×

zbMATH — the first resource for mathematics

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
PDF BibTeX Cite
Full Text: DOI EuDML
References:
[1] Abeasis S., Del Fra A., Kraft H.: The geometry of representations ofA n . Math. Ann.256, 401-418 (1981) · Zbl 0477.14027
[2] Boutot J.-F.: Frobenius et cohomologie locale, in séminaire Bourbaki, exposé 453, nov. 1974. Lect. Notes Math.514. Berlin Heidelberg New York: Springer (1976)
[3] Burns D.: On rational singularities in dimensions >2. Math. Ann.211, 237-244 (1974) · Zbl 0287.32010
[4] Elkik R.: Singularités rationnelles et déformations. Invent. math.47, 139-147 (1978) · Zbl 0383.14005
[5] Grauert H., Riemenschneider O.: Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen. Invent. math.11, 263-290 (1970) · Zbl 0202.07602
[6] Hartshorne R., Ogus A.: On the factoriality of local rings of small embedding codimension. Commun. Algebra1, 415-437 (1974) · Zbl 0286.13013
[7] Hironaka H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math.79, 109-326 (1964) · Zbl 0122.38603
[8] Hochster M.: Cohen-Macaulay rings and modules. In: Proceedings of the International Congress of Mathematicians, helsinki 1978, vol. 1, pp. 291-298 Helsinki: Academia Scientiarium Fennica 1980
[9] Hochster M., Roberts J.L.: Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay. Adv. Math.,13, 115-175 (1974) · Zbl 0289.14010
[10] Kraft H., Procesi C.: On the geometry of conjugacy classes in classical groups. Comment. Math. Helvetici57, 539-602 (1982) · Zbl 0511.14023
[11] Kempf G.: On the collapsing of homogeneous bundles. Invent. math.37, 229-239 (1976) · Zbl 0338.14015
[12] Kempf G.: Some quotient varieties have rational singularities, Michigan Math. J.24, 347-352 (1977) · Zbl 0385.14016
[13] Kempf G., Knudsen F., Mumford D., Saint-Donat B.: Toroidal embeddings I. Lect. Notes Math.339. Berlin Heidelberg New York: Springer 1973 · Zbl 0271.14017
[14] Viehweg E.: Rational singularities of higher dimensional schemes. Proc. Am. Math. Soc.63, 6-8 (1977) · Zbl 0352.14003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.