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Download as PDF; Printable version; ... The fiber of 5 under ... is a morphism of schemes, the fiber of a point in is the fiber product of schemes () ...
Then there is a morphism Spec(k(y)) → Y with image y, where k(y) is the residue field of y. The fiber of f over y is defined as the fiber product X × Y Spec(k(y)); this is a scheme over the field k(y). [3] This concept helps to justify the rough idea of a morphism of schemes X → Y as a family of schemes parametrized by Y.
A mapping : between total spaces of two fibrations : and : with the same base space is a fibration homomorphism if the following diagram commutes: . The mapping is a fiber homotopy equivalence if in addition a fibration homomorphism : exists, such that the mappings and are homotopic, by fibration homomorphisms, to the identities and . [2]: 405-406
The fiber over these zeroes in the resulting "eigenbundle" will be isomorphic to the fiber over them in E, while everywhere else the fiber is the trivial 0-dimensional vector space. The dual vector bundle E* is the Hom bundle Hom(E, R × X) of bundle homomorphisms of E and the trivial bundle R × X.
We make X into a fiber space over C by mapping (c,s) to s 2. We construct an isomorphism from X minus the fiber over 0 to E×C minus the fiber over 0 by mapping (c,s) to (c-log(s)/2πi,s 2). (The two fibers over 0 are non-isomorphic elliptic curves, so the fibration X is certainly not isomorphic to the fibration E×C over all of C.)
The choice of a (normalised) cleavage for a fibred -category specifies, for each morphism : in , a functor:; on objects is simply the inverse image by the corresponding transport morphism, and on morphisms it is defined in a natural manner by the defining universal property of cartesian morphisms.
In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely which category of fiber ...
Another example of a pullback comes from the theory of fiber bundles: given a bundle map π : E → B and a continuous map f : X → B, the pullback (formed in the category of topological spaces with continuous maps) X × B E is a fiber bundle over X called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.