Unramified morphism of algebraic schemes is a geometric generalization of a notion of an unramified field extension. The notion of ramification there, in turn is motivated by the branching phenomena in number fields, which in the Riemann surface picture involve branchings similar to the branching involved in Riemann surfaces over complex numbers.
A weaker (infinitesimal) version is the notion of formally unramified morphism.
The basic picture is one from the 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 occured again.
of residue fields is a finite and separable extension of fields.
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 thickennings 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.