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- All algebraic spaces contain a subspace that is open and dense in the Zariski topology and that is a scheme. One-dimensional and non-singular two-dimensional algebraic spaces are schemes, but this is not true of three-dimensional or singular two-dimensional algebraic spaces; a group in the category of algebraic spaces over a field is a scheme.
encyclopediaofmath.org/wiki/Algebraic_space
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This illustrates one difference between schemes and algebraic spaces: the quotient of an algebraic space by a discrete group acting freely is an algebraic space, but the quotient of a scheme by a discrete group acting freely need not be a scheme (even if the group is finite).
Michael Artin defined an algebraic space as a functor that is a sheaf in the étale topology and that, locally in the étale topology, is an affine scheme. Equivalently, an algebraic space is the quotient of a scheme by an étale equivalence relation.
topological space and describing which functions are smooth, a scheme can be presented by giving an appropriate topological space and describing which functions are regular. The tool we will use for managing the regular functions on the space is called a sheaf.
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Oct 9, 2017 · One-dimensional and non-singular two-dimensional algebraic spaces are schemes, but this is not true of three-dimensional or singular two-dimensional algebraic spaces; a group in the category of algebraic spaces over a field is a scheme.
Lemma 65.6.2. A scheme is an algebraic 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.
Classical algebraic geometry is the study of algebraic varieties, meaning spaces that can be described locally as solution sets of polynomial equations over an algebraically closed eld, such as the complex numbers C.
essential differences between algebraic geometry and the other fields, the inverse function theorem doesn’t hold in algebraic geometry. One other essential difference is that 1=Xis
