Nori's construction and the second Abel-Jacobi map (Q2474602)

Let \(X\) be an affine scheme of finite type over \(k\), a subfield of \(\mathbb{C}\). A sheaf \(F\) on \(X\) is said to be weakly constructible if \(X\) is the disjoint union of finite collection of locally closed subschemes \(Y_i\) defined over \(k\) such that the restrictions \(F|_{Y_i}\) are locally constant. \textit{M. V. Nori} [Constructive sheaves, Tata Inst. Fundam. Res. 16, 471--491 (2002; Zbl 1059.14025)] showed that there is a Zariski open \(U\) in \(X\) such that \(\dim(X- U)< \dim X\) and \(H^q(X(\mathbb{C}), j_!j^*F)= 0\) for \(q\neq\dim X\), where \(j: U\to X\) denotes the inclusion map. Based on this result, he also showed that affine \(k\)-varieties have a kind of ``cellular decomposition.'' In this paper the author gives an exposition of the outcome if one applies Nori's construction to étale cohomology, and shows that \(Rf_*\mathbb{Z}_\ell(a)\) for a variety \(X@>f>>\text{Spec\,}k\) is a complex each component of which comes from a mixed motive.