Finite Subsghemes of Group Schemes

Shatz, Stephen S.

Canadian journal of mathematics, Tome 22 (1970) no. 5, pp. 1079-1081

Résumé

If G is an ordinary group and H is a non-empty subset of G, then there are two elementary criteria for H to be a subgroup of G. The first and more general is that the mapping H × H → G × G → G, via 〈x, y〉 ⟼ xy –1 factor through H. The second is that H be finite and closed under multiplication.In the category of group schemes, if one writes down the hypotheses for the first criterion in diagram form, one can supply the proof by a suitable translation of the classical arguments. The only point that causes any difficulty whatsoever is that one must assume that the structure morphism πH: H → S (S is the base scheme) is an epimorphism in order to factor the identity section through H. The second criterion is also true for group schemes under a mild finite presentation hypothesis. It is our aim to provide a simple proof for the following theorem.

Détail

DOI : 10.4153/CJM-1970-124-3

Bibliographie

[1] 1. Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Rédigés avec la collaboration de J. Dieudonné, Inst. Hautes Études Sci. Publ. Math. No. 28 (1966). Google Scholar