1-motivic sheaves and the Albanese functor (Q1004484)

Let \(k\) be a perfect field, \(\tau\) a Grothendieck topology on the category of smooth \(k\)-schemes, \(Shv^{\tau}_{tr}(k)\) -- the category of \(\tau\)-sheaves with transfers on \(Sm/k.\) Let further \(DM_{eff}^{\tau}(k)\) be the full subcategory of the derived category \(D(Shv^{\tau}_{tr}(k))\) with objects the \({\mathbb A}^{1}\)-local complexes [cf. \textit{V. Voevodsky}, Triangulated categories of motives over a field. Cycles, transfers, and motivic homology theories. Princeton, NJ: Princeton University Press. Ann. Math. Stud. 143, 188--238 (2000; Zbl 1019.14009); \textit{C. Mazza, V. Voevodsky, C. Weibel}, Lecture notes on motivic cohomology. Clay Mathematics Monographs 2. (Providence), RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute. (2006; Zbl 1115.14010)]. To a smooth \(k\)-scheme \(X\) one can associate the representable sheaf \({\mathbb Z}_{tr}(X)\) and the homological motive \(M(X)\in DM_{eff}^{\tau}(k)\) -the \({\mathbb A}^{1}\)-localization of \({\mathbb Z}_{tr}(X).\) The smallest triangulated subcategory of \(DM_{eff}^{\tau}(k)\) that contains \(M(X)\) for \(X\in Sm/k\) and stable with respect to direct summands is called the category of geometric motives, and it is denoted \(DM_{eff,\, gm}^{\tau}(k).\) Under some assumptions, the canonical \(t\)-structure on \(D(Shv^{\tau}_{tr}(k))\) can be restricted to \(DM_{eff}^{\tau}(k)\) whose heart is the abelian category \(HI_{tr}^{\tau}(k)\subset Shv^{\tau}_{tr}(k)\) of the homotopy invariant \(\tau\)-sheaves with transfers. Denote by \(DM_{\leq n}^{\tau}(k)\subset DM_{eff}^{\tau}(k)\) the category of \(n\)-motivic complexes defined as the triangulated subcategory generated by \(M(X)\) for \(X\) of dimension \(\leq n\) and closed with respect to taking direct sums. Similar definitions can be adopted for appropriate subcategories. The authors define functors \(L{\pi}_{0}: DM_{eff}^{\tau}(k)\rightarrow D(HI_{\leq 0}(k))\) and \(LAlb :DM_{eff}^{\tau}(k)\rightarrow D(HI_{\leq 1}(k))\). These functors are used to construct higher Néron-Severi groups and higher Albanese sheaves. The functor \(LAlb\) is an extension of the one introduced by \textit{L. Barbieri-Viale} and \textit{B. Kahn} [On the derived category of \(1\)-motives, I. \url{arXiv:0706.1498}; Prépublication Mathématique de l'IHÉS (M/07/22)].