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Algebraic space. In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin [1] for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the ...
Oct 9, 2017 · Algebraic space. A generalization of the concepts of a scheme and an algebraic variety. This generalization is the result of certain constructions in algebraic geometry: Hilbert schemes, Picard schemes, moduli varieties, contractions, which are often not realizable in the category of schemes and require its extensions.
Space (mathematics) In mathematics, a space is a set (sometimes known as a universe) endowed with a structure defining the relationships among the elements of the set. A subspace is a subset of the parent space which retains the same structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces ...
May 27, 2020 · Wikipedia defines an algebraic space X X to be a sheaf on the big étale site (Sch/S)et (Sch / S) e t, such that: There is a surjective étale morphism hX → X h X → X. The diagonal morphism ΔX/S: X → X ×X Δ X / S: X → X × X is representable. What does the first condition actually mean? Possibly it makes sense by means of the second ...
e. In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space ...
n algebraic space over S. By Lemma 9.1 this means that F = hU/hR for some étale equivalence relation R . U ×S U in (Sch/S)fppf. Since f−1 is an exact functor we conclude th. t f−1F = h′ U/h′ R. Hence f−1F is an algebraic space o.
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i (see Spaces, Definition 12.5) such that eachX i is quasi-separated. Let S is a scheme contained in Sch fppf, and let X be an algebraic space over S. Then we say X is separated, locally separated, quasi-separated, or Zariski lo-cally quasi-separated if Xviewed as an algebraic space over Spec(Z) (see Spaces, Definition16.2 ...
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