An example concerning specialization of torsion subgroups of Chow groups (Q2906510)

Let \(S= \text{Spec\,}A\) be an affine, regular, integral scheme of finite type over \(\text{Spec\,}\mathbb{Z}\) and let \(f:{\mathcal X}\to S\) be a smooth, projective morphism with geometrically connected \(n\)-dimensional fibers. Let \(g: \{\eta\}\to S\) be the inclusion of the generic point \(\eta=\text{Spec\,}K\), and \(i_s: \{s\}\to S\) the inclusion of a non-generic point \(s= \text{Spec\,}F\). If \(l\) is a prime number there is a specialization map on \(l\)-primary torsion for Chow groups NEWLINE\[NEWLINE\sigma^r_{\overline s}: \mathrm{CH}^r(X_\eta)[l^\infty]\to \mathrm{CH}^r(X_{\overline s})[l^\infty]NEWLINE\]NEWLINE which is known to be injective for \(r= 1,2,n\). For \(2< r< n\), \textit{C. Schoen} [Math. Z. 252, No. 1, 11--17 (2006; Zbl 1085.14010)], has given are examples, with \(l\) prime to the characteristic of \(\mathbb{F}\), such that \(\sigma^r_s\) is not injective. In these examples \(l\in\{5,7,11,13,17\}\).NEWLINENEWLINE In this paper, the authors prove the following result:NEWLINENEWLINE Theorem 1. For all \(r< n\) there are examples where the map a is not injective for all but finitely many primes \(l\) prime to \(\text{char}(\mathbb{F})\).NEWLINENEWLINE These examples cover the case of mixed characteristic as well as the case of equal characteristic \(0\).NEWLINENEWLINEFor the entire collection see [Zbl 1242.14001].