Degenerations over \((A_{\infty})\)-singularities and construction of degenerations over commutative rings. (English) Zbl 1408.13022
Let \(k\) be a field, and let \(R\) be a \(k\)-algebra. Let \(M\) and \(N\) be \(R\)-modules with equal finite dimension as \(k\)-vector spaces. We say that \(M\) degenerates to \(N\) if in the module variety the point corresponding to \(M\) belongs to the Zariski closure of the orbit of the point corresponding to \(N\). The authors give a necessary condition of degeneration via matrix representations, and consider degenerations of indecomposable Cohen-Macaulay modules over hypersurface singularities of type \((A_{\infty})\). They also provide a method to construct degenerations of finitely generated modules over commutative rings.
Reviewer: Vladimir P. Kostov (Nice)
MSC:
| 13C14 | Cohen-Macaulay modules |
| 14D06 | Fibrations, degenerations in algebraic geometry |
| 16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
Keywords:
degeneration of modules; (maximal) Cohen-Macaulay module; hypersurface singularity of type \((A_{\infty})\); countable Cohen-Macaulay representation type; Knörrer’s periodicityReferences:
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