The Rost nilpotence theorem for geometrically rational surfaces
Let \(X\) be a smooth projective connected variety of dimension \(d\) over a field \(k\). We regard \(\mathrm{CH}_d(X \times X)\) as the ring of algebraic correspondences on \(X\). For a field extension \(E/k\), we write \(X_E\) for the base change of \(X\) to \(E\), and \(r_{E/k} : \mathrm{CH}_d(X \times X) \to \mathrm{CH}_d(X_E \times X_E)\) for the restriction map. We say that Rost nilpotence is true for \(X\) if for any \(f \in \mathrm{CH}_d(X \times X)\) for which there is a field extension \(E/k\) such that \(r_{E/k}(f)=0\), then \(f^d=0\) holds for some \(d>0\). The main result of the paper under review shows that the Rost nilpotence is true for any geometrically rational surface over a field of characteristic zero. A key (technical) step is to show that any correspondence is compatible with the \textit{S. Bloch} map [Ann. Sci. Éc. Norm. Supér. (4) 14, 41--59 (1981; Zbl 0524.14006)], which is a homomorphism connecting the Chow group and the Galois cohomology with coefficients in the Néron-Severi torus. To prove this, the Bloch map is examined in terms of \textit{M. Rost}'s cycle complex [Doc. Math., J. DMV 1, 319--393 (1996; Zbl 0864.14002)].
scientific article; zbMATH DE number 5723600
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