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
-dimensional topological
local field
, and define
a ring of operators
that acts on
,
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
. In particular we
show that
has
an
-dimensionalcubical 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.
Keywords
topological local fields, residues, Tate residue, Beilinson
adeles