  A concrete lower-bound for the Hochschild cohomological dimension of a
commutative $k$-algebra, in terms of three other homological invariants is
obtained. This result is then used to show that most $k$-algebras fail to be
quasi-free, even if they are smooth. This result generalizes a result of
\cite{cuntz1995algebra} to the case where the base-ring is no longer $\cc$ but
can be any commutative ring with unity.
