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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).
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, 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.
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.
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.
Schemes over a field of positive characteristic and the tame fundamental group [ edit ] For an algebraically closed field k {\displaystyle k} of positive characteristic, the results are different, since Artin–Schreier coverings exist in this situation.