Witt groups of Grassmann varieties
Authors:
Paul Balmer and Baptiste Calmès
Journal:
J. Algebraic Geom. 21 (2012), 601-642
DOI:
https://doi.org/10.1090/S1056-3911-2012-00613-4
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.
References
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- ---, Bases of total Witt groups and lax-similitude, J. Algebra Appl., to appear.
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References
- 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, Springer-Verlag, Berlin, 1971, Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6), Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Springer Lecture Notes in Mathematics, Vol. 225. MR 0354655 (50:7133)
- P. Balmer, Triangular Witt groups Part I: The 12-term localization exact sequence, K-Theory 4 (2000), no. 19, 311–363. MR 1763933 (2002h:19002)
- ---, Triangular Witt groups Part II: From usual to derived, Math. Z. 236 (2001), no. 2, 351–382. MR 1815833 (2002h:19003)
- ---, Products of degenerate quadratic forms, Compositio Mathematica 6 (2005), no. 141, 1374–1404. MR 2188441 (2006k:11059)
- P. Balmer and B. Calmès, Geometric description of the connecting homomorphism for Witt groups, Doc. Math. 14 (2009), 525–550. MR 2565903 (2010j:19009)
- ---, Bases of total Witt groups and lax-similitude, J. Algebra Appl., to appear.
- B. Calmès and J. Hornbostel, Push-forwards for Witt groups of schemes, Comment. Math. Helv. 86 (2011), no. 2, 437–468. MR 2775136 (2012e:19007)
- M. Demazure and P. Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeur, Paris, 1970. MR 0302656 (46:1800)
- W. Fulton, Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)
- S. Gille, Homotopy invariance of coherent Witt groups, Math. Z. 244 (2003), no. 2, 211–233. MR 1992537 (2004f:19009)
- 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. MR 1758562 (2001c:14076)
- M. Knebusch, Symmetric bilinear forms over algebraic varieties, Queen’s Papers in Pure and Appl. Math., No. 46, Queen’s Univ., Kingston, Ont., 1977, pp. 103–283. MR 0498378 (58:16506)
- D. Laksov, Algebraic Cycles on Grassmann Varieties, Adv. in Math. 9 (1972), no. 3, 267–295. MR 0318145 (47:6694)
- M. Levine and F. Morel, Algebraic cobordism, Springer Monographs in Mathematics, Springer, Berlin, 2007. MR 2286826 (2008a:14029)
- Q. Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. MR 1917232 (2003g:14001)
- I. Panin, Riemann-Roch theorems for oriented cohomology, Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II, vol. 131, Kluwer, Dordrecht, 2004, pp. 261–333. MR 2061857 (2005g:14025)
- R. W. Thomason, Les $K$-groupes d’un schéma éclaté et une formule d’intersection excédentaire, Invent. Math. 112 (1993), no. 1, 195–215. MR 1207482 (93k:19005)
- 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.
