Hochschild (co-)homology of schemes with tilting object

Source abstract

Given a k k –scheme X X that admits a tilting object T T , we prove that the Hochschild (co-)homology of X X is isomorphic to that of A = End X ⁡ ( T ) A=\operatorname {End}_{X}(T) . We treat more generally the relative case when X X is flat over an affine scheme Y = Spec ⁡ R Y=\operatorname {Spec} R , and the tilting object satisfies an appropriate Tor-independence condition over R R . Among applications, Hochschild homology of X X over Y Y is seen to vanish in negative degrees, smoothness of X X over Y Y is shown to be equivalent to that of A A over R R , and for X X a smooth projective scheme we obtain that Hochschild homology is concentrated in degree zero. Using the Hodge decomposition of Hochschild homology in characteristic zero, for X X smooth over Y Y the Hodge groups H q ( X , Ω X / Y p ) H^{q}(X,\Omega _{X/Y}^{p}) vanish for p > q p > q , while in the absolute case they even vanish for p ≠ q p\neq q . We illustrate the results for crepant resolutions of quotient singularities, in particular for the total space of the canonical bundle on projective space.