# Excellent connections in the motives of quadrics

Annales scientifiques de l'École Normale Supérieure (2011)

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

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topVishik, 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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- [3] O. Haution, Lifting of coefficients for Chow motives of quadrics, in Quadratic forms, linear algebraic groups, and cohomology, Dev. Math. 18, Springer, 2010, 239–247. Zbl1216.11041MR2648729
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- [7] N. A. Karpenko, On the first Witt index of quadratic forms, Invent. Math.153 (2003), 455–462. Zbl1032.11016MR1992018
- [8] M. Knebusch, Generic splitting of quadratic forms. I, Proc. London Math. Soc. 33 (1976), 65–93. Zbl0351.15016MR412101
- [9] M. Knebusch, Generic splitting of quadratic forms. II, Proc. London Math. Soc. 34 (1977), 1–31. Zbl0359.15013MR427345
- [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] M. Rost, The motive of a Pfister form, preprint http://www.math.uni-bielefeld.de/~rost/data/motive.pdf, 1998.
- [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] 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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