nLab
fiber

Contents

Idea

The fiber of a morphism or bundle f:EBf : E \to B over a point of BB is the collection of elements of EE that are mapped by ff to this point.

Definition

Lemma

For f:ABf : A \to B a morphism in a category and BB equipped with the structure of a pointed object pt:*Bpt : * \to B, the fiber of ff is the fiber product of ff with pp, hence the pullback

A× B* * A f B. \array{ A \times_B * &\to& * \\ \downarrow && \downarrow \\ A &\stackrel{f}{\to}& B } \,.

Examples

Bundles

Given a bundle p:EBp: E \to B and a global element x:1Bx: 1 \to B, the fibre or fiber E xE_x of (E,p)(E,p) over xx is the pullback

E x E p 1 x B \array { E_x & \to & E \\ \downarrow & & \downarrow_p \\ 1 & \stackrel{x}\to & B }

if it exists.

In a fiber bundle, all fibres are isomorphic to some standard fibre FF in a coherent way.

Kernels

In an additive category fibers over the zero object are called kernels.

Fibers of a sheaf of modules

The fiber of a sheaf \mathcal{E} of 𝒪\mathcal{O}-modules over a locally ringed space (X,𝒪)(X,\mathcal{O}) at a point xXx \in X is defined as the vector space (x) x 𝒪 xk(x)\mathcal{E}(x) \coloneqq \mathcal{E}_x \otimes_{\mathcal{O}_x} k(x) over the residue field k(x)k(x). If \mathcal{E} is quasicoherent, the associated vector bundle of the fiber is the pullback of the associated vector bundle of \mathcal{E}:

V((x))=SpecSym(x) Spec̲ XSym=V() Speck(x) X \array { V(\mathcal{E}(x)) = \mathrm{Spec} \mathrm{Sym} \mathcal{E}(x) & \to & \underline{Spec}_X \mathrm{Sym} \mathcal{E} = V(\mathcal{E}) \\ \downarrow & & \downarrow \\ \mathrm{Spec} k(x) & \to & X }

Revised on February 9, 2013 02:55:13 by Ingo Blechschmidt (79.219.164.73)