Editing Vector bundle (section)

==Sections and locally free sheaves==
[[File:Vector bundle with section.png|thumb|300px|A vector bundle <math>E</math> over a base <math>M</math> with section <math>s</math>.]]
[[File:Surface normals.svg|right|thumb|300px|The map associating a [[Normal vector|normal]] to each point on a [[Surface (topology)|surface]] can be thought of as a section. The surface is the space ''X'', and at each point ''x'' there is a vector in the vector space attached at ''x''.]]

Given a vector bundle {{pi}}: ''E'' → ''X'' and an open subset ''U'' of ''X'', we can consider [[Section (fiber bundle)|'''sections''']] of {{pi}} on ''U'', i.e. continuous functions ''s'': ''U'' → ''E'' where the composite {{pi}}&nbsp;∘&nbsp;''s'' is such that {{nowrap|1=({{pi}} ∘ ''s'')(''u'') = ''u''}} for all ''u'' in ''U''. Essentially, a section assigns to every point of ''U'' a vector from the attached vector space, in a continuous manner. As an example, sections of the tangent bundle of a differential manifold are nothing but [[vector field]]s on that manifold.

Let ''F''(''U'') be the set of all sections on ''U''. ''F''(''U'') always contains at least one element, namely the '''zero section''': the function ''s'' that maps every element ''x'' of ''U'' to the [[zero vector|zero element of the vector space]] {{pi}}<sup>−1</sup>({''x''}). With the [[pointwise]] addition and [[scalar multiplication]] of sections, ''F''(''U'') becomes itself a real vector space. The collection of these vector spaces is a [[sheaf (mathematics)|sheaf]] of vector spaces on ''X''.

If ''s'' is an element of ''F''(''U'') and α: ''U'' → '''R''' is a continuous map, then α''s'' (pointwise scalar multiplication) is in ''F''(''U''). We see that ''F''(''U'') is a [[module (mathematics)|module]] over the [[Ring (mathematics)|ring]] of continuous [[real-valued function]]s on ''U''. Furthermore, if O<sub>''X''</sub> denotes the structure sheaf of continuous real-valued functions on ''X'', then ''F'' becomes a sheaf of O<sub>''X''</sub>-modules.

Not every sheaf of O<sub>''X''</sub>-modules arises in this fashion from a vector bundle: only the [[locally free sheaf|locally free]] ones do. (The reason: locally we are looking for sections of a projection ''U'' × '''R'''<sup>''k''</sup> → ''U''; these are precisely the continuous functions ''U'' → '''R'''<sup>''k''</sup>, and such a function is a ''k''-[[tuple]] of continuous functions ''U'' → '''R'''.)

Even more: the category of real vector bundles on ''X'' is [[Equivalence of categories|equivalent]] to the category of locally free and [[Finitely generated module|finitely generate]]d sheaves of O<sub>''X''</sub>-modules.

So we can think of the category of real vector bundles on ''X'' as sitting inside the category of [[sheaf of modules|sheaves of O<sub>''X''</sub>-modules]]; this latter category is abelian, so this is where we can compute [[Kernel (algebra)|kernels]] and [[cokernel]]s of morphisms of vector bundles.

A rank ''n'' vector bundle is trivial [[if and only if]] it has ''n'' [[linearly independent]] global sections.