Complexes de chaînes étales et courbes de Deligne–Lusztig

In a first part, we lift the usual constructions of functors between derived categories of étale sheaves over schemes with a sheaf of algebras to pure derived categories. For varieties with finite group actions, we recover, in a more functorial way, Rickardʹs construction. We apply this to the case of Deligne–Lusztig varieties and show that Brouéʹs conjecture holds for curves. The additional ingredient we need is obtained from an easy property of the cohomology of etale covers of the affine line.