Let X be a smooth separated scheme over a noetherian base ring K. The decomposition we are interested in is an isomorphism R ℋomoX×KX(OX, OX) ≅ ⊕q (Λoxq τX/K)[-q] in the derived category D(Mod Ox × K x). Here τX/K is the relative tangent sheaf. Upon passing to cohomology sheaves such an isomorphism recovers the Hochschild-Kostant-Rosenberg Theorem. If K has characteristic 0 there is a decomposition that relies on a particular homomorphism of complexes from poly-tangents to continuous Hochschild cochains. We discuss sheaves of continuous Hochschild cochains on schemes and show why this approach to decomposition fails in positive characteristics. Our main result is a proof of the decomposition valid for any Gorenstein noetherian ring K of finite Krull dimension, regardless of characteristic. The proof is based on properties of minimal injective resolutions.
|Number of pages||31|
|Journal||Canadian Journal of Mathematics|
|State||Published - 1 Aug 2002|
- Derived categories
- Gorenstein rings
- Hochschild cohomology