Local Beilinson–Tate operators

Abstract

In 1968 Tate introduced a new approach to residues on algebraic curves, based on a certain ring of operators that acts on the completion at a point of the function field of the curve. This approach was generalized to higher-dimensional algebraic varieties by Beilinson in 1980. However, Beilinson’s paper had very few details, and his operator-theoretic construction remained cryptic for many years. Currently there is a renewed interest in the Beilinson–Tate approach to residues in higher dimensions.

Our paper presents a variant of Beilinson’s operator-theoretic construction. We consider an $n$-dimensional topological local field $K$, and define a ring of operators $E\left(K\right)$ that acts on $K$, which we call the ring of local Beilinson–Tate operators. Our definition is of an analytic nature (as opposed to the original geometric definition of Beilinson). We study various properties of the ring $E\left(K\right)$. In particular we show that $E\left(K\right)$ has an $n$-dimensional cubical decomposition, and this gives rise to a residue functional in the style of Beilinson and Tate. Presumably this residue functional coincides with the residue functional that we had constructed in 1992; but we leave this as a conjecture.