Finite locally free morphism

Definition

A morphism of schemes $f:X\to S$ is finite locally free if every point of $S$ has an affine neighborhood $U=\operatorname{Spec}R$ for which

f^{-1}(U)=\operatorname{Spec}A

and $A$ is a finite free $R$ -module. Equivalently, $f$ is finite, flat, and locally of finite presentation.

Rank and base change

The module rank defines a locally constant function on $S$ . If the rank is the constant $d$ , one says that $f$ has degree $d$ . Finite locally free morphisms are preserved by base change.

For a finite field extension $L/k$ , the map $\operatorname{Spec}L\to\operatorname{Spec}k$ is finite locally free of rank $[L:k]$ . In particular, $\operatorname{Spec}\mathbb C\to\operatorname{Spec}\mathbb R$ has rank $2$ ; this is the base morphism used in Weil restriction from $\mathbb C$ to $\mathbb R$ .

References

The Stacks Project Authors, Morphisms of Schemes, Lemma 29.48.2 and Section 29.48, “Finite locally free morphisms.” Stacks Project.
Alexander Grothendieck and Jean Dieudonné, Éléments de géométrie algébrique IV, §6.1.