- Let Xbe a scheme of finite type over a perfect field . In this paper we study the relation between two objects associated to X: the Grothendieck residue complex and the Beilinson adeles complex . The latter is a differential graded algebra (DGA). Our first main result (Theorem 0.1) is that is a right differential graded (DG) module over . We give an application to de Rham theory. Define graded sheaves ≔ ℋom X ( , ) and . It is known that is a DGA. Our second main result (Theorem 0.2) is that is a right DG -module. When Xis smooth then calculates de Rham homology, calculates cohomology, and the action induces the cap product. We extend these constructions to singular schemes in characteristic 0 using smooth formal embeddings. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.