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.

[1] Abramovich, D.; Corti, A.; Vistoli, A. Twisted bundles and admissible covers, Comm. Algebra, Volume 31 (2003), pp. 3547-3618 | DOI | MR | Zbl

[2] Abramovich, D.; Graber, T.; Vistoli, A. Gromov–Witten theory of Deligne–Mumford stacks, preprint (http://www.arxiv.org/abs/math.AG/0603151)

[3] Abramovich, D.; Vistoli, A. Compactifying the space of stable maps, J. Amer. Math. Soc, Volume 15 (2002), pp. 27-75 | DOI | MR | Zbl

[4] Artin, M. Versal deformations and algebraic stacks, Invent. Math., Volume 27 (1974), pp. 165-189 | DOI | MR | Zbl

[5] Behrend, K.; Noohi, B. Uniformization of Deligne–Mumford curves, J. Reine Angew. Math. (to appear) | Zbl

[6] Berthelot, P.; Grothendieck, A.; Illusie, L. Théorie des Intersections et Théorème de Riemann-Roch (SGA 6), 225, Springer Lecture Notes in Math, 1971 | MR | Zbl

[7] Conrad, B. Keel-Mori theorem via stacks (preprint available at http://www.math.lsa.umich.edu/ bdconrad)

[8] Deligne, P.; Mumford, D. The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math., Volume 36 (1969), pp. 75-109 | DOI | Numdam | MR | Zbl

[9] Demazure, M.; al. Schémas en groupes, Lecture Notes in Mathematics, 151, 152 and 153, Springer-Verlag, 1970 | MR | Zbl

[10] Dieudonné, J.; Grothendieck, A. Éléments de géométrie algébrique, 4, 8, 11, 17, 20, 24, 28, 32, Inst. Hautes Études Sci. Publ. Math., 1961–1967

[11] Giraud, J. Cohomologie non abélienne, Springer-Verlag, Berlin, 1971 (Die Grundlehren der mathematischen Wissenschaften, Band 179) | MR | Zbl

[12] Gorenstein, D. Finite groups, Chelsea Publishing Co., New York, 1980 (xvii+519 pp) | MR | Zbl

[13] Gruson, L.; Raynaud, M. Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math., Volume 13 (1971), pp. 1-89 | Zbl

[14] Illusie, L. Complexe cotangent et déformations. I, Lecture Notes in Mathematics, 239, Springer, Berlin, 1971 | MR | Zbl

[15] Jacobson, N. Lie algebras, Republication of the 1962 original, Dover Publications, Inc., New York, 1979 (ix+331 pp. ISBN: 0-486-63832-4) | MR

[16] Keel, S.; Mori, S. Quotients by groupoids, Ann. of Math. (2), Volume 145 (1997) no. 1, pp. 193-213 | DOI | MR | Zbl

[17] Kleiman, S. The Picard scheme, Fundamental algebraic geometry, Math. Surveys Monogr. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 235-321 | MR | Zbl

[18] Kresch, A. Geometry of Deligne–Mumford stacks, preprint

[19] Laumon, G.; Moret-Bailly, L. Champs Algébriques, Ergebnisse der Mathematik un ihrer Grenzgebiete, 39, Springer-Verlag, 2000 | MR | Zbl

[20] Milne, J. S. Étale cohomology, Princeton University Press, 1980 | MR | Zbl

[21] Mumford, D. Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5., Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970 (viii+242 pp) | MR | Zbl

[22] Olsson, M. A boundedness theorem for Hom-stacks, preprint, 2005 | MR

[23] Olsson, M. On proper coverings of Artin stacks, Advances in Mathematics, Volume 198 (2005), pp. 93-106 | DOI | MR | Zbl

[24] Olsson, M. Deformation theory of representable morphisms of algebraic stacks, Math. Zeit., Volume 53 (2006), pp. 25-62 | DOI | MR | Zbl

[25] Olsson, M. Hom-stacks and restriction of scalars, Duke Math. J., Volume 134 (2006), pp. 139-164 | DOI | MR | Zbl

[26] Olsson, M. Sheaves on Artin stacks, J. Reine Angew. Math. (Crelle’s Journal), Volume 603 (2007), pp. 55-112 | DOI

[27] Romagny, M. Group actions on stacks and applications, Michigan Math. J., Volume 53 (2005) no. 1, pp. 209-236 | DOI | MR | Zbl

[28] Saavedra Rivano, N. Catégories Tannakiennes, Lecture Notes in Mathematics, 265, Springer-Verlag, Berlin-New York, 1972 | MR | Zbl

[29] Thomason, R. W. Algebraic K-theory of group scheme actions, Algebraic Topology and Algebraic K-theory, 113, Annals of Mathematical Studies, Princeton University Press, Princeton, 1987 | MR | Zbl

[30] Vistoli, A. Grothendieck topologies, fibered categories and descent theory, Fundamental algebraic geometry (2005), pp. 1-104 (Math. Surveys Monogr., 123, Amer. Math. Soc., Providence, RI) | MR