Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-09T21:46:48.973Z Has data issue: false hasContentIssue false
  • English
  • Français

Reduced Steenrod operations and resolution of singularities

Published online by Cambridge University Press:  07 July 2011

Olivier Haution
Olivier Haution
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United [email protected]
Get access

Abstract

We give a new construction of a weak form of Steenrod operations for Chow groups modulo a prime number p for a certain class of varieties. This class contains projective homogeneous varieties which are either split or considered over a field admitting some form of resolution of singularities, for example any field of characteristic not p. These reduced Steenrod operations are sufficient for some applications to the theory of quadratic forms.

Keywords

Steenrod operationsRiemann-Roch theoremChow groups
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 2 , April 2012 , pp. 269 - 290
Copyright
Copyright © ISOPP 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

AKMW02.Abramovich, Dan, Karu, Kalle, Matsuki, Kenji, and Włodarczyk, Jarosław. Torification and factorization of birational maps. J. Amer. Math. Soc. 15 (3) (2002), 531–572 (electronic).Google Scholar
Ati66.Atiyah, Michael F.. Power operations in K-theory. Quart. J. Math. Oxford Ser. 17 (2) (1966), 165–193.CrossRefGoogle Scholar
Boi08.Boisvert, Alex. A new definition of the Steenrod operations in algebraic geometry. preprint, 2008. arXiv:0805.1414.Google Scholar
Bro03.Brosnan, Patrick. Steenrod operations in Chow theory. Trans. Amer. Math. Soc. 355 (5) (2003), 1869–1903 (electronic).Google Scholar
CP08.Cossart, Vincent and Piltant, Olivier. Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings. J. Algebra 320 (3) (2008), 1051–1082.CrossRefGoogle Scholar
CP09.Cossart, Vincent and Piltant, Olivier. Resolution of singularities of threefolds in positive characteristic II. J. Algebra 321 (7) (2009), 1836–1976.Google Scholar
Dem74.Demazure, Michel. Désingularisation des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. 7 (4) (1974), 53–88. Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I.Google Scholar
EKM08.Elman, Richard, Karpenko, Nikita, and Merkurjev, Alexander. The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications 56, American Mathematical Society, Providence, RI, 2008.Google Scholar
FL85.Fulton, William and Lang, Serge. Riemann-Roch algebra, Grundlehren der Mathematischen Wissenschaften 277 [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1985.CrossRefGoogle Scholar
Ful98.Fulton, William. Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics, 3. Folge, 2te Ausgabe [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 1998.Google Scholar
Gab.Gabber, Ofer. Finiteness theorems in étale cohomology. Abstract of a talk given at the IHES on January 12, 2009. http://www.ihes.fr/document?id=1713&id_attribute=48.Google Scholar
Gil05.Gillet, Henri. K-theory and intersection theory. In Handbook of K-theory. Vol. 1, 2, 235–293. Springer, Berlin, 2005.CrossRefGoogle Scholar
GS87.Gillet, Henri and Soulé, Christophe. Intersection theory using Adams operations. Invent. Math. 90 (2) (1987), 243–277.CrossRefGoogle Scholar
Hau.Haution, Olivier. On the first Steenrod square for Chow groups. Amer. J. Math., to appear. http://www.math.uiuc.edu/K-theory/0945/.Google Scholar
Hau10.Haution, Olivier. Duality and the topological filtration. preprint, 2010. arXiv:1006.1482.Google Scholar
Kar98.Karpenko, Nikita A.. Codimension 2 cycles on Severi-Brauer varieties. K-Theory 13 (4) (1998), 305–330.CrossRefGoogle Scholar
Köc91.Köck, Bernhard. Chow motif and higher Chow theory of G/P. Manuscripta Math. 70 (4) (1991), 363–372.CrossRefGoogle Scholar
Lev07.Levine, Marc. Steenrod operations, degree formulas and algebraic cobordism. Pure Appl. Math. Q. 3 (1) (2007), 283–306.CrossRefGoogle Scholar
LM07.Levine, Marc and Morel, Fabien. Algebraic cobordism. Springer Monographs in Mathematics. Springer, Berlin, (2007).Google Scholar
LP09.Levine, Marc and Pandharipande, Rahul. Algebraic cobordism revisited. Invent. Math. 176 (1) (2009), 63–130.CrossRefGoogle Scholar
Mer03.Merkurjev, Alexander. Steenrod operations and degree formulas. J. Reine Angew. Math. 565 (2003), 13–26.Google Scholar
Pan04.Panin, Ivan. Riemann-Roch theorems for oriented cohomology. In Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem. 131, 261–333. Kluwer Acad. Publ., Dordrecht, 2004.CrossRefGoogle Scholar
Ros06.Rost, Markus. On the basic correspondence of a splitting variety. preprint, 2006. http://www.math.uni-bielefeld.de/~rost/basic-corr.html.Google Scholar
Ros08.Rost, Markus. On Frobenius, K-theory, and characteristic numbers. preprint, 2008. http://www.math.uni-bielefeld.de/~rost/frobenius.html.Google Scholar
Sou85.Soulé, Christophe. Opérations en K-théorie algébrique. Canad. J. Math. 37 (3) (1985), 488–550.CrossRefGoogle Scholar
Voe03.Voevodsky, Vladimir. Reduced power operations in motivic cohomology. Publ. Math. Inst. Hautes Études Sci. 98 (2003), 1–57.CrossRefGoogle Scholar
Zai10.Zainoulline, Kirill. Degree formula for connective K-theory. Invent. Math. 179 (3) (2010), 507–522.CrossRefGoogle Scholar