Remarks on 1-motivic sheaves. (Q2874242)

Let \(k\) be a field of characteristic \(0\). Barbieri-Viale, Rosenschon, and Saito [\textit{L. Barbieri-Viale} et al., Ann. Math. (2) 158, No. 2, 593--633 (2003; Zbl 1124.14014)] introduced the category \({}^t\mathcal{M}_1\) of \(1\)-motives with torsion. The category \({}^t\mathcal{M}_1\) is an abelian category, and, when \(k=\mathbb{C}\), it is equivalent to the category of Mixed Hodge Structures of level \(\leq 1\). On the other hand, \textit{B. Kahn} and \textit{L. Barbieri-Viale} [``On the derived category of 1-motives'', preprint, \url{arXiv:1009.1900}]. have introduced the abelian category \(\mathrm{Shv}_1\) of \(1\)-motivic sheaves. The bounded derived categories \(D^b({}^t\mathcal{M}_1)\) and \(D^b(\mathrm{Shv}_1)\) are both equivalent to the full and thick subcategory of Voevodsky's triangulated category of motives \(\mathrm{DM}_{\mathrm{gm}}^{\mathrm{eff}}(k)\) generated by motives of smooth curves.NEWLINENEWLINEIn this paper, the author generalizes the constructions of L. Barbieri-Viale, A. Rosenschon, and M. Saito [loc.cit.] and B. Kahn and L. Barbieri-Viale [loc. cit.] to perfect fields. The author defines, on the one hand, a category of \(1\)-motives, still denoted \({}^t\mathcal{M}_1\), that is abelian, and on the other hand, a category of \(1\)-motivic sheaves \(\mathrm{Shv}_1^{\mathrm{fppf}}\) that is abelian, by using the fppf topology. The main result is then that when \(k\) is perfect but not algebraic over a finite field the bounded derived categories \(D^b({}^t\mathcal{M}_1)\) and \(D^b(\mathrm{Shv}^{\mathrm{fppf}}_1)\) are equivalent.