Smooth morphism - Knowlpedia

Definition

A morphism of schemes $f:X\to S$ is smooth if it is locally of finite presentation, flat, and every geometric fiber $X_{\overline s}$ is regular.

If every nonempty geometric fiber has pure dimension $n$ , then $f$ is smooth of relative dimension $n$ . Smoothness is preserved by base change and composition and is local on both source and target for the étale topology.

Infinitesimal criterion

For morphisms locally of finite presentation, smoothness is equivalent to formal smoothness: maps from a nilpotent closed subscheme lift locally across the thickening. This is the scheme-theoretic analogue of having no infinitesimal singularities in the fibers.

References

The Stacks Project Authors, “Smooth morphisms,” Tag 01V4.