A relative homology criteria of smoothness

We investigate the relationship between smoothness and the relative global dimension. We prove that a smooth ring map $B\to A$ between commutative rings implies the finiteness of the relative global dimension $\operatorname{gldim}(A,B)$. Conversely, we identify a sufficient condition on $B$ such that the finiteness of $\operatorname{gldim}(A,B)$ implies the smoothness of the map $B\to A$.