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We develop the Hochschild analogue of the coniveau spectral sequence and theGersten complex. Since Hochschild homology does not have devissage orA^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 inHartshorne's 1966 "Residues & Duality". Note that these are for coherentcohomology. We prove that they agree by an 'HKR isomorphism with supports'.  Using the close ties of Hochschild homology to Lie algebra homology, thisgives residue maps in Lie homology, which we show to agree with those \`a laTate-Beilinson.

Hochschild coniveau spectral sequence and the Beilinson residue