  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 $A= End_{X}(T)$. We
treat more generally the relative case when $X$ is flat over an affine scheme
$Y=\Spec R$ 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 $H^{q}(X,\Omega_{X/Y}^{p})$ vanish for $p
< q$, while in the absolute case they even vanish for $p\neq q$. We illustrate
the results for crepant resolutions of quotient singularities, in particular
for the total space of the canonical bundle on projective space.
