Hochschild coniveau spectral sequence and the Beilinson residue

TL;DR

This paper constructs a Hochschild analogue of the coniveau spectral sequence and Gersten complex, connecting Hochschild homology with residue maps in Lie algebra homology, differing from K-theory due to lack of devissage and A^1-invariance.

Contribution

It introduces a Hochschild coniveau spectral sequence and Gersten complex, establishing their relation to Cousin complexes and residue maps in Lie homology.

Findings

Spectral sequence rows resemble Cousin complexes in coherent cohomology

HKR isomorphism with supports confirms the complexes agree

Residue maps in Lie homology align with Tate-Beilinson residues

Abstract

We develop the Hochschild analogue of the coniveau spectral sequence and the Gersten complex. Since Hochschild homology does not have devissage or A^1-invariance, this is a little different from the K-theory story. In fact, the rows of our spectral sequence look a lot like the Cousin complexes in Hartshorne's 1966 "Residues & Duality". Note that these are for coherent cohomology. We prove that they agree by an 'HKR isomorphism with supports'. Using the close ties of Hochschild homology to Lie algebra homology, this gives residue maps in Lie homology, which we show to agree with those \`a la Tate-Beilinson.