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arXiv · 2404.08534

A relative homology criterion of smoothness

Kostiantyn Iusenko·Eduardo do Nascimento Marcos·Victor do Valle Pretti
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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.

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Kostiantyn Iusenko, Eduardo do Nascimento Marcos, Victor do Valle Pretti. 2024-04-12. A relative homology criterion of smoothness. https://arxiv.org/abs/2404.08534

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