The first part of the paper studies the torsion of the second Chow group \(\text{CH}^2\) of the product of Severi-Brauer varieties of an arbitrary finite collection of central simple algebras. The computation of the torsion of \(\text{CH}^2\) can be used to get a nontrivial information on the relative cohomology group \(H^3(K/F)\) for the function field \(K\) of the product of Severi-Brauer varieties of \(n\) central simple algebras. When \(n=2\) the authors get a definitive result (in Appendix A). The results in Part I are motivated by and extend the work of E. Peyre [Proc. Symp. Pure Math. 58, Part 2, 369–401 (1995; Zbl 0837.14006)].
Part two is concerned with the excellence property of field extensions. The extension \(L/F\) is excellent when the anisotropic part \(\varphi_L\) of any quadratic \(F\)-form \(\varphi\) is defined over \(F\). For the function field \(F(X)\) of a variety \(X\) the extension \(F(X)/F\) is said to be universally excellent when for every extension \(K/F\) the extension \(K(X)/K\) is excellent. For a quadric \(X\) a complete description of all the cases when \(F(X)/F\) is universally excellent is known from the work of several authors.
Then the authors take up the problem of describing the universal excellence of the extension \(F(SB(A))/F\) where \(SB(A)\) is the Severi-Brauer variety of a central simple algebra \(A\). They prove that this happens exactly when either the index of \(A\) is odd or \(A = Q \otimes D\), where \(Q\) is a quaternion algebra and \(D\) has odd index. Explicit examples of nonexcellent extensions are given, in particular, the function fields of Severi-Brauer varieties of biquaternion algebras over rational function fields in 4 variables produce such examples.
While this substantial paper deals with sophisticated matter, the authors make every effort to motivate their work and to present it as simply as they know how.