A relative homology criteria of smoothness

TL;DR

The paper develops a relative homological framework to characterize smoothness of a $B$-algebra $A$ via the relative global dimension $\operatorname{gldim}(A,B)$. It proves that smoothness implies finite $\operatorname{gldim}(A,B)$ and that, under mild conditions on $B$ (in particular with a perfect base field and $A$ flat of finite type over $B$), finite $\operatorname{gldim}(A,B)$ implies smoothness; it also establishes fiberwise and tensor-product formulas for $\operatorname{gldim}(A,B)$ and connects the criterion to relative Hochschild homology, offering an equivalent condition in terms of $\operatorname{HH}_j(A|B)$ vanishing in high degrees. The results provide a practical tool to assess smoothness via homological dimensions and illustrate the theory with examples, including étale cases and counterexamples where $A\otimes_B A$ is not projective. This advances the understanding of how relative homological dimensions govern geometric smoothness and offers a relative analogue of classical dimension criteria like Auslander–Buchsbaum and Serre.

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 characterization of smoothness in terms of relative Hochschild homology.

Theorems & Definitions (34)

• Remark 2.1
• Remark 2.2
• Example 2.4
• Remark 2.5
• Lemma 2.6
• proof
• Remark 2.7
• Remark 3.1
• Remark 3.2
• Proposition 3.3