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Motivic equivalence of affine quadrics

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Abstract

In this article we show that the motive of an affine quadric \(\{q=1\}\) determines the respective quadratic form.

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Notes

  1. Note that if \(p = aX^2\) is a one-dimensional quadratic form (with \(a \ne 0\)) then \(P \subset {\mathbb {P}}^0 = {\mathbb {P}}(k^1) = *\) is the closed projective subvariety given by \(\{aX^2 = 0\}\), which is empty. In particular \(M(P)=0\).

  2. There is more than one non-zero morphism if and only if \(q \cong X^2\), in which case \(M(A_q) = M(\{X^2 = 1\}) = M(* \coprod *)\).

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Acknowledgements

We are grateful to the Referee for useful remarks which improved the exposition.

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Authors and Affiliations

  1. Fakultät Mathematik, Universität Duisburg-Essen, Essen, Germany

    Tom Bachmann

  2. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Alexander Vishik

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  1. Tom Bachmann
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  2. Alexander Vishik
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Correspondence to Alexander Vishik.

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Communicated by Vasudevan Srinivas.

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Bachmann, T., Vishik, A. Motivic equivalence of affine quadrics. Math. Ann. 371, 741–751 (2018). https://doi.org/10.1007/s00208-018-1641-8

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  • DOI: https://doi.org/10.1007/s00208-018-1641-8

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