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Singularités rationnelles et quotients par les groupes réductifs. (Rational singularities and quotients by reductive groups). (French) Zbl 0619.14029
A ring homomorphism $$A\to A'$$ is called pure, if the map $$M\to A'\otimes_ AM$$ is injective for every A-module M. A morphism of affine schemes is pure, if so is the corresponding homomorphism of rings. In terms of a resolution $$Z\to X$$ of singularities, one defines the notion of rational singularities.
Theorem. Let X’$$\to X$$ be a pure morphism of affine schemes. Then if X’ has rational singularities, then so does X.
This implies that quotient spaces of algebraic actions of algebraic reductive groups (over fields of zero characteristic) on affine scheme with rational singularities have rational singularities. This generalizes a few previous results.
Reviewer: L.N.Vaserstein

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 14B05 Singularities in algebraic geometry 14M17 Homogeneous spaces and generalizations 14A15 Schemes and morphisms
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