The Brauer group of schemes
Time: 2015-11-03
Published By:
Speaker(s): Jean-Louis Colliot-Thelene (CNRS & Universite Paris Sud)
Time: November 3 - November 24, 2015
Venue: 一教208
Time: Nov. 3, 10, 17, 24 3:10-6:00
Summary:
The Brauer group of algebraic varieties features prominently in at least two directions of research:
birational problems (the L¨uroth problem) and arithmetic geometry (Brauer-Manin obstruction).
The November 2015 lectures will be devoted to the algebraic theory of the Brauer group. The first
part will be devoted to the general properties of the Brauer group. The second part will be concerned with
concrete computations of the Brauer group for various classes of algebraic varieties.
References:
J-P. Serre, Cohomologie galoisienne, Lecture Notes in Mathematics, vol. 5, 5`eme ´edition, r´evis´ee,
Springer-Verlag 1994. Translated into English as “Galois cohomology”, Springer monographes in Mathematics,
1997.
A. Grothendieck, Le groupe de Brauer I, II, III, in Dix expos´es sur la cohomologie des sch´emas, Masson
and North Holland, 1968.
Le groupe de Brauer : I. Alg`ebres d’Azumaya et interpr´etations diverses. S´eminaire Bourbaki, 9 (1964-
1966), Expos´e No. 290, available on NUMDAM, search :
http://www.numdam.org/numdam-bin/qrech
Also :
https://eudml.org/doc/109691
Le groupe de Brauer : II. Th´eories cohomologiques. S´eminaire Bourbaki, 9 (1964-1966), Expos´e No.
297, available on NUMDAM, use :
http://www.numdam.org/numdam-bin/qrech
Also :
https://eudml.org/doc/109698
The three talks GBI, GBII, GBIII, are available in the whole volume “Dix expos´es sur la cohomologie
des sch´emas ” available at:
webusers.imj-prg.fr/∼leila.schneps/grothendieckcircle/DixExp.pdf
J. S. Milne, Etale Cohomology, Princeton University Press, 1980. Chapters 1 to 4. ´
P. Gille and T. Szamuely, Central simple algebras and Galois cohomology, Cambridge studies in advanced
mathematics 101 (2006). Chapters 1 to 6.
J.-L. Colliot-Th´el`ene et J.-J. Sansuc, The rationality problem for fields of invariants under linear algebraic
groups, Proceedings of the International Colloquium on Algebraic groups and Homogeneous Spaces
(Mumbai 2004), ed. V. Mehta, TIFR Mumbai, Narosa Publishing House (2007), 113–186.
http://www.math.u-psud.fr/∼colliot/mumbai04.pdf
J.-L. Colliot-Th´el`ene, Notes on the Brauer group
http://www.math.u-psud.fr/∼colliot/CTnotesBrauer.pdf
Summary:
The Brauer group of algebraic varieties features prominently in at least two directions of research:
birational problems (the L¨uroth problem) and arithmetic geometry (Brauer-Manin obstruction).
The November 2015 lectures will be devoted to the algebraic theory of the Brauer group. The first
part will be devoted to the general properties of the Brauer group. The second part will be concerned with
concrete computations of the Brauer group for various classes of algebraic varieties.
References:
J-P. Serre, Cohomologie galoisienne, Lecture Notes in Mathematics, vol. 5, 5`eme ´edition, r´evis´ee,
Springer-Verlag 1994. Translated into English as “Galois cohomology”, Springer monographes in Mathematics,
1997.
A. Grothendieck, Le groupe de Brauer I, II, III, in Dix expos´es sur la cohomologie des sch´emas, Masson
and North Holland, 1968.
Le groupe de Brauer : I. Alg`ebres d’Azumaya et interpr´etations diverses. S´eminaire Bourbaki, 9 (1964-
1966), Expos´e No. 290, available on NUMDAM, search :
http://www.numdam.org/numdam-bin/qrech
Also :
https://eudml.org/doc/109691
Le groupe de Brauer : II. Th´eories cohomologiques. S´eminaire Bourbaki, 9 (1964-1966), Expos´e No.
297, available on NUMDAM, use :
http://www.numdam.org/numdam-bin/qrech
Also :
https://eudml.org/doc/109698
The three talks GBI, GBII, GBIII, are available in the whole volume “Dix expos´es sur la cohomologie
des sch´emas ” available at:
webusers.imj-prg.fr/∼leila.schneps/grothendieckcircle/DixExp.pdf
J. S. Milne, Etale Cohomology, Princeton University Press, 1980. Chapters 1 to 4. ´
P. Gille and T. Szamuely, Central simple algebras and Galois cohomology, Cambridge studies in advanced
mathematics 101 (2006). Chapters 1 to 6.
J.-L. Colliot-Th´el`ene et J.-J. Sansuc, The rationality problem for fields of invariants under linear algebraic
groups, Proceedings of the International Colloquium on Algebraic groups and Homogeneous Spaces
(Mumbai 2004), ed. V. Mehta, TIFR Mumbai, Narosa Publishing House (2007), 113–186.
http://www.math.u-psud.fr/∼colliot/mumbai04.pdf
J.-L. Colliot-Th´el`ene, Notes on the Brauer group
http://www.math.u-psud.fr/∼colliot/CTnotesBrauer.pdf
