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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Witt groups of Grassmann varieties

Authors: Paul Balmer and Baptiste Calmès
Journal: J. Algebraic Geom. 21 (2012), 601-642
Published electronically: May 23, 2012
MathSciNet review: 2957690
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Abstract | References | Additional Information

Abstract: We compute the Witt groups of split Grassmann varieties, over any regular base $X$. The answer is that the total Witt group of the Grassmannian is a free module over the total Witt ring of $X$. We provide an explicit basis for this free module, which is indexed by a special class of Young diagrams, that we call even Young diagrams.

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  • Schémas en groupes I, II et III, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 151, 152 and 153, Springer-Verlag, Berlin, 1962/1970.
  • Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre. MR 0354655
  • Paul Balmer, Triangular Witt groups. I. The 12-term localization exact sequence, $K$-Theory 19 (2000), no. 4, 311–363. MR 1763933, DOI
  • Paul Balmer, Triangular Witt groups. II. From usual to derived, Math. Z. 236 (2001), no. 2, 351–382. MR 1815833, DOI
  • Paul Balmer, Products of degenerate quadratic forms, Compos. Math. 141 (2005), no. 6, 1374–1404. MR 2188441, DOI
  • Paul Balmer and Baptiste Calmès, Geometric description of the connecting homomorphism for Witt groups, Doc. Math. 14 (2009), 525–550. MR 2565903
  • ---, Bases of total Witt groups and lax-similitude, J. Algebra Appl., to appear.
  • Baptiste Calmès and Jens Hornbostel, Push-forwards for Witt groups of schemes, Comment. Math. Helv. 86 (2011), no. 2, 437–468. MR 2775136, DOI
  • Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 0302656
  • William Fulton, Intersection theory, 2nd ed., 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. 2, Springer-Verlag, Berlin, 1998. MR 1644323
  • Stefan Gille, Homotopy invariance of coherent Witt groups, Math. Z. 244 (2003), no. 2, 211–233. MR 1992537, DOI
  • A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. (1961), no. 11, 167.
  • N. A. Karpenko, Cohomology of relative cellular spaces and of isotropic flag varieties, Algebra i Analiz 12 (2000), no. 1, 3–69 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 12 (2001), no. 1, 1–50. MR 1758562
  • Manfred Knebusch, Symmetric bilinear forms over algebraic varieties, Conference on Quadratic Forms—1976 (Proc. Conf., Queen’s Univ., Kingston, Ont., 1976) Queen’s Univ., Kingston, Ont., 1977, pp. 103–283. Queen’s Papers in Pure and Appl. Math., No. 46. MR 0498378
  • Dan Laksov, Algebraic cycles on Grassmann varieties, Advances in Math. 9 (1972), 267–295. MR 318145, DOI
  • M. Levine and F. Morel, Algebraic cobordism, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2286826
  • Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232
  • I. Panin, Riemann-Roch theorems for oriented cohomology, Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, Kluwer Acad. Publ., Dordrecht, 2004, pp. 261–333. MR 2061857, DOI
  • R. W. Thomason, Les $K$-groupes d’un schéma éclaté et une formule d’intersection excédentaire, Invent. Math. 112 (1993), no. 1, 195–215 (French). MR 1207482, DOI
  • C. Walter, Grothendieck-Witt groups of projective bundles, preprint, 2003.

Additional Information

Paul Balmer
Affiliation: Department of Mathematics, UCLA, Los Angeles, California 90095-1555
MR Author ID: 652084

Baptiste Calmès
Affiliation: Université d’Artois, Laboratoire de Mathématiques de Lens, France

Received by editor(s): May 23, 2009
Received by editor(s) in revised form: April 29, 2011
Published electronically: May 23, 2012
Additional Notes: The first author is supported by NSF grant DMS-0969644.