Excellent connections in the motives of quadrics

Alexander Vishik

Annales scientifiques de l'École Normale Supérieure (2011)

  • Volume: 44, Issue: 1, page 183-195
  • ISSN: 0012-9593

Abstract

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In this article we prove the conjecture claiming that the motive of a real quadric is the “most decomposable” among anisotropic quadrics of given dimension over all fields. This imposes severe restrictions on the motive of arbitrary anisotropic quadric. As a corollary we estimate from below the rank of indecomposable direct summand in the motive of a quadric in terms of its dimension. This generalizes the well-known Binary Motive Theorem. Moreover, we have the description of the Tate motives involved. This, in turn, gives another proof of Karpenko’s Theorem on the value of the first higher Witt index. But also other new relations among higher Witt indices follow.

How to cite

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Vishik, Alexander. "Excellent connections in the motives of quadrics." Annales scientifiques de l'École Normale Supérieure 44.1 (2011): 183-195. <http://eudml.org/doc/272213>.

@article{Vishik2011,
abstract = {In this article we prove the conjecture claiming that the motive of a real quadric is the “most decomposable” among anisotropic quadrics of given dimension over all fields. This imposes severe restrictions on the motive of arbitrary anisotropic quadric. As a corollary we estimate from below the rank of indecomposable direct summand in the motive of a quadric in terms of its dimension. This generalizes the well-known Binary Motive Theorem. Moreover, we have the description of the Tate motives involved. This, in turn, gives another proof of Karpenko’s Theorem on the value of the first higher Witt index. But also other new relations among higher Witt indices follow.},
author = {Vishik, Alexander},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {quadratic forms; motives; Chow groups; Steenrod operations},
language = {eng},
number = {1},
pages = {183-195},
publisher = {Société mathématique de France},
title = {Excellent connections in the motives of quadrics},
url = {http://eudml.org/doc/272213},
volume = {44},
year = {2011},
}

TY - JOUR
AU - Vishik, Alexander
TI - Excellent connections in the motives of quadrics
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2011
PB - Société mathématique de France
VL - 44
IS - 1
SP - 183
EP - 195
AB - In this article we prove the conjecture claiming that the motive of a real quadric is the “most decomposable” among anisotropic quadrics of given dimension over all fields. This imposes severe restrictions on the motive of arbitrary anisotropic quadric. As a corollary we estimate from below the rank of indecomposable direct summand in the motive of a quadric in terms of its dimension. This generalizes the well-known Binary Motive Theorem. Moreover, we have the description of the Tate motives involved. This, in turn, gives another proof of Karpenko’s Theorem on the value of the first higher Witt index. But also other new relations among higher Witt indices follow.
LA - eng
KW - quadratic forms; motives; Chow groups; Steenrod operations
UR - http://eudml.org/doc/272213
ER -

References

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  2. [2] R. Elman, N. A. Karpenko & A. Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications 56, Amer. Math. Soc., 2008. Zbl1165.11042MR2427530
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  4. [4] J. Hurrelbrink, N. A. Karpenko & U. Rehmann, The minimal height of quadratic forms of given dimension, Arch. Math. (Basel) 87 (2006), 522–529. Zbl1109.11023MR2283683
  5. [5] J. Hurrelbrink & U. Rehmann, Splitting patterns of excellent quadratic forms, J. reine angew. Math. 444 (1993), 183–192. Zbl0791.11016MR1241799
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  7. [7] N. A. Karpenko, On the first Witt index of quadratic forms, Invent. Math.153 (2003), 455–462. Zbl1032.11016MR1992018
  8. [8] M. Knebusch, Generic splitting of quadratic forms. I, Proc. London Math. Soc. 33 (1976), 65–93. Zbl0351.15016MR412101
  9. [9] M. Knebusch, Generic splitting of quadratic forms. II, Proc. London Math. Soc. 34 (1977), 1–31. Zbl0359.15013MR427345
  10. [10] M. Rost, Some new results on the Chow groups of quadrics, preprint http://www.mathematik.uni-bielefeld.de/~rost/data/chowqudr.pdf, 1990. 
  11. [11] M. Rost, The motive of a Pfister form, preprint http://www.math.uni-bielefeld.de/~rost/data/motive.pdf, 1998. 
  12. [12] A. Vishik, Motives of quadrics with applications to the theory of quadratic forms, in Geometric methods in the algebraic theory of quadratic forms, Lecture Notes in Math. 1835, Springer, 2004, 25–101. Zbl1047.11033
  13. [13] A. Vishik, Symmetric operations, Tr. Mat. Inst. Steklova 246 (2004), 92–105, English translation: Proc. of the Steklov Institute of Math. 246 (2004), 79–92. Zbl1117.14007

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