Geometric motives and the \(h\)-topology
Let \({\mathbf{Sch}}\) be the category of schemes of finite type over a field \(k\). Let further \({\mathbb Q}{\mathbf{Sch}}\) be the \({\mathbb Q}\)-linear hull of \({\mathbf{Sch}}.\) This means that morphisms are \({\mathbb Q}\)-linear combinations of morphisms of schemes. Denote by \(\overline{{\mathbb Q}{\mathbf{Sch}}}\) the closure of \({{\mathbb Q}{\mathbf{Sch}}}\) under countable direct sums and by \({\mathbf{K}}^{-}({\overline{{\mathbb Q}{\mathbf{Sch}}}})\) the homotopy category of bounded above complexes in \(\overline{{\mathbb Q}{\mathbf{Sch}}}.\) The category of effective quasigeometric motives \({\mathbf{DM}}^{\text{eff}}_{\overline{gm}, h}(\mathbb Q)\) is the localization of \({\mathbf{K}}^{-}({\overline{{\mathbb Q}{\mathbf{Sch}}}})\) with respect to the following complexes: - for a smooth \(X\) the \({\mathbb A}^{1}\)-complexes: \(X\times {\mathbb A}^{1} \rightarrow X\) - Čech nerves: \(\quad \cdots\rightarrow U{\times}_{X}U \rightarrow U \rightarrow X\) of \(h\)-coverings, \(U\)-smooth, \(X \in {\mathbf{Sch}}.\) The subcategory of bounded complexes in \({\mathbb Q}{\mathbf{Sch}}\) is the category of effective motives \({\mathbf{DM}}^{\text{eff}}_{{gm}, h}(\mathbb Q).\) The category of effective geometric motives with transfers is defined in the following way: Let \({\mathbf{DM}}^{\text{eff}}_{\overline{gm}}(\mathbb Q)\) be the localization of \({\mathbf{K}}^{-}({\overline{{\mathbb Q}{\mathbf{SmCor}}}})\) with respect to \(A^{1}\)-compexes and Mayer-Vietoris complexes for smooth \(U\) and \(V\): \(U\cap V \rightarrow U \sqcup V \rightarrow U \cup V .\) Here \({{\mathbb Q}{\mathbf{SmCor}}}\) is the category whose objects are smooth schemes and morphisms are \({\mathbb Q}\)-linear combinations of cycles \(W\subset X\times Y\) finite over \(X.\) The Voevodsky category of effective motives with transfers \({\mathbf{DM}}^{\text{eff}}_{{gm}}(\mathbb Q)\) is defined analogously via bounded complexes. The author proves that there is an equivalence of categories of effective geometric motives with and without transfers: \({\mathbf{DM}}^{\text{eff}}_{{gm}}(\mathbb Q)\cong {\mathbf{DM}}^{\text{eff}}_{{gm}, h}(\mathbb Q).\)
scientific article; zbMATH DE number 6116866