#### Vol. 9, No. 1, 2015

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors' Addresses Editors' Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Ethics Statement Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Other MSP Journals
Local Beilinson–Tate operators

### Amnon Yekutieli

Vol. 9 (2015), No. 1, 173–224
##### 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.

Warning: We have not been able to recognize your IP address 54.226.23.160 as that of a subscriber to this journal. Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form.

Or, visit our subscription page for instructions on purchasing a subscription. You may also contact us at contact@msp.org or by using our contact form.

Or, you may purchase this single article for USD 40.00:

##### 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