sheaf of sections

0.1 Presheaf Definition

Consider a rank r vector bundleMathworldPlanetmath EM, whose typical fibre is defined with respect to a field k. Let {Uα} constitute a cover for M. Then, sectionsPlanetmathPlanetmathPlanetmath of the bundle over some UM are defined as continuous functions UE, which commute with the natural projectionMathworldPlanetmath map π:EM; πs=idM. Denote the space of sections of the bundle over U to be Γ(U,E). The space of sections is a vector spaceMathworldPlanetmath over the field k by defining addition and scalar multiplication pointwise: for s,tΓ(U,E), pU and ak

(s+t)(p)s(p)+t(p)    (as)(p)as(p).

Then, this forms a presheaf , a functor from ((topM)) to the category of vector spaces, with restrictionPlanetmathPlanetmathPlanetmath maps the natural restriction of functions.

0.2 Sheaf Axioms

It is easy to see that it satisfies the sheaf axioms: for U open and {Vi} a cover of U,

  1. 1.

    if s(U) and s|Vi=0 for all i, then s=0.

  2. 2.

    if si(Vi) for all i, such that for each i,j with ViVj, si|ViVj=sj|ViVj, then there is an s(U) with s|Vi=si for all i.

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

1 Sheafification

We may also see the vector bundle by applying associated sheaf construction to the presheaf UΓ(U,E). First though, we show that the stalk of the sheaf at a point is isomorphicPlanetmathPlanetmathPlanetmath to the fibre of the bundle E at the point. Let [s,U] be a germ at pM (pUM), and define a map ψ:pEp by


First, we show that the map is a vector space homomorphism. Consider two germs [s,U] and [t,V] in p. These map to sp and tp respectively. We add the germs by finding an open set WUV and adding the restrictions of the sections;


Of course, pW, so we have ψ(s|W+t|W)=sp+tp, since the restriction maps are simply restriction of functions. Now, it is easy to show that ψ is injectivePlanetmathPlanetmath. Assume ψ([t,V])=ψ([s,U])=sp. Then

ψ([t,V])-ψ([s,U]) =sp-sp
ψ([t,V]-[s,U]) =0

Now, we show that ψ is surjectivePlanetmathPlanetmath. For spEp, let UM open be isomorphic to some subset U of m. Then, Γ(U,E) is the set of continuous maps UVE, where VE is the typical fibre of E;


Then let [s,U] be the constant function s:Usx, and we have constructed an isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ψ between p and Ep.

To construct the Étalé space, take the disjoint unionMathworldPlanetmath of stalks, Spé()=pMp, and endow it with the following topology: the open sets shall be of the form


collectionMathworldPlanetmath of germs of sections at points in UM.

Then, the associated sheaf to is the presheaf which assigns continuous maps Γ(U,Spé()) to each open U. These are maps where the preimageMathworldPlanetmath of Us is open. Clearly, this implies that Γ(U,E)Γ(U,Spé()). To go the other way, note that open sets of Spé() are the images of continuous maps UE. An open subset of Spé() may be written as a union of Ut; Uts{tp,sp|pU}. Then, by single-valuedness of maps, a continuous map USpé() must map to Ut for some tΓ(U,E), so we have Γ(U,E)Γ(U,Spé()).

Title sheaf of sections
Canonical name SheafOfSections
Date of creation 2013-03-22 15:46:36
Last modified on 2013-03-22 15:46:36
Owner guffin (12505)
Last modified by guffin (12505)
Numerical id 7
Author guffin (12505)
Entry type Definition
Classification msc 55R25
Related topic VectorBundle
Defines Sheaf of Sections