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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 morphism is quasi-finite if it is of finite type and has finite fibers. quasi-projective A quasi-projective variety is a locally closed subvariety of a projective space. quasi-separated A morphism f : Y → X is called quasi-separated or (Y is quasi-separated over X) if the diagonal morphism Y → Y × X Y is quasi-compact.
If is a morphism of , then those morphisms of that project to are called -morphisms, and the set of -morphisms between objects and in is denoted by (,). A morphism m : x → y {\displaystyle m:x\to y} in F {\displaystyle F} is called ϕ {\displaystyle \phi } -cartesian (or simply cartesian ) if it satisfies the following condition:
A function between topological spaces is called monotone if every fiber is a connected subspace of its domain. A function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is monotone in this topological sense if and only if it is non-increasing or non-decreasing , which is the usual meaning of " monotone function " in real analysis .
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.
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
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed field such as the complex numbers, these fibers are elliptic curves, perhaps without a chosen origin.)
A morphism : is of finite type if there exists a cover such that the fibers can be covered by finitely many affine schemes making the induced ring morphisms into finite-type morphisms. A typical example of a finite-type morphism is a family of schemes.