Hochschild (Co-)Homology of Schemes with Tilting Object

Given a $k$–scheme $X$ that admits a tilting object $T$, we prove that the Hochschild (co-)homology of $X$ is isomorphic to that of $A=En{d}_X(T)$. We treat more generally the relative case when $X$ is flat over an affine scheme $Y=\text{Spec}R$ and the tilting object satisfies an appropriate Tor-independence condition over $R$. Among applications, Hochschild homology of $X$ over $Y$ is seen to vanish in negative degrees, smoothness of $X$ over $Y$ is shown to be equivalent to that of $A$ over $R$, and for $X$ a smooth projective scheme we obtain that Hochschild homology is concentrated in degree zero. Using the Hodge decomposition \cite{BFl2} of Hochschild homology in characteristic zero, for $X$ smooth over $Y$ the Hodge groups $H^q(X,{\mathrm{\Omega }}_{X\mathrm{/}Y}^p)$ vanish for $p < q$,whileintheabsolutecasetheyevenvanishfor$p\neq q$. We illustrate the results for crepant resolutions of quotient singularities, in particular for the total space of the canonical bundle on projective space.