Traces and Differential Operators over Beilinson Completion Algebras

TL;DR

This paper develops a duality theory for Beilinson completion algebras, linking dual modules and differential operators, with applications to constructing the Grothendieck residue complex on algebraic schemes.

Contribution

It introduces dual modules for BCAs, establishes their functorial properties, and shows that duality respects differential operators, advancing the theory of residues in algebraic geometry.

Findings

Dual modules K(A) exist for every BCA A.

Duality respects continuous differential operators.

Framework supports construction of Grothendieck residue complexes.

Abstract

A Beilinson completion algebra (BCA) A is a complete semilocal algebra over a perfect field k, whose residue fields are high dimensional local fields. In addition A is a semi-topological algebra. The completion of the structure sheaf of an algebraic k-variety along a saturated chain of points is the prototypical example of a BCA. We single out two kinds of homomorphisms between BCAs: morphisms and intensifications. The first kind includes residually finite local homomorphisms, whereas the second kind is a sort of localization. We prove that every BCA A has a dual module K(A), and these dual modules are contravariant w.r.t. morphisms and covariant w.r.t. intensifications. For any semi-topological A-module M we define its dual Dual_{A} M := Hom_{A}^{cont}(M, K(A)). This duality operation has the remarkable property of respecting differential operators: given a continuous DO D : M --> N, there is a dual DO Dual_{A}(D) : Dual_{A} N --> Dual_{A} M. The results above are used (in a subsequent paper) to construct the Grothendieck residue complex K_{X}^{.} on any finite type k-scheme X, and to derive many of its properties.