Rost projectors and Steenrod operations
Let \(F\) be a field of characteristic \(\neq 2\), and let \(\varphi\) be a nondegenerate quadratic form over \(F\) with associated projective quadric \(X\) of dimension \(n=\dim\varphi -2\geq 1\) defined by the equation \(\varphi =0\). An element \(\rho\) in the integral Chow group \({\mathbb C}{\mathbb H}^n(X\times X)\) is called a Rost correspondence if, over an algebraic closure \(\overline{F}\) of \(F\), one has a decomposition \[ \rho_{\overline{F}}= [X_ {\overline{F}}\times x]+[x\times X_{\overline{F}}]\in {\mathbb C}{\mathbb H}^n(X\times X), \] where \(x\in X_{\overline{F}}\) is a rational point, and if this element is an idempotent with respect to composition of correspondences, then it is called a Rost projector. It follows from Rost's nilpotence theorem that the quadric \(X\) possesses a Rost projector if and only if it has a Rost correspondence. Isotropic quadrics always have a Rost projector, and an important problem which is also addressed in the present paper is to characterize those anisotropic quadrics which have a Rost projector. Let \(X\) be an anisotropic quadric possessing a Rost projector \(\rho\). The authors compute the \(0\)-dimensional component of the total Steenrod operation on the modulo \(2\) Chow group of the Rost motive given by the projector \(\rho\) using results by \textit{P. Brosnan} who constructed Steenrod operations on Chow groups in an elementary way [Trans. Am. Math. Soc. 355, 1869--1903 (2003; Zbl 1045.55005)]. (Steenrod operations in motivic cohomology have been defined earlier by V. Voevodsky.) The authors use their computations to determine the Chow group of the Rost motive, a result announced earlier (without proof) by Rost. This is then applied to compute the Chow group of a quadric associated to an excellent quadratic form. Recall that a quadratic form is called excellent if it is a Pfister neighbor (i.e. isometric to a subform of more than half the dimension of a scalar-multiple of a Pfister form) whose complement (i.e. its orthogonal complement inside this scalar-multiple of the Pfister form) is again a Pfister neighbor, and so on. Another application is a new and simpler proof of a theorem originally due to Vishik in characteristic \(0\) [cf. \textit{O. Izhboldin} and \textit{A. Vishik}, Quadratic forms and their applications (Dublin, 1999), Contemp. Math. 272, 103--125 (2000; Zbl 0972.11017)] which states that if the quadric \(X\) possesses a Rost projector, then \(\dim X +1=2^m\) for some \(m\geq 1\). An important open problem is whether in this case the quadratic form \(\varphi\) is actually given by a Pfister neighbor. So far, this is known to be true for \(n\leq 3\) due to the first author [J. Pure Appl. Algebra 160, 195--227 (2001; Zbl 0998.11016)].
scientific article; zbMATH DE number 1862892