de Rham Theory for Tame Stacks and Schemes with Linearly Reductive Singularities
[Théorie de De Rham pour les champs non sauvages et schémas avec des singularités linéairement réductives]
Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2013-2051.

Nous démontrons que la suite spectrale de Hodge-De Rham d’un champ d’Artin propre modéré en caractéristique p (d’après Abramovich, Olsson et Vistoli) qui se relève mod p 2 dégénère. Nous étendons ce résultat à des schémas quotients d’un schéma lisse par un schéma en groupes linéaires réductifs.

We prove that the Hodge-de Rham spectral sequence for smooth proper tame Artin stacks in characteristic p (as defined by Abramovich, Olsson, and Vistoli) which lift mod p 2 degenerates. We push the result to the coarse spaces of such stacks, thereby obtaining a degeneracy result for schemes which are étale locally the quotient of a smooth scheme by a finite linearly reductive group scheme.

DOI : 10.5802/aif.2741
Classification : 14A20, 14F40
Keywords: de Rham, Hodge, tame stack, linearly reductive
Mot clés : De Rham, Hodge, champs modéré, linéaire réductif
Satriano, Matthew 1

1 University of Michigan Department of Mathematics 2074 East Hall, Ann Arbor, MI 48109-1043 (USA)
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Satriano, Matthew. de Rham Theory for Tame Stacks and Schemes with Linearly Reductive Singularities. Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2013-2051. doi : 10.5802/aif.2741. http://www.numdam.org/articles/10.5802/aif.2741/

[1] Abramovich, Dan; Olsson, Martin; Vistoli, Angelo Tame stacks in positive characteristic, Ann. Inst. Fourier (Grenoble), Volume 58 (2008) no. 4, pp. 1057-1091 | Numdam | MR | Zbl

[2] Abramovich, Dan; Vistoli, Angelo Compactifying the space of stable maps, J. Amer. Math. Soc., Volume 15 (2002) no. 1, p. 27-75 (electronic) | DOI | MR | Zbl

[3] Behrend, K. Cohomology of stacks, Intersection theory and moduli (ICTP Lect. Notes, XIX), Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, p. 249-294 (electronic) | MR | Zbl

[4] Conrad, Brian Cohomological descent, 2009 (http://math.stanford.edu/~conrad/papers/cohdescent.pdf)

[5] Deligne, Pierre; Illusie, Luc Relèvements modulo p 2 et décomposition du complexe de de Rham, Invent. Math., Volume 89 (1987) no. 2, pp. 247-270 | DOI | MR | Zbl

[6] Faltings, Gerd p-adic Hodge theory, J. Amer. Math. Soc., Volume 1 (1988) no. 1, pp. 255-299 | DOI | MR | Zbl

[7] Fantechi, Barbara; Mann, Etienne; Nironi, Fabio Smooth toric DM stacks, 2009 (arXiv:0708.1254v2)

[8] Hochster, Melvin; Roberts, Joel L. Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Advances in Math., Volume 13 (1974), pp. 115-175 | MR | Zbl

[9] Illusie, Luc Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin, 1971 | MR | Zbl

[10] Katz, Nicholas M. Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. (1970) no. 39, pp. 175-232 | Numdam | MR | Zbl

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

[12] Laumon, Gérard; Moret-Bailly, Laurent Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 39, Springer-Verlag, Berlin, 2000 | MR | Zbl

[13] Matsuki, Kenji; Olsson, Martin Kawamata-Viehweg vanishing as Kodaira vanishing for stacks, Math. Res. Lett., Volume 12 (2005) no. 2-3, pp. 207-217 | MR | Zbl

[14] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34, Springer-Verlag, Berlin, 1994 | MR | Zbl

[15] Olsson, Martin C. Hom ̲-stacks and restriction of scalars, Duke Math. J., Volume 134 (2006) no. 1, pp. 139-164 | DOI | MR | Zbl

[16] Olsson, Martin C. Sheaves on Artin stacks, J. Reine Angew. Math., Volume 603 (2007), pp. 55-112 | DOI | MR | Zbl

[17] Satriano, Matthew A generalization of the Chevalley-Shephard-Todd theorem to the case of linearly reductive group schemes, 2009 (arXiv:0911.2058v1)

[18] Steenbrink, J. H. M. Mixed Hodge structure on the vanishing cohomology, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 525-563 | MR | Zbl

[19] Toen, B. K-théorie et cohomologie des champs algébriques: Théorèmes de Riemann-Roch, D-modules et théorèmes GAGA, 1999 (arXiv:math/9908097v2)

[20] Vistoli, Angelo Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math., Volume 97 (1989) no. 3, pp. 613-670 | DOI | MR | Zbl

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