Annals of K-Theory

Cohomologie non ramifiée de degré 3 : variétés cellulaires et surfaces de del Pezzo de degré au moins 5

Yang Cao

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Abstract

Dans cet article, où le corps de base est un corps de caractéristique zéro quelconque, pour X une variété géométriquement cellulaire, on étudie le quotient du troisième groupe de cohomologie non ramifiée Hnr3(X,(2)) par sa partie constante. Pour X une compactification lisse d’un torseur universel sur une surface géométriquement rationnelle, on montre que ce quotient est fini. Pour X une surface de del Pezzo de degré 5, on montre que ce quotient est trivial, sauf si X est une surface de del Pezzo de degré 8 d’un type particulier.

We consider geometrically cellular varieties X over an arbitrary field of characteristic zero. We study the quotient of the third unramified cohomology group Hnr3(X,(2)) by its constant part. For X a smooth compactification of a universal torsor over a geometrically rational surface, we show that this quotient is finite. For X a del Pezzo surface of degree 5, we show that this quotient is zero, unless X is a del Pezzo surface of degree 8 of a special type.

Article information

Source
Ann. K-Theory, Volume 3, Number 1 (2018), 157-171.

Dates
Received: 25 August 2016
Revised: 15 March 2017
Accepted: 2 April 2017
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.akt/1513774605

Digital Object Identifier
doi:10.2140/akt.2018.3.157

Mathematical Reviews number (MathSciNet)
MR3695367

Zentralblatt MATH identifier
06775614

Subjects
Primary: 14E08: Rationality questions [See also 14M20] 19E15: Algebraic cycles and motivic cohomology [See also 14C25, 14C35, 14F42]

Keywords
del Pezzo surface unramified cohomology

Citation

Cao, Yang. Cohomologie non ramifiée de degré 3 : variétés cellulaires et surfaces de del Pezzo de degré au moins 5. Ann. K-Theory 3 (2018), no. 1, 157--171. doi:10.2140/akt.2018.3.157. https://projecteuclid.org/euclid.akt/1513774605


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References

  • A. A. Albert, “A note on normal division algebras of order sixteen”, Bull. Amer. Math. Soc. 38:10 (1932), 703–706.
  • A. A. Albert, “Tensor products of quaternion algebras”, Proc. Amer. Math. Soc. 35 (1972), 65–66.
  • A. Auel and M. Bernardara, “Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields”, preprint, 2015.
  • J.-L. Colliot-Thélène, “Surfaces de Del Pezzo de degré $6$”, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A109–A111.
  • J.-L. Colliot-Thélène, “Birational invariants, purity and the Gersten conjecture”, pp. 1–64 in $K$-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), edited by B. Jacob and A. Rosenberg, Proc. Sympos. Pure Math. 58, Amer. Math. Soc., Providence, RI, 1995.
  • J.-L. Colliot-Thélène, “Points rationnels sur les variétés non de type général, chapitre II: Surfaces rationnelles”, course notes, Institut Henri Poincaré, 1999, https://tinyurl.com/CT99ChII.
  • J.-L. Colliot-Thélène, “Exposant et indice d'algèbres simples centrales non ramifiées”, Enseign. Math. $(2)$ 48:1-2 (2002), 127–146.
  • J.-L. Colliot-Thélène, “Descente galoisienne sur le second groupe de Chow: mise au point et applications”, Doc. Math. Extra vol.: Alexander S. Merkurjev's sixtieth birthday (2015), 195–220.
  • J.-L. Colliot-Thélène and J.-J. Sansuc, “La descente sur les variétés rationnelles, II”, Duke Math. J. 54:2 (1987), 375–492.
  • J.-L. Colliot-Thélène and C. Voisin, “Cohomologie non ramifiée et conjecture de Hodge entière”, Duke Math. J. 161:5 (2012), 735–801.
  • J.-L. Colliot-Thélène, D. Harari, and A. N. Skorobogatov, “Compactification équivariante d'un tore (d'après Brylinski et Künnemann)”, Expo. Math. 23:2 (2005), 161–170.
  • W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, Princeton University Press, 1993.
  • N. Jacobson, Finite-dimensional division algebras over fields, Springer, 1996.
  • B. Kahn, “Applications of weight-two motivic cohomology”, Doc. Math. 1 (1996), No. 17, 395–416.
  • B. Kahn, “Motivic cohomology of smooth geometrically cellular varieties”, pp. 149–174 in Algebraic $K$-theory (Seattle, WA, 1997), edited by W. Raskind and C. Weibel, Proc. Sympos. Pure Math. 67, Amer. Math. Soc., Providence, RI, 1999.
  • B. Kahn, Formes quadratiques sur un corps, Cours Spécialisés 15, Société Mathématique de France, Paris, 2008.
  • B. Kahn, “Cohomological approaches to $\mathit{SK}_1$ and $\mathit{SK}_2$ of central simple algebras”, Doc. Math. Extra vol.: Andrei A. Suslin sixtieth birthday (2010), 317–369.
  • B. Kahn, “Classes de cycles motiviques étales”, Algebra Number Theory 6:7 (2012), 1369–1407.
  • J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik $($3$)$ 32, Springer, 1996.
  • C. Mazza, V. Voevodsky, and C. Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs 2, American Mathematical Society, Providence, RI, 2006.
  • J. McCleary, A user's guide to spectral sequences, 2nd ed., Cambridge Studies in Advanced Mathematics 58, Cambridge University Press, 2001.
  • J. S. Milne, Étale cohomology, Princeton Mathematical Series 33, Princeton University Press, 1980.
  • J.-P. Serre, Cohomologie galoisienne, 3e ed., Lecture Notes in Math. 5, Springer, 1965.
  • A. Várilly-Alvarado, “Arithmetic of del Pezzo surfaces”, pp. 293–319 in Birational geometry, rational curves, and arithmetic, edited by F. Bogomolov et al., Springer, 2013.