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
DOI: https://doi.org/10.1090/jag/674
Published electronically: February 10, 2016
MathSciNet review: 3466353
Full-text PDF

Abstract | References | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] 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 (2009k:14108), https://doi.org/10.1353/ajm.0.0017
  • [2] Kai A. Behrend, The Lefschetz trace formula for algebraic stacks, Invent. Math. 112 (1993), no. 1, 127-149. MR 1207479 (94d:14023), https://doi.org/10.1007/BF01232427
  • [3] Emili Bifet, Corrado De Concini, and Claudio Procesi, Cohomology of regular embeddings, Adv. Math. 82 (1990), no. 1, 1-34. MR 1057441 (91h:14052), https://doi.org/10.1016/0001-8708(90)90082-X
  • [4] Francis Borceux, Handbook of categorical algebra. 1, Basic category theory. Encyclopedia of Mathematics and its Applications, vol. 50, Cambridge University Press, Cambridge, 1994. MR 1291599 (96g:18001a)
  • [5] 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 0051508 (14,490e)
  • [6] Armand Borel, Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, Tôhoku Math. J. (2) 13 (1961), 216-240 (French). MR 0147579 (26 #5094)
  • [7] 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 (91i:14034)
  • [8] 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
  • [9] Brian Conrad, A modern proof of Chevalley's theorem on algebraic groups, J. Ramanujan Math. Soc. 17 (2002), no. 1, 1-18. MR 1906417 (2003f:20078)
  • [10] Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5-77 (French). MR 0498552 (58 #16653b)
  • [11] P. Deligne, Cohomologie étale, Séminaire de Géométrie Algébrique du Bois-Marie SGA 4 $ {1\over 2}$, avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier. Lecture Notes in Mathematics, Vol. 569, Springer-Verlag, Berlin-New York, 1977. MR 0463174 (57 #3132)
  • [12] Pierre Deligne, Lettre à Luc Illusie, June 26, 2012.
  • [13] Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, avec un appendice Corps de classes local par Michiel Hazewinkel. Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). MR 0302656 (46 #1800)
  • [14] Dan Edidin and William Graham, Equivariant intersection theory, Invent. Math. 131 (1998), no. 3, 595-634. MR 1614555 (99j:14003a), https://doi.org/10.1007/s002220050214
  • [15] 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 (2002f:14002)
  • [16] D. B. A. Epstein, Steenrod operations in homological algebra, Invent. Math. 1 (1966), 152-208. MR 0199240 (33 #7389)
  • [17] Daniel Ferrand, Conducteur, descente et pincement, Bull. Soc. Math. France 131 (2003), no. 4, 553-585 (French, with English and French summaries). MR 2044495 (2005a:13016)
  • [18] Ofer Gabber and Lorenzo Ramero, Almost ring theory, Lecture Notes in Mathematics, vol. 1800, Springer-Verlag, Berlin, 2003. MR 2004652 (2004k:13027)
  • [19] Jean Giraud, Méthode de la descente, Bull. Soc. Math. France Mém. 2 (1964), viii+150 (French). MR 0190142 (32 #7556)
  • [20] Jean Giraud, Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften, Band 179, Springer-Verlag, Berlin-New York, 1971 (French). MR 0344253 (49 #8992)
  • [21] A. Grothendieck, Classes de Chern et représentations linéaires des groupes discrets, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam; Masson, Paris, 1968, pp. 215-305 (French). MR 0265370 (42 #280)
  • [22] A. Grothendieck (avec la collaboration de J. Dieudonné), Éléments de géométrie algébrique. IV. Étude globale élémentaire de quelques classes de morphismes,
    Publ. Math. Inst. Hautes Études Sci. 20, 24, 28, 32 (1964-1967). MR 0173675, MR 0199181, MR 0217086, MR 0238860
  • [23] Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, Vol. 239, Springer-Verlag, Berlin-New York, 1971 (French). MR 0491680 (58 #10886a)
  • [24] Luc Illusie, Travaux de Quillen sur la cohomologie des groupes, Séminaire Bourbaki, 24e année (1971/1972), Exp. No. 405. Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973, pp. 89-105 (French). MR 0488055 (58 #7627b)
  • [25] Luc Illusie, Elementary abelian $ \ell $-groups and mod $ \ell $ equivariant étale cohomology algebras, Astérisque 370 (2015), 177-195 (English, with English and French summaries). MR 3364747
  • [26] 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
  • [27] 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 (2006k:18001)
  • [28] Seán Keel and Shigefumi Mori, Quotients by groupoids, Ann. of Math. (2) 145 (1997), no. 1, 193-213. MR 1432041 (97m:14014), https://doi.org/10.2307/2951828
