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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 ...
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.
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).
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.
abelian 1. An abelian variety is a complete group variety. For example, consider the complex variety / or an elliptic curve over a finite field . 2. An abelian scheme is a (flat) family of abelian varieties.
Motivic cohomology provides a rich invariant already for fields. (Note that a field k determines a scheme Spec(k), for which motivic cohomology is defined.)Although motivic cohomology H i (k, Z(j)) for fields k is far from understood in general, there is a description when i = j:
The (topological) fundamental group associated with a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. Although it is still being studied for the classification of algebraic varieties even in algebraic geometry, for many applications the fundamental group has been found to be inadequate for the classification of objects, such as ...