Chow rings of excellent quadrics
Excellent quadratic forms of height \(h\) over a field of characteristic not \(2\) are defined as follows: \(0\)- and \(1\)-dimensional forms are excellent of height \(0\), and a form \(\xi\), \(\dim\xi\geq 2\), is excellent of height \(h\geq 1\) if there exists a form \(\xi'\) such that \(\xi\perp -\xi'\) is similar to a Pfister form, \(\dim\xi>\dim\xi'\), and \(\xi'\) is excellent of height \(h-1\). In particular, Pfister forms and Pfister neighbors of codimension \(1\) are excellent. Other important examples are quadratic forms given by a sum of squares. The definition goes back to \textit{M. Knebusch} [Proc. Lond. Math. Soc., III. Ser. 34, 1--31 (1977; Zbl 0359.15013)], and excellent quadratic forms have been studied extensively ever since because of their many nice properties. Let now \(k\) be a field of characteristic \(0\), and let \(X_\xi\) be the quadric defined by an anisotropic excellent quadratic form \(\xi\) over \(k\). Consider \(CH^*(X_\xi)\), the Chow ring of the quadric \(X_\xi\). The additive structure of this ring has been known due to previous work and results by Rost, Karpenko-Merkurjev and the reviewer, and it can be expressed in terms of so-called Rost motives. In the present paper, the author determines the multiplicative structure of \(CH^*(X_\xi)\) in the case \(\dim(X_\xi)\not\equiv 2\bmod 4\) in which case it turns out that it only depends on \( \dim(X_\xi)\). A crucial ingredient is the study of the structure of \(\Omega^*(X_\xi)\), which denotes the algebraic cobordism as defined by Levine-Morel or the motivic cobordism as defined by Voevodsky, and which is intimately related to \(CH^*(X_\xi)\).
scientific article; zbMATH DE number 5321419