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Moduli of representations of finite dimensional algebras. (English) Zbl 0837.16005
Let $$A$$ be a finite dimensional algebra over an algebraically closed field $$k$$ and let $$Q$$ be its associated quiver (the set $$Q_0$$ of vertices of $$Q$$ consists of isomorphism classes of indecomposable projective modules). Let mod-$$A$$, resp. mod-$$kQ$$, denote the abelian category of finite dimensional representations of $$A$$, resp. $$Q$$. Fix a dimension vector $$\alpha\in Z^{Q_0}$$. Then isomorphism classes of representations of $$Q$$ with dimension vector $$\alpha$$ are in bijective correspondence with the orbits of the group $$\text{GL}(\alpha)$$ in a certain vector space $${\mathcal R}(Q,\alpha)$$. Let $$\Delta$$ denote the kernel of the representation of $$\text{GL}(\alpha)$$ in $${\mathcal R}(Q,\alpha)$$. Let $$V_A(\alpha)$$ denote the representations in $${\mathcal R}(Q,\alpha)$$ associated to $$A$$-modules with dimension vector $$\alpha$$; it is a closed $$\text{GL}(\alpha)$$-invariant subvariety of $${\mathcal R}(Q,\alpha)$$. Any $$\theta\in Z^{Q_0}$$ can be interpreted either as a character $$\chi_\theta$$ of $$\text{GL}(\alpha)$$ or as a homomorphism $$\theta:K_0(\text{mod-}kQ)\to Z$$. The main purpose of this paper is to show that the notions of stability and semistability arising from Mumford’s geometric invariant theory [see D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory (Springer-Verlag, 3rd edition 1993; Zbl 0797.14004)] applied to the action of $$\text{GL}(\alpha)$$ on $$V_A(\alpha)$$ can be translated into algebraic, $$K$$-theoretical, properties of $$\text{mod-}A$$. More concretely, a point $$x \in V_A (\alpha)$$ corresponding to an $$A$$-module $$M$$ is $$\chi_\theta$$-stable (or semistable) for the action of $$\text{GL} (\alpha)$$ if and only if $$M$$ is a $$\theta$$-stable (or semistable) module. Here $$M$$ is $$\theta$$-semistable (resp., stable) if $$\theta (M) = 0$$ and $$\theta (M') \geq 0$$ for any $$M' \subseteq M$$ (resp., semistable and $$\theta (M') = 0$$ for $$M' \subseteq M$$ implies $$M' = 0$$ or $$M' = M$$). The algebraic quotient $${\mathcal M}_A (\alpha, \theta) := V_A (\alpha) // (\text{GL} (\alpha), \chi_\theta)$$ is a coarse moduli space for families of $$\theta$$-semistable modules with dimension vector $$\alpha$$ up to $$S$$-equivalence [see also C. S. Seshadri, Ann. Math., II. Ser. 85, 303-336 (1967; Zbl 0173.230)] and is in fact a projective variety.

##### MSC:
 16D90 Module categories in associative algebras 16G20 Representations of quivers and partially ordered sets 14L30 Group actions on varieties or schemes (quotients) 16G30 Representations of orders, lattices, algebras over commutative rings
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