Betti realization of Voevodsky motives (Q952125)

Let \(k\) be a subfield of \(\mathbb{C}\) and let \(DM^-(k)\) be the triangulated category of motivic complexes constructed by Voevodsky. This paper deals with the construction of a realization functor from \(DM^-(k)\) to Betti cohomology. More precisely: there exists a functor \[ \tau_k: DM^-(k)\to D({\mathcal A}b) \] to the derived category of abelian groups, such that the cohomological functor \[ H_B(-,\mathbb{Z}):DM^-(k)\to Ab \] where \(H_B(M,\mathbb{Z})=\Hom_{D({\mathcal A}b)}(\tau_k(M), \mathbb{Z})\), generalizes the functor of singular cohomology of \(X(\mathbb{C})\) for a smooth and quasi-projective scheme \(X\). Moreover, by taking \(\mathbb{Q}\)-coefficients, the functor \(H_B(-,\mathbb{Z}\otimes\mathbb{Q}\), when restricted to the subcategory \(DM_{gm}(k)\) of geometrical motives in \(DM^-(k)\), coincides with the component corresponding to singular cohomology of the realization functor constructed by \textit{A. Huber} [J. Algebr. Geom. 9, No. 4 (2000; Zbl 0994.14014)]. The construction given here is based on the results of Suslin and Voevodky, showing that the sheaves with transfers \(\mathbb{Z}_{tr}(X)\) are represented, in the category \(Sch(k)\) of schemes of finite type over \(k\), by the group associated to the monoid of symmetric powers of \(X\). Then the sheaf obtained by extension to the strong topology of \(X(\mathbb{C})\) may be used to compute the singular homology of \(X(\mathbb{C})\).