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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 finer étale topology .
Oct 9, 2017 · An algebraic space is a sheaf of sets $F$ in the étale topology of schemes satisfying the condition of local representability (in the étale topology): There exists a scheme $U$ and a sheaf morphism $\tilde U \rightarrow F$ such that for any scheme $V$ and morphism $\tilde V \rightarrow F$ the fibred product $\tilde U \times_F \tilde V$ is ...
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.
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 ...
More precisely, given a scheme T ∈Ob((Sch/S)fppf) the representable functor hT is an algebraic space. Proof. The functor hT is a sheaf by our remarks in Section 65.2. The diagonal hT → hT ×hT =hT×T is representable because (Sch/S)fppf has fibre products. The identity map hT → hT is surjective étale. .
Apr 4, 2021 · Definition 1.1. A vector space (over ) consists of a set along with two operations " " and " " subject to these conditions. For any . There is a zero vector such that for all . Each has an additive inverse such that . If is a scalar, that is, a member of and then the scalar multiple is in . If and then . If and , then . For any , . Remark 1.2.
Algebraic spaces. Fibre products of algebraic spaces. Glueing algebraic spaces. Presentations of algebraic spaces. Algebraic spaces and equivalence relations. Algebraic spaces, retrofitted. Immersions and Zariski coverings of algebraic spaces. Separation conditions on algebraic spaces. Examples of algebraic spaces. Change of big site.
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