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Reductivity and finiteness of pseudo-reflections of algebraic groups and homogeneous fiber bundles. (English) Zbl 1296.13006

Let \(G\) be an affine algebraic group defined over an algebraically closed field \(K\), and suppose \(X\) is an affine \(K\)-scheme upon which \(G\) acts (i.e., \(G\) acts \(K\)-rationally by \(K\)-algebra homomorphisms on the ring \(\mathcal{O}_X(X)\)). It is a fundamental result in Geometric Invariant Theory that \(G\) is reductive if and only if the ring \(\mathcal{O}_X(X)^{G^0}\) is finitely generated for all actions of \(G^0\) on affine varieties \(X\) (see, e.g., D. Mumford et al. [Geometric invariant theory. 3rd enl. ed. Berlin: Springer-Verlag (1993; Zbl 0797.14004)] for a full description of the history). The main result of this paper provides new geometric characterisations of reductivity in terms of properties of \(G\)-actions on Krull schemes, where the scheme \(X\) is said to be Krull if \(\mathcal{O}_X(X)\) is a Krull domain.
Let \(\mathfrak{P}\) denote a prime ideal of \(\mathcal{O}_X(X)\), and let \(\ell_{\mathfrak{P}}(G)\) denote the inertia group. Then the pseudo-reflection group \(\mathfrak{R}(X,G)\) of the action is the subgroup of \(G\) generated by all \(\ell_{\mathfrak{P}}(G)\) as \(\mathfrak{P}\) runs over the prime ideals of \(\mathcal{O}_X(X)\) of height \(1\) whose intersection with \(\mathcal{O}_X(X)^G\) also has height one. This pseudo-reflection group acts on \(X\) so we can look at the image of \(\mathcal{R}(X,G)\) in Aut \(X\). The main result shows that \(G\) is reductive if and only if the image of \(\mathfrak{R}(X,G^0)\) in Aut \(X\) is finite for all \(G^0\)-actions on affine Krull schemes \(X\).

MSC:

13A50 Actions of groups on commutative rings; invariant theory
14R20 Group actions on affine varieties
20G05 Representation theory for linear algebraic groups
14L30 Group actions on varieties or schemes (quotients)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies

Citations:

Zbl 0797.14004
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References:

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