 99%

“…A scalarly multiplicative basis is called normed if it satisfies the following condition: (Ra(a, a)), all steps of ~ are not higher than (2, 1) and (3,2). Since ~q~ ~ M (IPva(a, a) …”

Section: I E (S A))i-lm(a)\(~(a A))im(a) Is a Basis In M(a) It Imentioning

confidence: 99%

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Existence of a multiplicative basis for a finitely spaced module over an aggregate

Roiter

Sergeichuk

1994

Ukr Math J

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It is proved that a finitely spaced module over a k-category admits a multiplicative basis (such a module gives rise to a matrix problem in which the allowed column transformations are determined by a module structure, the row transformations are arbitrary, and the number of canonical matrices is finite).It was proved in [1] that a finite-dimensional algebra having finitely many isoclasses of indecomposable representations admits a multiplicative basis. In [2, Secs. 4.10-4.12], an analogous hypothesis was formulated for finitely spaced modules over an aggregate and an approach to its proof was proposed. The purpose of the present paper is to prove this hypothesis. Below, k always denotes an algebraically closed field.Let us recall some definitions given in [2] (see also [3]).By definition, an aggregate A over k is a category that satisfies the following conditions: As a consequence, each X ~ A is a finite direct sum of indecomposables and the algebra of endomorphisms of each indecomposable is local.We denote by flA a spectroid of N, i.e., a full subcategory formed by chosen representatives of the isoclasses of indecomposables; let RA be the radical of A. Suppose that flA has finitely many objects. For each a, b ~ fiN,

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Tame and wild subspace problems

Cited by 44 publications

References 6 publications

Module Varieties over Canonical Algebras

Module Varieties over Canonical Algebras

Elementary and multielementary representations of vectroids

Existence of a multiplicative basis for a finitely spaced module over an aggregate

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