The action of Adeles on the residue complex

Let X be a scheme of finite type over a perfect field double-struck k sign. In this paper we study the relation between two objects associated to X: the Grothendieck residue complex. script K sign_X^. and the Beilinson adeles complex double-struck A sign_-red ^.((script O sign_X). The latter is a differential graded algebra (DGA). Our first main result (Theorem 0.1) is that script K sign _X^. is a right differential graded (DG) module over double-struck A sign_-red^.(script O sign_X). We give an application to de Rham theory. Define graded sheaves ℱ _X^.:= ℋom_{script O sign X} (Ω _{X/double-struck k sign}^., script K sign_X ^.) and script A sign_X^.:= double-struck A sign_-red^.(script O sign_X) ⊗_{script O sign X} Ω_{X/double-struck k sign}^.. It is known that script A sign_X^. is a DGA. Our second main result (Theorem 0.2) is that ℱ_X^. is a right DG script A sign_X^.-module. When X is smooth then ℱ _X^. calculates de Rham homology, script A sign _X^. calculates cohomology, and the action induces the cap product. We extend these constructions to singular schemes in characteristic 0 using smooth formal embeddings.