The notion of unramified morphism of algebraic schemes is a geometric generalization of the notion of an unramified field extension. The notion of ramification there is in turn motivated by branching phenomena in number fields, which involve branchings similar to those occurring in Riemann surfaces over the complex numbers.
A weaker (infinitesimal) version is the notion of formally unramified morphism.
The basic picture is one from Riemann surfaces: the power has a branching point around . Dedekind and Weber in 19th century considered more generally algebraic curves over more general fields, and proposed a generalization of a Riemann surface picture by considering valuations and in this analysis the phenomenon of branching occurred again.
is injective, where as the open set, but as a scheme it is the open subscheme of the thickening .
In this condition, it is sufficient to consider the thickenings of affine -schemes. Thus is formally unramified if for each morphism of -schemes which has an extension to a morphism the extension is unique.
The notion of unramified morphism is stable under base change and composition.
Every open immersion of schemes is formally etale hence a fortiori formally unramified. A morphism which is locally of finite type is unramified iff the diagonal morphism is an open immersion.
Characterization: a morphism is formally unramified iff the module of relative Kahler differentials is zero.
Ogus, Unramified morphisms, pdf, notes from a course on algebraic geometry at Berkeley