Vol. 300, No. 2, 2019

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Hochschild coniveau spectral sequence and the Beilinson residue

Oliver Braunling and Jesse Wolfson

Vol. 300 (2019), No. 2, 257–329
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
Authors
Oliver Braunling
Freiburg Institute for Advanced Studies — FRIAS
Freiburg
Germany
Jesse Wolfson
Department of Mathematics
University of California, Irvine
Irvine, CA
United States