de Rham Theory for Tame Stacks and Schemes with Linearly Reductive Singularities
Annales de l'Institut Fourier, Volume 62 (2012) no. 6, p. 2013-2051

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.

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.

DOI : https://doi.org/10.5802/aif.2741
Classification:  14A20,  14F40
Keywords: de Rham, Hodge, tame stack, linearly reductive
@article{AIF_2012__62_6_2013_0,
     author = {Satriano, Matthew},
     title = {de Rham Theory for Tame Stacks and Schemes with Linearly Reductive Singularities},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {6},
     year = {2012},
     pages = {2013-2051},
     doi = {10.5802/aif.2741},
     mrnumber = {3060750},
     zbl = {pre06159904},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2012__62_6_2013_0}
}
Satriano, Matthew. de Rham Theory for Tame Stacks and Schemes with Linearly Reductive Singularities. Annales de l'Institut Fourier, Volume 62 (2012) no. 6, pp. 2013-2051. doi : 10.5802/aif.2741. http://www.numdam.org/item/AIF_2012__62_6_2013_0/

[1] Abramovich, Dan; Olsson, Martin; Vistoli, Angelo Tame stacks in positive characteristic, Ann. Inst. Fourier (Grenoble), Tome 58 (2008) no. 4, pp. 1057-1091 http://aif.cedram.org/item?id=AIF_2008__58_4_1057_0 | Numdam | MR 2427954 | Zbl 1222.14004

[2] Abramovich, Dan; Vistoli, Angelo Compactifying the space of stable maps, J. Amer. Math. Soc., Tome 15 (2002) no. 1, p. 27-75 (electronic) | Article | MR 1862797 | Zbl 0991.14007

[3] Behrend, K. Cohomology of stacks, Intersection theory and moduli, Abdus Salam Int. Cent. Theoret. Phys., Trieste (ICTP Lect. Notes, XIX) (2004), p. 249-294 (electronic) | MR 2172499 | Zbl 1081.58003

[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., Tome 89 (1987) no. 2, pp. 247-270 | Article | MR 894379 | Zbl 0632.14017

[6] Faltings, Gerd p-adic Hodge theory, J. Amer. Math. Soc., Tome 1 (1988) no. 1, pp. 255-299 | Article | MR 924705 | Zbl 0764.14012

[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., Tome 13 (1974), pp. 115-175 | MR 347810 | Zbl 0289.14010

[9] Illusie, Luc Complexe cotangent et déformations. I, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Vol. 239 (1971) | MR 491680 | Zbl 0224.13014

[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 291177 | Zbl 0221.14007

[11] Keel, Seán; Mori, Shigefumi Quotients by groupoids, Ann. of Math. (2), Tome 145 (1997) no. 1, pp. 193-213 | Article | MR 1432041 | Zbl 0881.14018

[12] Laumon, Gérard; Moret-Bailly, Laurent Champs algébriques, Springer-Verlag, Berlin, 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], Tome 39 (2000) | MR 1771927 | Zbl 0945.14005

[13] Matsuki, Kenji; Olsson, Martin Kawamata-Viehweg vanishing as Kodaira vanishing for stacks, Math. Res. Lett., Tome 12 (2005) no. 2-3, pp. 207-217 | MR 2150877 | Zbl 1080.14023

[14] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], Tome 34 (1994) | MR 1304906 | Zbl 0797.14004

[15] Olsson, Martin C. Hom ̲-stacks and restriction of scalars, Duke Math. J., Tome 134 (2006) no. 1, pp. 139-164 | Article | MR 2239345 | Zbl 1114.14002

[16] Olsson, Martin C. Sheaves on Artin stacks, J. Reine Angew. Math., Tome 603 (2007), pp. 55-112 | Article | MR 2312554 | Zbl 1137.14004

[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 485870 | Zbl 0373.14007

[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., Tome 97 (1989) no. 3, pp. 613-670 | Article | MR 1005008 | Zbl 0694.14001