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Top-stable degenerations of finite dimensional representations. II. (English) Zbl 1303.16017

Let \(\Lambda\) be a finite dimensional algebra over an algebraically closed field. The authors prove that, for any semisimple object \(T\in\Lambda\text{-mod}\), the class of those \(\Lambda\)-modules with fixed dimension vector \(d\) and top \(T\) which do not permit any proper top-stable degenerations possesses a fine moduli space \(\mathfrak{ModuliMax}_d^T\) that is a projective variety. The authors show that any projective variety arises as \(\mathfrak{ModuliMax}_d^T\) for suitable \(\Lambda\), \(T\) and \(d\) (Example 5.4).
The following classification theorem is proved. Theorem B. For any semisimple \(T\in\Lambda\text{-mod}\), the modules of dimension vector \(d\) which are degeneration-maximal among those with top \(T\) have a fine moduli space \(\mathfrak{ModuliMax}_d^T\), that classifies them up to isomorphism. The variety \(\mathfrak{ModuliMax}_d^T\) is projective. Moreover, given any module \(M\) whose top \(M/JM\) is contained in \(T\), the closed sub-variety of \(\mathfrak{ModuliMax}_d^T\) consisting of the points that correspond to degenerations of \(M\) is a fine moduli space for the maximal top-\(T\) degenerations of \(M\).
In Theorem A a structural characterization of modules with no proper top-stable degenerations is given.
For part I see B. Huisgen-Zimmermann [Proc. Lond. Math. Soc. (3) 96, No. 1, 163-198 (2008; Zbl 1207.16010)].

MSC:

16G10 Representations of associative Artinian rings
14D06 Fibrations, degenerations in algebraic geometry
14D20 Algebraic moduli problems, moduli of vector bundles
16G20 Representations of quivers and partially ordered sets
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)

Citations:

Zbl 1207.16010

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References:

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