nLab unramified morphism

Idea

General

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.

So “unramified” means “not branching”. In the context of differential geometry unramified maps correspond to immersions.

A weaker (infinitesimal) version is the notion of formally unramified morphism.

Historical remarks

The basic picture is one from 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 occurred again.

Definition

Definition

A morphism $f \colon X\longrightarrow Y$ of schemes is unramified if it is locally of finite presentation, and if for every point $x\in X$ we have $\mathfrak{m}_{f(x)}\mathcal{O}_{X,x}=\mathfrak{m}_x$ and the induced morphism of residue fields

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

is a finite and separable extension of fields.

Definition

A morphism $f:X\to Y$ of schemes is formally unramified if for every infinitesimal thickening $T\to T'$ of schemes over $Y$, the canonical morphism of sheaves of sets over $T$

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

to

$$U\mapsto (Sch_{/Y})(U',X)$$

is injective, where $U' = U$ as the open set, but as a scheme it is the open subscheme of the thickening $T'$.

Remark

In this condition, it is sufficient to consider the thickenings 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 $T'\to X$ the extension is unique.

Properties and usage

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

An etale morphism of schemes is, by one of the equivalent definitions, a morphism that 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.

Related concepts

• decidable object

• ramification of ideals

• formally unramified morphism

Literature

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

• EGA IV

• James Milne, p. 20 of Lectures on Étale Cohomology