Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


Cohomological invariants of algebraic stacks

Author: Roberto Pirisi
Journal: Trans. Amer. Math. Soc. 370 (2018), 1885-1906
MSC (2010): Primary 14D23, 14F20
Published electronically: November 22, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to lay the foundations of a theory of invariants in étale cohomology for smooth Artin stacks. We compute the invariants in the case of the stack of elliptic curves, and we use the theory we developed to get some results regarding Brauer groups of algebraic spaces.

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

  • [BF03] Grégory Berhuy and Giordano Favi, Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003), 279-330. MR 2029168
  • [BO74] Spencer Bloch and Arthur Ogus, Gersten's conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4) 7 (1974), 181-201 (1975). MR 0412191
  • [BR97] J. Buhler and Z. Reichstein, On the essential dimension of a finite group, Compositio Math. 106 (1997), no. 2, 159-179. MR 1457337,
  • [BRV11] Patrick Brosnan, Zinovy Reichstein, and Angelo Vistoli, Essential dimension of moduli of curves and other algebraic stacks, J. Eur. Math. Soc. (JEMS) 13 (2011), no. 4, 1079-1112. MR 2800485,
  • [CTO89] Jean-Louis Colliot-Thélène and Manuel Ojanguren, Variétés unirationnelles non rationnelles: au-delà de l'exemple d'Artin et Mumford, Invent. Math. 97 (1989), no. 1, 141-158. MR 999316,
  • [EG96] Dan Edidin and William Graham, Equivariant intersection theory, Invent. Math. 131 (1998), no. 3, 595-634. MR 1614555,
  • [GMS03] Skip Garibaldi, Alexander Merkurjev, and Jean-Pierre Serre, Cohomological invariants in Galois cohomology, University Lecture Series, vol. 28, American Mathematical Society, Providence, RI, 2003. MR 1999383
  • [Gui08] Pierre Guillot, Geometric methods for cohomological invariants, Doc. Math. 12 (2007), 521-545. MR 2365912
  • [Knu71] Donald Knutson, Algebraic spaces, Lecture Notes in Mathematics, Vol. 203, Springer-Verlag, Berlin-New York, 1971. MR 0302647
  • [Kre99] Andrew Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495-536. MR 1719823,
  • [Liu02] Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. MR 1917232
  • [LMB99] 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
  • [Nis89] Ye. A. Nisnevich, The completely decomposed topology on schemes and associated descent spectral sequences in algebraic $ K$-theory, Algebraic $ K$-theory: connections with geometry and topology (Lake Louise, AB, 1987) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 279, Kluwer Acad. Publ., Dordrecht, 1989, pp. 241-342. MR 1045853
  • [Pir15] Roberto Pirisi, Cohomological invariants of algebraic curves, Ph.D. thesis, Scuola Normale Superiore di Pisa, 2015.
  • [Ros96] Markus Rost, Chow groups with coefficients, Doc. Math. 1 (1996), No. 16, 319-393. MR 1418952
  • [Sal84] David J. Saltman, Noether's problem over an algebraically closed field, Invent. Math. 77 (1984), no. 1, 71-84. MR 751131,
  • [Sta15] The Stacks Project authors, Stacks project,, 2015.
  • [Wit37] Ernst Witt, Theorie der quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math. 176 (1937), 31-44. MR 1581519,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14D23, 14F20

Retrieve articles in all journals with MSC (2010): 14D23, 14F20

Additional Information

Roberto Pirisi
Affiliation: Department of Mathematics and Statistics, University of Ottawa, King Edward Street, K1N6N5, Ottawa, Canada
Address at time of publication: Department of Mathematics, University of British Columbia, 1984 Mathematics Road, V6T 1Z2, British Columbia, Canada

Keywords: Algebraic stack, cohomological invariant
Received by editor(s): March 14, 2016
Received by editor(s) in revised form: June 12, 2016, and June 22, 2016
Published electronically: November 22, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society