A Lower-Bound on the Hochschild Cohomological Dimension

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 $\text{cc}$ but can be any commutative ring with unity.