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The sheaf of rational functions K X of a scheme X is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry. In the case of algebraic varieties , such a sheaf associates to each open set U the ring of all rational functions on that open set; in other words, K X ( U ) is the ...
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
Hom(−,X) : (Affine schemes) op Sets. sending an affine scheme Y to the set of scheme maps. [4] A scheme is determined up to isomorphism by its functor of points. This is a stronger version of the Yoneda lemma, which says that a X is determined by the map Hom(−,X) : Schemes op → Sets.
If a finite-dimensional vector space V over a field is viewed as a scheme over the field, [note 1] then the dimension of the scheme V is the same as the vector-space dimension of V. Let = [,,] / (,), k a field.
In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determines a variety over a bigger field, or the pullback of a family of varieties, or a fiber of a family of varieties.
In algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V.In classical algebraic geometry they are ratios of polynomials; in complex geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.
Proper algebraic spaces over a field of dimension one (curves) are schemes. Non-singular proper algebraic spaces of dimension two over a field (smooth surfaces) are schemes. Quasi-separated group objects in the category of algebraic spaces over a field are schemes, though there are non quasi-separated group objects that are not schemes.
In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology.