nLab geometric fibre

Idea

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

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

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

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

Definition

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

Definition

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

$$E_x \coloneqq E \times_X Spec (\overline{k}) \,.$$

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 $p$ is a finite étale cover, and then the geometric fibre is just a finite set.

References

• EGA I 3.4.5, p. 112.