Geometry of directing modules over tame algebras. (English) Zbl 0965.16009
Let \(A\) be a finite dimensional associative \(K\)-algebra with an identity over an algebraically closed field \(K\), let \(\text{mod}_A\) be the category of all finite dimensional left \(A\)-modules, and let \(\text{mod}_A({\mathbf d})\) be the module variety of \(A\)-modules of dimension \(\mathbf d\), where \(\mathbf d\) is a non-negative vector in the Grothendieck group \(K_0(A)\) of \(A\) [see A. Skowroński, G. Zwara, Contemp. Math. 229, 343-356 (1998; Zbl 0955.16014)].
The main aim of the paper is to describe conditions under which the module variety \(\text{mod}_A({\mathbf d})\) for the dimension-vector \(\mathbf d\) of arbitrary directing modules over the tame algebra \(A\) is normal, smooth, complete intersection, or it has other specific geometric properties. First some preliminary material concerning the theory of tame tilted algebras and their module categories is given. Then the irreducible components of \(\text{mod}_A({\mathbf d})\) and its maximal \(G({\mathbf d})\)-orbits are described, where \(A\) is a tame algebra, \(\mathbf d\) is the dimension-vector of a directing (indecomposable) \(A\)-module, \(G({\mathbf d})\) is the product of general linear groups acting on the affine variety \(\text{mod}_A({\mathbf d})\). Among other things it is proved that \(\text{mod}_A({\mathbf d})\) is a complete intersection. In conclusion some interesting examples are considered. The authors remark that the main results of this paper can be applied to the study of geometry of modules over tame quasi-tilted algebras [G. Bobiński, A. Skowroński, Colloq. Math. 79, No. 1, 85-118 (1999; Zbl 0994.16009)].
The main aim of the paper is to describe conditions under which the module variety \(\text{mod}_A({\mathbf d})\) for the dimension-vector \(\mathbf d\) of arbitrary directing modules over the tame algebra \(A\) is normal, smooth, complete intersection, or it has other specific geometric properties. First some preliminary material concerning the theory of tame tilted algebras and their module categories is given. Then the irreducible components of \(\text{mod}_A({\mathbf d})\) and its maximal \(G({\mathbf d})\)-orbits are described, where \(A\) is a tame algebra, \(\mathbf d\) is the dimension-vector of a directing (indecomposable) \(A\)-module, \(G({\mathbf d})\) is the product of general linear groups acting on the affine variety \(\text{mod}_A({\mathbf d})\). Among other things it is proved that \(\text{mod}_A({\mathbf d})\) is a complete intersection. In conclusion some interesting examples are considered. The authors remark that the main results of this paper can be applied to the study of geometry of modules over tame quasi-tilted algebras [G. Bobiński, A. Skowroński, Colloq. Math. 79, No. 1, 85-118 (1999; Zbl 0994.16009)].
Reviewer: Aleksandr G.Aleksandrov (Moskva)
MSC:
| 16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |
| 14L30 | Group actions on varieties or schemes (quotients) |
| 14M10 | Complete intersections |
| 14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |
| 16P10 | Finite rings and finite-dimensional associative algebras |
| 16D90 | Module categories in associative algebras |
Keywords:
finite dimensional modules; finitely generated algebras; indecomposable Auslander-Reiten quivers; degenerations of modules; directing modules; tame tilted algebras; quiver representations; indecomposable modules; complete intersections; categories of modulesReferences:
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