Tame stacks in positive characteristic

Abramovich, Dan; Olsson, Martin; Vistoli, Angelo

Annales de l'Institut Fourier, Tome 58 (2008) no. 4, pp. 1057-1091.

Nous introduisons et étudions, en caractéristique positive et mixte, une classe de champs algébriques à champs d’inertie finis, appelés champs algébriques modérés. Cette classe inclue les champs modérés de Deligne–Mumford, et on peut dire qu’elle se comporte mieux que la classe des champs généraux de Deligne–Mumford. En outre, nous donnons une caractérisation complète des schémas en groupes finis plats et linéairement réductifs sur une base quelconque. Notre résultat principal est le suivant  : un champ algébrique modéré est, localement dans la topologie étale, un quotient par une action d’un schéma en groupes fini linéairement réductif.

We introduce and study a class of algebraic stacks with finite inertia in positive and mixed characteristic, which we call tame algebraic stacks. They include tame Deligne-Mumford stacks, and are arguably better behaved than general Deligne-Mumford stacks. We also give a complete characterization of finite flat linearly reductive schemes over an arbitrary base. Our main result is that tame algebraic stacks are étale locally quotient by actions of linearly reductive finite group schemes.

Classification : 14A20, 14L15
Keywords: Algebraic stacks, moduli spaces, group schemes
Mot clés : champs algébriques, espaces des modules, schémas en groupes

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