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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Quotient stacks and equivariant étale cohomology algebras: Quillen’s theory revisited

Authors: Luc Illusie and Weizhe Zheng
Journal: J. Algebraic Geom. 25 (2016), 289-400
Published electronically: February 10, 2016
MathSciNet review: 3466353
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Abstract: Let $k$ be an algebraically closed field. Let $\Lambda$ be a noetherian commutative ring annihilated by an integer invertible in $k$ and let $\ell$ be a prime number different from the characteristic of $k$. We prove that if $X$ is a separated algebraic space of finite type over $k$ endowed with an action of a $k$-algebraic group $G$, the equivariant étale cohomology algebra $H^*([X/G],\Lambda )$, where $[X/G]$ is the quotient stack of $X$ by $G$, is finitely generated over $\Lambda$. Moreover, for coefficients $K \in D^+_c([X/G],\mathbb {F}_{\ell })$ endowed with a commutative multiplicative structure, we establish a structure theorem for $H^*([X/G],K)$, involving fixed points of elementary abelian $\ell$-subgroups of $G$, which is similar to Quillen’s theorem in the case $K = \mathbb {F}_{\ell }$. One key ingredient in our proof of the structure theorem is an analysis of specialization of points of the quotient stack. We also discuss variants and generalizations for certain Artin stacks.

