A relative homology criterion of smoothness

We investigate the relationship between smoothness and the relative global dimension of a ring extension.

We prove that a smooth commutative algebra $A$ over $B$ has finite relative global dimension $\text{gdim}(A,B)$.

Conversely, under a mild condition on $B$, the finiteness of $\text{gdim}(A,B)$ implies that the map $B \to A$ is smooth.

We also relate the relative global dimension to the usual global dimension of the fibers of $B \to A$, and establish a formula for the relative global dimension of tensor products of extensions.

Finally, we present examples and an alternative characterisation of smoothness in terms of relative Hochschild homology.