Journal of the Ramanujan Mathematical Society

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The Units-Picard Complex of a Reductive Group Scheme


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1 Departamento de Matematicas, Universidad de La Serena, Cisternas 1200, La Serena 1700000, Chile
     

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Let S be a locally noetherian regular scheme. We compute the units-Picard complex of a reductive S-group scheme G in terms of the dual algebraic fundamental complex of G. To this end, we establish a units-Picard-Brauer exact sequence for a torsor under a smooth S-group scheme.
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  • [BBD] A. Beilinson, J. Bernstein and P. Deligne, Faisceaux pervers, in: Analysis and topology on singular spaces I (Luminy 1981), Ast´risque, 100 (1982) 5–171.

  • [Bey] R. Beyl, The connecting morphism in the kernel-cokernel sequence, Arch. der Math., 32 no. 4, (1979) 305–308.

  • [BvH] M. Borovoi, and J. van Hamel, Extended Picard complexes and linear algebraic groups, J. reine angew. Math., 627 (2009) 53–82.

  • [BGA] M. Borovoi, and C. D. Gonzalez-Aviles, The algebraic fundamental group of a reductive group scheme over an arbitrary base scheme, Cent. Eur. J. Math., 12(4) (2014) 545–558.

  • [CE] H. Cartan and S. Eilenberg, Homological Algebra. Princeton U. Press, Princeton (1956).

  • [CGP] B. Conrad, O. Gabber and G. Prasad, Pseudo-reductive groups, Second Ed. New Math. Monograps, Cambridge U. Press, 26 (2015).

  • [CTS] J.-L. Colliot-Thelene and J.-J. Sansuc, La descente sur les vari´et´es rationnelles, II. Duke Math. J., 54 no. 2, (1987) 375–492.

  • [CT08] J.-L. Colliot-Thelene, Resolutions flasques des groupes lin´eaires connexes, J. Reine Angew. Math., 618 (2008) 77–133.

  • [SGA3new] M. Demazure and A. Grothendieck, (Eds.): Schemas en groupes. Seminaire de Geometrie Algebrique du Bois Marie 1962-64 (SGA 3). Augmented and corrected 2008-2011 re-edition of the original by P. Gille and P. Polo. Available at http://www.math.jussieu.fr/ polo/SGA3. Reviewed at http://www.jmilne.org/math/xnotes/SGA3r.pdf.

  • [Gi] J. Giraud, Cohomologie non abelienne, Grundlehren Math. Wiss., Springer-Verlag, Berlin, New York 179 (1971).

  • [GA1] C. D. Gonzalez-Aviles, Flasque resolutions of reductive group schemes, Cent. Eur. J. Math., 11(7) (2013) 1159–1176.

  • [GA2] C. D. Gonzalez-Aviles, On the group of units and the Picard group of a product, Eur. J. Math., 3 (2017) 471–506.

  • [GA3] C. D. Gonzalez-Aviles, The units-Picard complex and the Brauer group of a product, J. Pure Appl. Algebra, 222 no. 9, (2018) 2746–2772.

  • [EGA Inew] A. Grothendieck and J. Dieudonne, ´ Elements de geometrie algebrique I. Le langage des sch´emas, Grund. der Math. Wiss., 166 (1971).

  • [EGA] A. Grothendieck and J. Dieudonne, Elements de geometrie algebrique, Publ. Math. IHES, 8 (= EGA II) (1961), 11 III1 (1961), 20 (= EGA IV1) (1964), 24 (= EGA IV2) (1965), 32 (= EGA IV4) (1967).

  • [FI] R. Fossum, and B. Iversen, On Picard groups of algebraic fibre spaces, J. Pure Appl. Alg., 3 (1973) 269–280.

  • [HSk] D. Harari and A. Skorobogatov, Descent theory for open varieties. In: Torsors, etale homotopy and applications to rational points, 250–279, London Math. Soc. Lecture Note Ser., 405, Cambridge Univ. Press, Cambridge (2013).

  • [Klei] S. Kleiman, The Picard scheme. arXiv:math/0504020v1 [math.AG].

  • [Liu] Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, 6 (2002).

  • [May] P. J. May, The geometry of iterated loop spaces, Lecture Notes in Math., Springer-Verlag, 271 (1975).

  • [MiEt] J. S. Milne, ´Etale Cohomology. Princeton University Press, Princeton (1980).

  • [ADT] J. S. Milne, Arithmetic Duality Theorems. Second Ed. (electronic version) (2006).

  • [Nfdc23-1] Comment 2 at https://mathoverflow.net/questions/273503/.

  • [Nfdc23-2] Answer to https://mathoverflow.net/questions/273762.

  • [Pic] G. Picavet, Recent progress on submersions: a survey and new properties. Algebra (Hindawi Publishing Corporation). Volume 2013, http://dx.doi.org/10.1155/2013/128064.

  • [Ray] M. Raynaud, Faisceaux amples sur les schemas en groupes et les espaces homogenes, Lecture Notes in Math., Springer, Berlin-Heidelberg-New York, 119 (1970).

  • [San81] J.-J. Sansuc, Groupe de Brauer et arithmetique des groupes algebriques lineaires sur un corps de nombres, J. Reine Angew. Math., 327 (1981) 12–80.

  • [SP] The Stacks Project, http://stacks.math.columbia.edu.

  • [T] G. Tamme, Introduction to E tale Cohomology. Translated from the German by Manfred Kolster. Universitext. Springer-Verlag, Berlin (1994).

  • [Ver] J.-L. Verdier, Des categories derivees des categories ab´eliennes, Asterisque. Soc. Math., France, 239 (1996).


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  • The Units-Picard Complex of a Reductive Group Scheme

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Authors

Cristian D. Gonzalez-Aviles
Departamento de Matematicas, Universidad de La Serena, Cisternas 1200, La Serena 1700000, Chile

Abstract


Let S be a locally noetherian regular scheme. We compute the units-Picard complex of a reductive S-group scheme G in terms of the dual algebraic fundamental complex of G. To this end, we establish a units-Picard-Brauer exact sequence for a torsor under a smooth S-group scheme.

References