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.