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  • Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398. MR 2450211, DOI
  • Kai A. Behrend, The Lefschetz trace formula for algebraic stacks, Invent. Math. 112 (1993), no. 1, 127–149. MR 1207479, DOI
  • Emili Bifet, Corrado De Concini, and Claudio Procesi, Cohomology of regular embeddings, Adv. Math. 82 (1990), no. 1, 1–34. MR 1057441, DOI
  • Francis Borceux, Handbook of categorical algebra. 1, Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge, 1994. Basic category theory. MR 1291599
  • Armand Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115–207 (French). MR 51508, DOI
  • Armand Borel, Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, Tohoku Math. J. (2) 13 (1961), 216–240 (French). MR 147579, DOI
  • Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822
  • Michel Brion, Preena Samuel, and V. Uma, Lectures on the structure of algebraic groups and geometric applications, CMI Lecture Series in Mathematics, vol. 1, Hindustan Book Agency, New Delhi; Chennai Mathematical Institute (CMI), Chennai, 2013. MR 3088271
  • Brian Conrad, A modern proof of Chevalley’s theorem on algebraic groups, J. Ramanujan Math. Soc. 17 (2002), no. 1, 1–18. MR 1906417
  • Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77 (French). MR 498552
  • P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174
  • Pierre Deligne, Lettre à Luc Illusie, June 26, 2012.
  • Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 0302656
  • Dan Edidin and William Graham, Equivariant intersection theory, Invent. Math. 131 (1998), no. 3, 595–634. MR 1614555, DOI
  • Dan Edidin, Brendan Hassett, Andrew Kresch, and Angelo Vistoli, Brauer groups and quotient stacks, Amer. J. Math. 123 (2001), no. 4, 761–777. MR 1844577
  • D. B. A. Epstein, Steenrod operations in homological algebra, Invent. Math. 1 (1966), 152–208. MR 199240, DOI
  • Daniel Ferrand, Conducteur, descente et pincement, Bull. Soc. Math. France 131 (2003), no. 4, 553–585 (French, with English and French summaries). MR 2044495, DOI
  • Ofer Gabber and Lorenzo Ramero, Almost ring theory, Lecture Notes in Mathematics, vol. 1800, Springer-Verlag, Berlin, 2003. MR 2004652
  • Jean Giraud, Méthode de la descente, Bull. Soc. Math. France Mém. 2 (1964), viii+150 (French). MR 190142
  • Jean Giraud, Cohomologie non abélienne, Springer-Verlag, Berlin-New York, 1971 (French). Die Grundlehren der mathematischen Wissenschaften, Band 179. MR 0344253
  • A. Grothendieck, Classes de Chern et representations linearies des groupes discrets, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 215–305 (French). MR 265370
  • A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 173675
  • Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680
  • Luc Illusie, Travaux de Quillen sur la cohomologie des groupes, Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 405, Springer, Berlin, 1973, pp. 89–105. Lecture Notes in Math., Vol. 317 (French). MR 0488055
  • Luc Illusie, Elementary abelian $\ell $-groups and ${\rm mod}\,\ell $ equivariant étale cohomology algebras, Astérisque 370 (2015), 177–195 (English, with English and French summaries). MR 3364747
  • Luc Illusie and Weizhe Zheng, Odds and ends on finite group actions and traces, Int. Math. Res. Not. IMRN 1 (2013), 1–62. MR 3041694, DOI
  • Masaki Kashiwara and Pierre Schapira, Categories and sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 332, Springer-Verlag, Berlin, 2006. MR 2182076
  • Seán Keel and Shigefumi Mori, Quotients by groupoids, Ann. of Math. (2) 145 (1997), no. 1, 193–213. MR 1432041, DOI
  • G. M. Kelly, Basic concepts of enriched category theory, Repr. Theory Appl. Categ. 10 (2005), vi+137. Reprint of the 1982 original [Cambridge Univ. Press, Cambridge; MR0651714]. MR 2177301
  • Andrew Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495–536. MR 1719823, DOI
  • Gérard Laumon and Laurent Moret-Bailly, 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], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927
  • Yifeng Liu and Weizhe Zheng, Enhanced six operations and base change theorem for Artin stacks, arXiv:1211.5294.
  • Saunders MacLane, Categories for the working mathematician, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. MR 0354798
  • Martin Olsson, Sheaves on Artin stacks, J. Reine Angew. Math. 603 (2007), 55–112. MR 2312554, DOI
  • Sam Payne, Equivariant Chow cohomology of toric varieties, Math. Res. Lett. 13 (2006), no. 1, 29–41. MR 2199564, DOI
  • Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR 298694, DOI
  • Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR 298694, DOI
  • Michèle Raynaud, Modules projectifs universels, Invent. Math. 6 (1968), 1–26 (French). MR 236164, DOI
  • Joël Riou, Exposé XVI. Classes de Chern, morphismes de Gysin, pureté absolue, Astérisque 363-364 (2014), 301–349 (French). Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. MR 3329786
  • David Rydh, Existence and properties of geometric quotients, J. Algebraic Geom. 22 (2013), no. 4, 629–669. MR 3084720, DOI
  • Jean-Pierre Serre, Arbres, amalgames, ${\rm SL}_{2}$, Société Mathématique de France, Paris, 1977 (French). Avec un sommaire anglais; Rédigé avec la collaboration de Hyman Bass; Astérisque, No. 46. MR 0476875
  • Jean-Pierre Serre, Groupes algébriques et corps de classes, 2nd ed., Publications de l’Institut Mathématique de l’Université de Nancago [Publications of the Mathematical Institute of the University of Nancago], vol. 7, Hermann, Paris, 1984 (French). Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], 1264. MR 907288
  • Jean-Pierre Serre, Sous-groupes finis des groupes de Lie, Astérisque 266 (2000), Exp. No. 864, 5, 415–430 (French, with French summary). Séminaire Bourbaki, Vol. 1998/99. MR 1772682
  • The Stacks Project Authors, Stacks Project,
  • Burt Totaro, Group cohomology and algebraic cycles, Cambridge Tracts in Mathematics, vol. 204, Cambridge University Press, Cambridge, 2014. MR 3185743
  • Jean-Louis Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239 (1996), xii+253 pp. (1997) (French, with French summary). With a preface by Luc Illusie; Edited and with a note by Georges Maltsiniotis. MR 1453167
  • Weizhe Zheng, Sur l’indépendance de $l$ en cohomologie $l$-adique sur les corps locaux, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 2, 291–334 (French, with English and French summaries). MR 2518080, DOI
  • Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 3, Société Mathématique de France, Paris, 2003 (French). Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960-61]; Directed by A. Grothendieck; With two papers by M. Raynaud; Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)]. MR 2017446
  • Schémas en groupes. I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274458
  • Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354652
  • Cohomologie $l$-adique et fonctions $L$, Lecture Notes in Mathematics, Vol. 589, Springer-Verlag, Berlin-New York, 1977 (French). Séminaire de Géometrie Algébrique du Bois-Marie 1965–1966 (SGA 5); Edité par Luc Illusie. MR 0491704
  • Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, Vol. 288, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I); Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim. MR 0354656

Additional Information

Luc Illusie
Affiliation: Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
MR Author ID: 90990
ORCID: 0000-0002-6634-6325

Weizhe Zheng
Affiliation: Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
MR Author ID: 733496

Received by editor(s): May 31, 2013
Received by editor(s) in revised form: November 12, 2013, July 29, 2014, November 5, 2014, and February 8, 2015
Published electronically: February 10, 2016
Additional Notes: The second author was partially supported by China’s Recruitment Program of Global Experts; National Natural Science Foundation of China Grant 11321101; Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences; National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences.
Dedicated: To the memory of Daniel Quillen
Article copyright: © Copyright 2016 University Press, Inc.