Title: Traces and Differential Operators over Beilinson Completion Algebras

Publication status: appeared in Compositio Math. 99 (1995).

A Beilinson Completion Algebra (BCA) is a generalization of a complete semi-local ring. A BCA A over a perfect field k is a finite product of local BCAs. Algebraically, a local BCA is a complete local k-algebra whose  residue field is a high dimensional local field. In addition A has a topology, and we require that there is a surjection F((s_{1}, ..., s_{n}))[[t_{1}, ..., t_{m}]] --> A respecting all this structure, with F a finitely generated extension of k.

This is an abstraction of the algebras one gets by applying Beilinson completion to the sheaf  \cal{O}_{X} along a chain of points in X, when X is a finite type scheme over k.

There are two kinds of important maps between BCAs, called morphisms and intensifications. These again are extracted from geometric situations.

The main result is the existence of a dual module \cal{K}(A). This is an injective module, with topology, which varies functorially with morphisms and intensifications.

The theory of BCAs is the basic tool for the explicit construction of the residue complex in  Residues and differential operators on schemes.

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