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.