The Action of Adeles on the Residue Complex

Abstract

Let X be a scheme of finite type over a perfect field k. In this paper we study the relation between two important objects associated to X: the Grothendieck residue complex and the Beilinson adeles complex. It is known that the complex of adeles is a DGA (differential graded algebra). Our first main result is that the residue complex is a right DG module over the adeles complex. The second main result is that the de Rham residue complex is a DG module over the de Rham adeles complex. This action gives rise to the cap product in de Rham (co)homology.