Local Beilinson–Tate operators

Yekutieli, Amnon

doi:10.2140/ant.2015.9.173

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Local Beilinson–Tate operators

Amnon Yekutieli

Vol. 9 (2015), No. 1, 173–224

DOI: 10.2140/ant.2015.9.173

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 $E (K)$ 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 $E (K)$ . In particular we show that $E (K)$ 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.

Keywords

topological local fields, residues, Tate residue, Beilinson adeles

Mathematical Subject Classification 2010

Primary: 12J25

Secondary: 32A27, 13J05, 11R56, 46A13, 46H30

Milestones

Received: 25 June 2014

Revised: 20 October 2014

Accepted: 25 December 2014

Published: 18 February 2015

Authors

Amnon Yekutieli
Department of Mathematics Ben Gurion University Be’er Sheva 84105 62381 Beersheva Israel

Vol. 9, No. 1, 2015