Residues and Differential Operators on Schemes

TL;DR

This paper uses Beilinson Completion Algebras to explicitly construct the Grothendieck residue complex on schemes, revealing new properties and interactions with differential operators, with applications to algebraic structures and homology.

Contribution

It provides an explicit construction of the residue complex using BCAs, uncovering new properties and applications in algebraic geometry and D-modules.

Findings

New properties of the residue complex revealed

Analysis of De Rham homology spectral sequence

Algebraic description of intersection cohomology D-module

Abstract

Beilinson Completion Algebras (BCAs) are generalizations of complete local rings, and have a rich algebraic-analytic structure. These algebras were introduced in my paper "Traces and Differential Operators over Beilinson Completion Algebras", Compositio Math. 99 (1995). In the present paper BCAs are used to give an explicit construction of the Grothendieck residue complex on an algebraic scheme. This construction reveals new properties of the residue complex, and in particular its interaction with differential operators. Applications include: (i) results on the algebraic structure of rings of differential operators; (ii) an analysis of the niveau spectral sequence of De Rham homology; (iii) a proof of the contravariance of De Rham homology w.r.t. etale morphisms; (iv) an algebraic description of the intersection cohomology D-module of a curve.