Hochschild coniveau spectral sequence and the Beilinson residue

Oliver Braunling and Jesse Wolfson

Vol. 300 (2019), No. 2, 257–329

DOI: 10.2140/pjm.2019.300.257

Abstract

Higher-dimensional residues can be constructed either following Grothendieck–Hartshorne using local cohomology, or following Tate–Beilinson using Lie algebra homology. We show that there is a natural link: we develop the Hochschild analogue of the coniveau spectral sequence. The rows of our spectral sequence look a lot like the Cousin complexes in Hartshorne’s Residues and duality, which live in the framework of coherent cohomology. We prove that the complexes agree by an “HKR isomorphism with supports”. Using the close ties of Hochschild homology to Lie algebra homology, this yields a direct comparison.

Keywords

adeles, Tate residue, Beilinson residue, Tate central extension, residue symbol, Cousin complex, Tate object

Mathematical Subject Classification 2010

Primary: 19D55

Milestones

Received: 7 March 2018

Revised: 28 August 2018

Accepted: 29 August 2018

Published: 30 July 2019