0. Introduction 305 1. Review of Beilinson completion algebras 307 2. Construction of the residue complex;: Jc 310 3. Duality for proper morphisms 316 4. Duals of differential operators 319 5. The de Rham residue complex 323 6. de Rham homology and theniveau spectral sequence 325 7. The intersection cohomology-module of a curve 332

0. Introduction. Suppose X is a finite-type scheme over a field k, with struc-tural morphism rr. Consider the twisted inverse image functor re!" Dc+(k) Dc+(X) of Grothendieck duality theory (see [Hall). The residue complex:’Jc is defined to be the Cousin complex of zr! k. It is a bounded complex of quasi-coherent 60x-modules, possessing remarkable functorial properties. In this paper we provide an explicit construction of Jc. This construction reveals some new properties of o: jc and also has applications in other areas of algebraic geometry. Grothendieck duality, as developed by Hartshorne in [Hall, is an abstract theory, statedin the language ofderived categories. Even though this abstraction is suitable formany important applications, one often wants moreexplicit infor-mation. Thus, a significant amount of work was directed at finding a presenta-tion of duality in terms of differential forms and residues. Mostly, the focus was on thedualizing sheaf COx, in various circumstances. The structure of COx as a coherent (gx-module and its variance properties are thoroughly understood by now, thanks to an extended effort including [K1],[KW],[Li],[HK1],[HK2],[LS], and [HS]. Regarding an explicit presentation of the full duality theory of dualizing complexes, there have been some advances in recent years, notably in the papers [Yell,[SY],[Hu],[Hg], and [Sa]. In this paper we give a totally new construction of the residue complex when k isa perfect field of any characteristic and X is any finite-type k-scheme. The main idea is the use of Beilinson completion algebras (BCAs), introduced in