Unramified morphism

A morphism of schemes $f\text{\hspace{0.17em}⁣}:X\to S$ is unramified if it is locally of finite type (Locally finite type and finite type morphisms) and formally unramified (Formally unramified morphism). By Formal unramifiedness iff Omega vanishes this is equivalent to asking that $f$ be locally of finite type and that ${\mathrm{\Omega }}_{X\mathrm{/}S}=0$ (Sheaf of relative Kähler differentials); either formulation may be used.

Convention: finite type versus finite presentation. The convention here is the one for which "unramified" requires only locally of finite type, in accordance with the Stacks Project. Some authors (and the older terminology of EGA) use the stronger convention, asking for local finite presentation instead of local finite type; a morphism with that stronger property is sometimes called G-unramified. The two notions coincide when the source and target are locally Noetherian, but not in general. This page uses the finite-type convention throughout; where a later page needs the finite-presentation notion, it says so explicitly.