Vector Bundle - Sections and Locally Free Sheaves

Sections and Locally Free Sheaves

Given a vector bundle π : EX and an open subset U of X, we can consider sections of π on U, i.e. continuous functions s : UE 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 : UR 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 OX denotes the structure sheaf of continuous real-valued functions on X, then F becomes a sheaf of OX-modules.

Not every sheaf of OX-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 × RkU; these are precisely the continuous functions URk, and such a function is an k-tuple of continuous functions UR.)

Even more: the category of real vector bundles on X is equivalent to the category of locally free and finitely generated sheaves of OX-modules. So we can think of the category of real vector bundles on X as sitting inside the category of sheaves of OX-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.

Read more about this topic:  Vector Bundle

Famous quotes containing the words sections, locally, free and/or sheaves:

    That we can come here today and in the presence of thousands and tens of thousands of the survivors of the gallant army of Northern Virginia and their descendants, establish such an enduring monument by their hospitable welcome and acclaim, is conclusive proof of the uniting of the sections, and a universal confession that all that was done was well done, that the battle had to be fought, that the sections had to be tried, but that in the end, the result has inured to the common benefit of all.
    William Howard Taft (1857–1930)

    To see ourselves as others see us can be eye-opening. To see others as sharing a nature with ourselves is the merest decency. But it is from the far more difficult achievement of seeing ourselves amongst others, as a local example of the forms human life has locally taken, a case among cases, a world among worlds, that the largeness of mind, without which objectivity is self- congratulation and tolerance a sham, comes.
    Clifford Geertz (b. 1926)

    The free man is a warrior.—How is freedom measured among individuals, among peoples? According to the resistance that must be overcome, according to the trouble it takes to stay on top. The highest type of free man must be sought where the highest resistance is constantly overcome: five steps away from tyranny, close to the threshold of the danger of servitude.
    Friedrich Nietzsche (1844–1900)

    A thousand golden sheaves were lying there,
    Shining and still, but not for long to stay—
    As if a thousand girls with golden hair
    Might rise from where they slept and go away.
    Edwin Arlington Robinson (1869–1935)