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Dan Abramovich; Martin Olsson; Angelo Vistoli
Tame stacks in positive characteristic
(Champs modérés en caractéristique positive)
Annales de l'institut Fourier, 58 no. 4 (2008), p. 1057-1091, doi: 10.5802/aif.2378
Article PDF | Reviews MR 2427954 | Zbl pre05303669
Class. Math.: 14A20, 14L15
Keywords: Algebraic stacks, moduli spaces, group schemes

Résumé - Abstract

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.

Bibliography

[1] D. Abramovich, A. Corti & A. Vistoli, “Twisted bundles and admissible covers”, Comm. Algebra 31 (2003), p. 3547-3618 Article |  MR 2007376 |  Zbl 1077.14034
[2] D. Abramovich, T. Graber & A. Vistoli, “Gromov–Witten theory of Deligne–Mumford stacks”, preprint, http://www.arxiv.org/abs/math.AG/0603151 arXiv
[3] D. Abramovich & A. Vistoli, “Compactifying the space of stable maps”, J. Amer. Math. Soc 15 (2002), p. 27-75 Article |  MR 1862797 |  Zbl 0991.14007
[4] M. Artin, “Versal deformations and algebraic stacks”, Invent. Math. 27 (1974), p. 165-189 Article |  MR 399094 |  Zbl 0317.14001
[5] K. Behrend & B. Noohi, “Uniformization of Deligne–Mumford curves”, J. Reine Angew. Math., to appear  Zbl 1124.14004
[6] P. Berthelot, A. Grothendieck & L. Illusie, Théorie des Intersections et Théorème de Riemann-Roch (SGA 6) 225, Springer Lecture Notes in Math, 1971  MR 354655 |  Zbl 0218.14001
[7] B. Conrad, “Keel-Mori theorem via stacks”, preprint available at http://www.math.lsa.umich.edu/ bdconrad
[8] P. Deligne & D. Mumford, “The irreducibility of the space of curves of given genus”, Inst. Hautes Études Sci. Publ. Math. 36 (1969), p. 75-109 Numdam |  MR 262240 |  Zbl 0181.48803
[9] M. Demazure & al., Schémas en groupes, Lecture Notes in Mathematics 151, 152 and 153, Springer-Verlag, 1970  MR 274458 |  Zbl 0207.51401
[10] J. Dieudonné & A. Grothendieck, Éléments de géométrie algébrique 4, 8, 11, 17, 20, 24, 28, 32, Inst. Hautes Études Sci. Publ. Math., 1961–1967
[11] J. Giraud, Cohomologie non abélienne, Springer-Verlag, 1971, Die Grundlehren der mathematischen Wissenschaften, Band 179  MR 344253 |  Zbl 0226.14011
[12] D. Gorenstein, Finite groups, Chelsea Publishing Co., 1980, xvii+519 pp  MR 569209 |  Zbl 0463.20012
[13] L. Gruson & M. Raynaud, “Critères de platitude et de projectivité. Techniques de “platification” d’un module”, Invent. Math. 13 (1971), p. 1-89  Zbl 0227.14010
[14] L. Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics 239, Springer, 1971  MR 491680 |  Zbl 0224.13014
[15] N. Jacobson, Lie algebras, Republication of the 1962 original, Dover Publications, Inc., 1979, ix+331 pp. ISBN: 0-486-63832-4  MR 559927
[16] S. Keel & S. Mori, “Quotients by groupoids”, Ann. of Math. (2) 145 (1997) no. 1, p. 193-213 Article |  MR 1432041 |  Zbl 0881.14018
[17] S. Kleiman, The Picard scheme, Fundamental algebraic geometry, Math. Surveys Monogr. 123, 2005, p. 235–321  MR 2223410 |  Zbl 1085.14001
[18] A. Kresch, “Geometry of Deligne–Mumford stacks”, preprint
[19] G. Laumon & L. Moret-Bailly, Champs Algébriques, Ergebnisse der Mathematik un ihrer Grenzgebiete 39, Springer-Verlag, 2000  MR 1771927 |  Zbl 0945.14005
[20] J. S. Milne, Étale cohomology, Princeton University Press, 1980  MR 559531 |  Zbl 0433.14012
[21] D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5., Published for the Tata Institute of Fundamental Research, 1970, viii+242 pp  MR 282985 |  Zbl 0223.14022
[22] M. Olsson, “A boundedness theorem for Hom-stacks”, preprint, 2005  MR 2357471 |  Zbl pre05243870
[23] M. Olsson, “On proper coverings of Artin stacks”, Advances in Mathematics 198 (2005), p. 93-106 Article |  MR 2183251 |  Zbl 1084.14004
[24] M. Olsson, “Deformation theory of representable morphisms of algebraic stacks”, Math. Zeit. 53 (2006), p. 25-62 Article |  MR 2206635 |  Zbl 1096.14007
[25] M. Olsson, “Hom-stacks and restriction of scalars”, Duke Math. J. 134 (2006), p. 139-164 Article |  MR 2239345 |  Zbl 1114.14002
[26] M. Olsson, “Sheaves on Artin stacks”, J. Reine Angew. Math. (Crelle’s Journal) 603 (2007), p. 55-112 Article |  Zbl pre05151128
[27] M. Romagny, “Group actions on stacks and applications”, Michigan Math. J. 53 (2005) no. 1, p. 209-236 Article |  MR 2125542 |  Zbl 1100.14001
[28] N. Saavedra Rivano, Catégories Tannakiennes, Lecture Notes in Mathematics 265, Springer-Verlag, 1972  MR 338002 |  Zbl 0241.14008
[29] R. W. Thomason, Algebraic K-theory of group scheme actions, Algebraic Topology and Algebraic K-theory 113, Annals of Mathematical Studies, 1987  MR 921490 |  Zbl 0701.19002
[30] A. Vistoli, “Grothendieck topologies, fibered categories and descent theory”, Fundamental algebraic geometry (2005), p. 1-104, Math. Surveys Monogr., 123, Amer. Math. Soc., Providence, RI  MR 2223406
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