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Tame stacks in positive characteristic. (English) Zbl 1222.14004

Ann. Inst. Fourier 58, No. 4, 1057-1091 (2008); corrigendum ibid. 64, No. 3, 945-946 (2014).
A tame stack is an algebraic stack \({\mathcal M}\) with finite inertia and such that the pushforward functor \(\rho_{*} :\mathrm{QCoh}({\mathcal M}) \to \mathrm{QCoh}(M)\) from quasi-coherent sheaves on \({\mathcal M}\) to quasi-coherent sheaves on its coarse moduli space \(M\), is exact. The exactness of the pushforward functor \(\rho_{*}\) is a property that Deligne-Mumford stacks enjoy in characteristic zero, but which fails in positive characteristic.
Another, equivalent, description of a tame stack is given in the main result of the article, Theorem 3.2, stating that a stack with finite inertia is tame if and only if the coarse moduli space of the stack is étale locally the quotient of a scheme by a finite, flat, linearly reductive group. It follows from the main result that tame stacks have several other nice properties as well, such as flatness of its moduli space and that the formation of the coarse moduli commutes with base change.
An important ingredient in the proof of the main result is the classification of finite, flat, linearly reductive groups that the authors give. They show that a finite, flat group \(G\) is linearly reductive if and only if each geometric fiber of \(G\) is an extension of a tame and étale group by a diagonalizable group.
In positive characteristic several interesting moduli problems are not covered by Deligne-Mumford stacks, and the fact that Deligne-Mumford stacks do not always behave well with respect to the above mentioned properties, makes the notion of tame stacks interesting and important.

MSC:

14A20 Generalizations (algebraic spaces, stacks)
14L15 Group schemes
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References:

[1] Abramovich, D.; Corti, A.; Vistoli, A., Twisted bundles and admissible covers, Comm. Algebra, 31, 3547-3618 (2003) · Zbl 1077.14034 · doi:10.1081/AGB-120022434
[2] Abramovich, D.; Graber, T.; Vistoli, A., Gromov-Witten theory of Deligne-Mumford stacks, preprint · Zbl 1193.14070
[3] Abramovich, D.; Vistoli, A., Compactifying the space of stable maps, J. Amer. Math. Soc, 15, 27-75 (2002) · Zbl 0991.14007 · doi:10.1090/S0894-0347-01-00380-0
[4] Artin, M., Versal deformations and algebraic stacks, Invent. Math., 27, 165-189 (1974) · Zbl 0317.14001 · doi:10.1007/BF01390174
[5] Behrend, K.; Noohi, B., Uniformization of Deligne-Mumford curves, J. Reine Angew. Math. · Zbl 1124.14004
[6] Berthelot, P.; Grothendieck, A.; Illusie, L., Théorie des Intersections et Théorème de Riemann-Roch (SGA 6), 225 (1971) · Zbl 0218.14001
[7] Conrad, B., Keel-Mori theorem via stacks
[8] Deligne, P.; Mumford, D., The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math., 36, 75-109 (1969) · Zbl 0181.48803 · doi:10.1007/BF02684599
[9] Demazure, M.; al., Schémas en groupes, 151, 152 and 153 (1970) · Zbl 0207.51401
[10] Dieudonné, J.; Grothendieck, A., Éléments de géométrie algébrique, 4, 8, 11, 17, 20, 24, 28, 32 (19611967) · Zbl 0203.23301
[11] Giraud, J., Cohomologie non abélienne (1971) · Zbl 0226.14011
[12] Gorenstein, D., Finite groups (1980) · Zbl 0463.20012
[13] Gruson, L.; Raynaud, M., Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math., 13, 1-89 (1971) · Zbl 0227.14010
[14] Illusie, L., Complexe cotangent et déformations. I, 239 (1971) · Zbl 0224.13014
[15] Jacobson, N., Lie algebras (1979) · JFM 61.1044.02
[16] Keel, S.; Mori, S., Quotients by groupoids, Ann. of Math. (2), 145, 1, 193-213 (1997) · Zbl 0881.14018 · doi:10.2307/2951828
[17] Kleiman, S., Fundamental algebraic geometry, 235-321 (2005) · Zbl 1085.14001
[18] Kresch, A., Geometry of Deligne-Mumford stacks, preprint · Zbl 1169.14001
[19] Laumon, G.; Moret-Bailly, L., Champs Algébriques, 39 (2000) · Zbl 0945.14005
[20] Milne, J. S., Étale cohomology (1980) · Zbl 0433.14012
[21] Mumford, D., Abelian varieties (1970) · Zbl 0223.14022
[22] Olsson, M., A boundedness theorem for Hom-stacks, preprint (2005)
[23] Olsson, M., On proper coverings of Artin stacks, Advances in Mathematics, 198, 93-106 (2005) · Zbl 1084.14004 · doi:10.1016/j.aim.2004.08.017
[24] Olsson, M., Deformation theory of representable morphisms of algebraic stacks, Math. Zeit., 53, 25-62 (2006) · Zbl 1096.14007 · doi:10.1007/s00209-005-0875-9
[25] Olsson, M., Hom-stacks and restriction of scalars, Duke Math. J., 134, 139-164 (2006) · Zbl 1114.14002 · doi:10.1215/S0012-7094-06-13414-2
[26] Olsson, M., Sheaves on Artin stacks, J. Reine Angew. Math. (Crelle’s Journal), 603, 55-112 (2007) · Zbl 1137.14004 · doi:10.1515/CRELLE.2007.012
[27] Romagny, M., Group actions on stacks and applications, Michigan Math. J., 53, 1, 209-236 (2005) · Zbl 1100.14001 · doi:10.1307/mmj/1114021093
[28] Saavedra Rivano, N., Catégories Tannakiennes, 265 (1972) · Zbl 0241.14008
[29] Thomason, R. W., Algebraic K-theory of group scheme actions, 113 (1987) · Zbl 0701.19002
[30] Vistoli, A., Grothendieck topologies, fibered categories and descent theory, Fundamental algebraic geometry, 1-104 (2005)
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