Vector Bundle - Sections and Locally Free Sheaves

Sections and Locally Free Sheaves

Given a vector bundle π : E → X and an open subset U of X, we can consider sections of π on U, i.e. continuous functions s : U → E where the composite π∘s is such that (π∘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 fields 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 element of the vector space π−1({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 of vector spaces on X.

If s is an element of F(U) and α : U → R is a continuous map, then αs (pointwise scalar multiplication) is in F(U). We see that F(U) is a module over the ring of continuous real-valued functions on U. Furthermore, if O_X denotes the structure sheaf of continuous real-valued functions on X, then F becomes a sheaf of O_X-modules.

Not every sheaf of O_X-modules arises in this fashion from a vector bundle: only the locally free ones do. (The reason: locally we are looking for sections of a projection U × Rk → U; these are precisely the continuous functions U → Rk, and such a function is an k-tuple of continuous functions U → R.)

Even more: the category of real vector bundles on X is equivalent to the category of locally free and finitely generated sheaves of O_X-modules. So we can think of the category of real vector bundles on X as sitting inside the category of sheaves of O_X-modules; this latter category is abelian, so this is where we can compute kernels and cokernels of morphisms of vector bundles.

Note that a rank n vector bundle is trivial if and only if it has n linearly independent global sections.