nLab relative dimension

For morphisms of schemes

Definition

Let $f : X \to Y$ be a morphism locally of finite type between two schemes $X$ and $Y$. The relative dimension of $f$ at a point $y \in Y$ is the dimension of the fiber $f^{-1}(y)$. If all the nonempty fibers $f^{-1}(y)$ ($y \in Y$) are purely? of the same dimension $n$, then one says that $f$ is of relative dimension $n$.

References

• Stacks project, 02NI

Though the definition does not seem to appear in EGA, there are some relevant propositions in (EGA, IV_2, 5.6):

• Alexandre Grothendieck, Éléments de géométrie algébrique, IV. Étude locale des schémas et des morphismes de schémas, Seconde partie. Publications Mathématiques de l’IHÉS, 24 (1965). See (5.6.5 - 5.6.8).