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Fundamentals of Specialization Theory

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Specialization of Quadratic and Symmetric Bilinear Forms

Part of the book series: Algebra and Applications ((AA,volume 11))

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Abstract

We also allow the case dim φ = 0, standing for the unique bilinear form on the zero vector space, the form φ = 0. We agree that the form φ = 0 has good reduction and set λ *(φ) = 0. Problem 1.2. Is this definition meaningful? Up to isometry λ *(φ) should be independent of the choice of the matrix (c ij ). We shall later see that this is indeed the case, provided 2 ∉ m, so that L has characteristic ≠ 2. If L has characteristic 2, then λ *(φ) is well-defined up to “stable isometry” (see §1.3).

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Notes

  1. 1.

    The case where K has only two elements, \(K = \mathbb{F}_2\), is not covered by the more general theorems there. The statement of Theorem1.11 for \(K = \mathbb{F}_2\) is trivial however, since K has only one square class \(\langle 1 \rangle\) and \(\langle 1,1 \rangle \sim 0\).

  2. 2.

    The letter W in the notation \({\lambda_W}\) refers to “Witt” or “weak”, see §1.1 and §1.7.

  3. 3.

    If \(E=(E,B)\) is a bilinear space, then \(-E\) denotes the space \((E,-B)\).

  4. 4.

    \(\varphi \otimes K\) is the form φ, considered over K instead of over k. If E is a bilinear space over k corresponding to \({\varphi}\), then \({K \otimes_k E}\)—as described in §1.3—is a bilinear space corresponding to \({\varphi \otimes K}\).

  5. 5.

    We do not make a notational distinction between the forms \({\left( {0 \atop 1} {1 \atop 0 } \right)}\) over the different fields occurring here.

  6. 6.

    In keeping with our earlier conventions, it would perhaps be better to write \(\varphi (x) = \frac{1}{2} \varphi (x, x)\). However, the factor \(\frac{1}{2}\) is not important for now.

  7. 7.

    In earlier works (especially [33]) \(K_{0}\) was always chosen to be k. From a technical point of view it is favourable to allow \(K_{0}\) to be an inessential extension of k, just as in [37].

  8. 8.

    The form \(\langle 1\rangle\) also counts as a Pfister form, more precisely a 0-fold Pfister form.

  9. 9.

    Strictly speaking, square classes are isomorphism classes of one-dimensional spaces.

  10. 10.

    In accordance with our earlier agreement, every \(E_{s}\) is considered as a quadratic subspace of E, \(E_s = (E_s, q \vert{E_s})\).

  11. 11.

    See § 1.2 for the term “stably isometric”.

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Authors and Affiliations

  1. Department of Mathematics, University of Regensburg, Universitätsstr. 31, 93040, Regensburg, Germany

    Prof. Dr. Manfred Knebusch

  2. School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, Ireland

    Dr. Thomas Unger

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  1. Prof. Dr. Manfred Knebusch
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Correspondence to Manfred Knebusch .

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Knebusch, M., Unger, T. (2010). Fundamentals of Specialization Theory. In: Specialization of Quadratic and Symmetric Bilinear Forms. Algebra and Applications, vol 11. Springer, London. https://doi.org/10.1007/978-1-84882-242-9_1

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