Fiber (mathematics)

Last updated

In mathematics, the term fiber (US English) or fibre (British English) can have two meanings, depending on the context:


  1. In naive set theory, the fiber of the element y in the set Y under a map f : XY is the inverse image of the singleton under f.
  2. In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because, in general, not every point is closed.


Fiber in naive set theory

Let f : XY be a map. The fiber of an element commonly denoted by is defined as

That is, the fiber of y under f is the set of elements in the domain of f that are mapped to y.

The inverse image or preimage generalizes the concept of the fiber to subsets of the codomain. The notation is still used to refer to the fiber, as the fiber of an element y is the preimage of the singleton set , as in . That is, the fiber can be treated as a function from the codomain to the powerset of the domain: while the preimage generalizes this to a function between powersets:

If f maps into the real numbers, so is simply a number, then the fiber is also called the level set of y under f: If f is a continuous function and y is in the image of f, then the level set of y under f is a curve in 2D, a surface in 3D, and, more generally, a hypersurface of dimension d − 1.

Fiber in algebraic geometry

In algebraic geometry, if f : XY is a morphism of schemes, the fiber of a point p in Y is the fiber product of schemes

where k(p) is the residue field at p.

See also

Related Research Articles

In mathematics, a partial functionf from a set X to a set Y is a function from a subset S of X to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition of f. If S equals X, the partial function is said to be total.

Surjective function Function such that every element has a preimage (mathematics)

In mathematics, a function f from a set X to a set Y is surjective, if for every element y in the codomain Y of f, there is at least one element x in the domain X of f such that f(x) = y. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y.

In algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by , is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an affine scheme.

Injective function Function that preserves distinctness

In mathematics, an injective function is a function that maps distinct elements of its domain to distinct elements of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

In algebra, the kernel of a homomorphism is generally the inverse image of 0. An important special case is the kernel of a linear map. The kernel of a matrix, also called the null space, is the kernel of the linear map defined by the matrix.

Function (mathematics) Mapping that associates a single output value to each input

In mathematics, a function is a binary relation between two sets that associates every element of the first set to exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.

In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphismf from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,

Image (mathematics) The set of all values of a function

In mathematics, the image of a function is the set of all output values it may produce.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.

In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.

In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves and the moduli stack of elliptic curves. Originally, they were introduced by Grothendieck to keep track of automorphisms on moduli spaces, a technique which allows for treating these moduli spaces as if their underlying schemes or algebraic spaces are smooth. But, through many generalizations the notion of algebraic stacks was finally discovered by Michael Artin.

Fibred categories are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which inverse images of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space X to another topological space Y is associated the pullback functor taking bundles on Y to bundles on X. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stacks, which are fibered categories with "descent". Fibrations also play an important role in categorical semantics of type theory, and in particular that of dependent type theories.

In mathematics, a branched covering is a map that is almost a covering map, except on a small set.

In mathematics, in the field of sheaf theory and especially in algebraic geometry, the direct image functor generalizes the notion of a section of a sheaf to the relative case.

In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes.

In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist.

This is a glossary of algebraic geometry.

In algebraic geometry, the tangent space to a functor generalizes the classical construction of a tangent space such as the Zariski tangent space. The construction is based on the following observation. Let X be a scheme over a field k.

In algebraic geometry, a sheaf of algebras on a ringed space X is a sheaf of commutative rings on X that is also a sheaf of -modules. It is quasi-coherent if it is so as a module.