nLab
geometric fibre

Contents

Idea

In algebraic geometry a geometric fiber is a fiber of a bundle over a geometric point.

For a bundle p:EXp: E\to X of topological spaces, the fibre over a point, xXx\in X may be thought of as the preimage E x=p 1(x)E_x = p^{-1}(x) equipped with its subspace topology. More abstract, this is the pullback x× XE{x} \times_X E of pp along the map sending a singleton space ** to xXx \in X, the object which is universal with the property of making this diagram commute:

E x E p * x X. \array{ E_x &\to& E \\ \downarrow && \downarrow^{\mathrlap{p}} \\ {*} &\stackrel{x}{\to}& X } \,.

Adapting this to the algebraic context we get the following definition.

Definition

Let XX be a scheme over some base field kk. Fix an algebraic closure k¯\overline{k} of kk and let ξ¯:Spec(k¯)X\overline{\xi} : Spec(\overline{k}) \to X be a geometric point in XX.

Definition

For a morphism, p:EXp: E\to X, the geometric fibre over the geometric point xx is the pullback

E xE× XSpec(k¯). E_x \coloneqq E \times_X Spec (\overline{k}) \,.
Remark

In EGA, the case is also considered in which the field is replaced by a local ring, (see EGA p. 112), in which case the word ‘geometric’ is dropped. (This needs checking.)

Example and application

An important case is where pp is a finite étale cover, and then the geometric fibre is just a finite set.

References

  • EGA I 3.4.5, p. 112.

Revised on September 25, 2012 13:26:25 by Urs Schreiber (131.174.191.188)