nLab
unramified morphism

Contents

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 zz nz\mapsto z^n has a branching point around z=0z=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:XYf \colon X\longrightarrow Y of schemes is unramified if it is locally of finite presentation, and if for every point yYy\in Y the induced morphism of residue fields

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

is a finite and separable extension of fields.

Definition

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

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

to

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

is injective, where U=UU' = U as the open set, but as a scheme it is the open subscheme of the thickening TT'.

Remark

In this condition, it is sufficient to consider the thickenings of affine YY-schemes. Thus ff is formally unramified if for each morphism of YY-schemes TXT\to X which has an extension to a morphism TXT'\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 XX× YXX\to X\times_Y X is an open immersion.

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

Literature

Revised on December 9, 2013 06:06:39 by David Corfield (46.208.144.193)