Some new examples in the theory of quadratic forms
The aim of this paper is to answer some questions in the algebraic theory of quadratic forms. In the first part of the paper, the authors obtain some important computations of the topological filtration on the Grothendieck group of some projective homogeneous varieties. In the second part of the paper, which is a consequence of the first one, the authors deduce some results on the isotropy problem and the structure of certain forms in \(I^2\) and \(I^3\) over a field of characteristic not \(2\), where \(I^n\) is the \(n\)-th power of the ideal of even-dimensional quadratic forms. Recall that the isotropy problem over function fields of quadrics is to classify the pairs of anisotropic quadratic forms \(\varphi\), \(\psi\) such that \(\varphi\) becomes isotropic over the function field \(F(\psi)\) of the projective quadric defined by \(\psi=0\). When \(\varphi\) is isotropic over \(F(\psi)\), the isotropy is called standard (in the sense of Definition 18.1) when one of the two following conditions is satisfied: \(\psi\) is a sub-form of \(\varphi\) (up to a scalar); or \(\varphi\) contains a Pfister neighbour form which is isotropic over \(F(\psi)\). The authors treat the isotropy in the case where \(\dim \varphi=6\). It was shown before that the isotropy is always standard when \(\dim \varphi \leq 6\), except for the case: \(\dim \varphi=6\), \(\dim \psi=4\), \(1\neq d_{\pm}\varphi \neq d_{\pm}\psi\neq 1\) and \(\text{ind}(C_0(\varphi))= \text{ind}(C_0(\psi)\otimes C_0(\varphi))=2\), where \(``d_{\pm}"\), \(``C_0"\) and ``ind'' mean respectively the signed discriminant, the even Clifford algebra and the Schur index. In this paper, the authors produce an example where the isotropy is not standard in the above exceptional case. After this, they construct an example of an \(8\)-dimensional quadratic form with trivial discriminant such that the Schur index of its Clifford algebra is \(4\) but the form is not isometric to a sum of two quadratic forms of dimension \(4\) with trivial discriminants. This last example allowed the construction of a \(14\)-dimensional quadratic form in \(I^3\) which is not a difference of two \(3\)-fold Pfister forms. The examples on \(8\) and \(14\)-dimensional forms are obtained independently by \textit{D. W. Hoffmann} and \textit{J.-P. Tignol} by using different arguments [Doc. Math., J. DMV 3, 189-214 (1998; Zbl 0914.11014)].
Grothendieck group and Chow group of a smooth algebraic variety
scientific article; zbMATH DE number 1515376