(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (198)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

sheaf of sections

(Definition)

Presheaf Definition

Consider a rank vector bundle $E\rightarrow M$ , whose typical fibre is defined with respect to a field . Let $\{U_\alpha\}$ constitute a cover for . Then, sections of the bundle over some $U\subset M$ are defined as continuous functions $U\rightarrow E$ , which commute with the natural projection map $\pi \!:\!E \rightarrow M$ ; $\pi \circ s = id_M$ . Denote the space of sections of the bundle over U to be $\Gamma(U,E)$ . The space of sections is a vector space over the field by defining addition and scalar multiplication pointwise: for $s,t \in \Gamma(U,E)$ , $p\in U$ and $s\in k$

$\displaystyle (s+t)(p) \equiv s(p) + t(p) \qquad \qquad (a \cdot s) (p) \equiv a \cdot s(p).$

Then, this forms a presheaf $\mathcal E$ , a functor from top to the category of vector spaces, with restriction maps the natural restriction of functions.

Sheaf Axioms

It is easy to see that it satisfies the sheaf axioms: for

open and $\{V_i\}$ a cover of

if $s\in \mathcal E(U)$ and $s\vert_{V_i} =0$ for all , then .
if $s_i\in \mathcal E(V_i)$ for all , such that for each with $V_i\cap V_j \ne \emptyset$ , $s_i\vert_{V_i\cap V_j} = s_j\vert_{V_i\cap V_j}$ , then there is an $s \in \mathcal E(U)$ with $s\vert_{V_i} = s_i$ for all .

The first follows from the fact that for any , there is always at least one element of $\mathcal E(U)$ , the zero section, and that the transition functions of the bundle are linear maps. The second follows by the construction of the bundle.

Sheafification

We may also see the vector bundle by applying associated sheaf construction to the presheaf $U \mapsto \Gamma(U,E)$ . First though, we show that the stalk of the sheaf $\mathcal E$ at a point is isomorphic to the fibre of the bundle at the point. Let be a germ at $p\in M$ $(p\in U \subset M)$ , and define a map $\psi \!:\!\mathcal E_p \rightarrow E_p$ by

$\displaystyle \psi:[s,U] \mapsto s_p.$

First, we show that the map is a vector space homomorphism. Consider two germs and in $\mathcal E_p$ . These map to and respectively. We add the germs by finding an open set $W\in U\cap V$ and adding the restrictions of the sections;

$\displaystyle [s,U] + [t,V] \equiv [s\vert _W + t\vert _W, W].$

Of course, $p\in W$ , so we have $\psi(s\vert _W + t\vert _W) = s_p + t_p$ , since the restriction maps are simply restriction of functions. Now, it is easy to show that $\psi$ is injective. Assume $\psi([t,V]) = \psi([s,U]) = s_p$ . Then

$\displaystyle \psi([t,V]) - \psi([s,U])$	$\displaystyle = s_p - s_p$
$\displaystyle \psi([t,V] - [s,U])$	$\displaystyle = 0$
$\displaystyle [t,V]$	$\displaystyle = [s,U]$

Now, we show that $\psi$ is surjective. For $s_p\in E_p$ , let $U\subset M$ open be isomorphic to some subset $U_\ensuremath{\mathbb{R}}$ of $\ensuremath{\mathbb{R}}^m$ . Then, $\Gamma(U,E)$ is the set of continuous maps $U\rightarrow V_E$ , where is the typical fibre of ;

$\displaystyle \Gamma(U,E) = \bigoplus_{i=1}^r \mathcal C_{U_\ensuremath{\mathbb{R}}}^\infty.$

Then let

be the constant function $s:U_\ensuremath{\mathbb{R}}\mapsto s_x$ , and we have constructed an isomorphism $\psi$ between $\mathcal E_p$ and

To construct the Étalé space, take the disjoint union of stalks, Spé $(\mathcal E) = \coprod_{p\in M} \mathcal E_p$ , and endow it with the following topology: the open sets shall be of the form

$\displaystyle U_s = \bigl\{s_p \vert s\in \Gamma(U,\mathcal E), p\in U\subset M\bigr\},$

collection of germs of sections at points in $U\subset M$ .

Then, the associated sheaf to $\mathcal E$ is the presheaf which assigns continuous maps $\Gamma(U,\ensuremath{ \text{Sp\'e}(\mathcal E)})$ to each open . These are maps where the preimage of is open. Clearly, this implies that $\Gamma(U,E) \subset \Gamma(U,\ensuremath{ \text{Sp\'e}(\mathcal E)})$ . To go the other way, note that open sets of $\ensuremath{ \text{Sp\'e}(\mathcal E)}$ are the images of continuous maps $U\rightarrow E$ . An open subset of $\ensuremath{ \text{Sp\'e}(\mathcal E)}$ may be written as a union of ; $U_{ts} \equiv \{t_p, s_p \vert p\in U\}$ . Then, by single-valuedness of maps, a continuous map $U\rightarrow \ensuremath{ \text{Sp\'e}(\mathcal E)}$ must map to for some $t\in \Gamma(U,E)$ , so we have $\Gamma(U,E) \supset \Gamma(U,\ensuremath{ \text{Sp\'e}(\mathcal E)})$ .

"sheaf of sections" is owned by guffin.

(view preamble)