nLab
unramified morphism

Idea

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.

Historical remarks

The basic picture is one from the Riemann surfaces: the power $z\mapsto z^n$ has a branching point around $z=0$. 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.

Definition

A morphism $f:X\to Y$ of schemes is unramified if it is locally of finite presentation, and if for every point $y\in Y$ the induced morphism

$${𝒪}_{X,f(y)}/{𝔪}_{f(y)}\to {𝒪}_{Y,y}/{𝔪}_y$$\mathcal{O}_{X,f(y)}/\mathfrak{m}_{f(y)}\to\mathcal{O}_{Y,y}/\mathfrak{m}_y

of residue fields is a finite and separable extension of fields.

A morphism $f:X\to Y$ of schemes is formally unramified if for every infinitesimal thickenning $T\to T\prime$ of schemes over $Y$, the canonical morphism of sheaves of sets over $T$

$$U\mapsto (\mathrm{Scheme}/Y)(U,X)$$U\mapsto (Scheme/Y)(U,X)

to

$$U\mapsto (\mathrm{Scheme}/Y)(U\prime ,X)$$U\mapsto (Scheme/Y)(U',X)

is injective, where $U\prime =U$ as the open set, but as a scheme it is the open subscheme of the thickening $T\prime$. In this condition, it is sufficient to consider the thickennings of affine $Y$-schemes. Thus $f$ is formally unramified if for each morphism of $Y$-schemes $T\to X$ which has an extension to a morphism $T\prime \to X$ the extension is unique.

Properties and usage

The notion of unramified morphism is stable under base change and composition.

Etale morphism is by (one of the equivalent definitions) a morphism of schemes which is flat and unramified.

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 $X\to X{\times }_{Y}X$ is an open immersion.

Characterization: a morphism is formally unramified iff the module ${\Omega }_{Y/X}$ of relative Kahler differentials is zero.

Literature

• Ogus, Unramified morphisms, pdf, notes from a course on algebraic geometry at Berkeley

• EGA IV