  • [29] 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
  • [30] Andrew Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495-536. MR 1719823 (2001a:14003), https://doi.org/10.1007/s002220050351
  • [31] 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 (2001f:14006)
  • [32] Yifeng Liu and Weizhe Zheng, Enhanced six operations and base change theorem for Artin stacks, arXiv:1211.5294.
  • [33] Saunders Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798 (50 #7275)
  • [34] Martin Olsson, Sheaves on Artin stacks, J. Reine Angew. Math. 603 (2007), 55-112. MR 2312554 (2008b:14002), https://doi.org/10.1515/CRELLE.2007.012
  • [35] Sam Payne, Equivariant Chow cohomology of toric varieties, Math. Res. Lett. 13 (2006), no. 1, 29-41. MR 2199564 (2007f:14052), https://doi.org/10.4310/MRL.2006.v13.n1.a3
  • [36] Daniel Quillen, The spectrum of an equivariant cohomology ring. I, Ann. of Math. (2) 94 (1971), 549-572. MR 0298694 (45 #7743)
  • [37] Daniel Quillen, The spectrum of an equivariant cohomology ring. II, Ann. of Math. (2) 94 (1971), 573-602. MR 0298694 (45 #7743)
  • [38] Michèle Raynaud, Modules projectifs universels, Invent. Math. 6 (1968), 1-26 (French). MR 0236164 (38 #4462)
  • [39] Joël Riou, Exposé XVI. Classes de Chern, morphismes de Gysin, pureté absolue, Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasi-excellents. Astérisque 363-364 (2014), 301-349 (French). MR 3329786
  • [40] David Rydh, Existence and properties of geometric quotients, J. Algebraic Geom. 22 (2013), no. 4, 629-669. MR 3084720, https://doi.org/10.1090/S1056-3911-2013-00615-3
  • [41] Jean-Pierre Serre, Arbres, amalgames, $ {\rm SL}_{2}$, avec un sommaire en anglais; Rédigé avec la collaboration de Hyman Bass; Astérisque 46 (1977) (French). MR 0476875 (57 #16426)
  • [42] 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], 7, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], 1264, Hermann, Paris, 1984 (French). MR 907288 (88g:14044)
  • [43] Jean-Pierre Serre, Sous-groupes finis des groupes de Lie, Séminaire Bourbaki, Volume 1998/99, no. 864, Astérisque 266 (2000), 415-430. MR 1772682 (2001j:20075)
  • [44] The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu
  • [45] Burt Totaro, Group cohomology and algebraic cycles, Cambridge Tracts in Mathematics, vol. 204, Cambridge University Press, Cambridge, 2014. MR 3185743
  • [46] Jean-Louis Verdier, Des catégories dérivées des catégories abéliennes, with a preface by Luc Illusie; edited and with a note by Georges Maltsiniotis. Astérisque 239 (1996), xii+253 pp. (1997) (French. French summary). MR 1453167 (98c:18007)
  • [47] 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 (2010i:14032)
  • [48] Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 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 (2004g:14017)
  • [49] Schémas en groupes. I, II, III, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vols. 151-153, Springer-Verlag, Berlin-New York, 1970 (French). MR 0274458 (43 #223a), MR 0274459 (43 #223b), MR 0274460 (43 #223c)
  • [50] Théorie des topos et cohomologie étale des schémas. I, II, III, 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. Lecture Notes in Mathematics, Vols. 269, 270, 305, Springer-Verlag, Berlin-New York, 1972, 1973 (French). MR 0354652 (50 #7130); MR 0354653 (50 #7131); MR 0354654 (50 #7132)
  • [51] Cohomologie $ l$-adique et fonctions $ L$, Séminaire de Géométrie Algébrique du Bois-Marie 1965-1966 (SGA 5), dirigé par A. Grothendieck, édité par Luc Illusie. Lecture Notes in Mathematics, Vol. 589, Springer-Verlag, Berlin-New York, 1977 (French). MR 0491704
  • [52] Groupes de monodromie en géométrie algébrique. I, Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969. I (SGA 7 I), dirigé par A. Grothendieck, avec la collaboration de M. Raynaud et D. S. Rim. Lecture Notes in Mathematics, Vol. 288, Springer-Verlag, Berlin, 1972; II (SGA 7 II), dirigé par P. Deligne et N. Katz. Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin, 1973. MR 0354656 (50 #7134); MR 0354657 (50 #7135)


Additional Information

Luc Illusie
Affiliation: Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
Email: Luc.Illusie@math.u-psud.fr

Weizhe Zheng
Affiliation: Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: wzheng@math.ac.cn

DOI: https://doi.org/10.1090/jag/674
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.

American Mathematical Society