A relative homology criterion of smoothness

Abstract

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.

Final version, to be published in Proceedings of the American Mathematical Society. Some corrections following the referee report