Алгебралық кеңістіктер: Схемалардың жалпыламасы

In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin 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. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology. The resulting category of algebraic spaces extends the category of schemes and allows one to carry out several natural constructions that are used in the construction of moduli spaces but are not always possible in the smaller category of schemes, such as taking the quotient of a free action by a finite group (cf. the Keel–Mori theorem).

There are two common ways to define algebraic spaces: they can be defined as either quotients of schemes by étale equivalence relations, or as sheaves on a big étale site that are locally isomorphic to schemes. These two definitions are essentially equivalent.

Algebraic spaces are similar to schemes, and much of the theory of schemes extends to algebraic spaces. For example, most properties of morphisms of schemes also apply to algebraic spaces, one can define cohomology of quasicoherent sheaves, this has the usual finiteness properties for proper morphisms, and so on. 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. Commutative group objects in the category of algebraic spaces over an arbitrary scheme which are proper, locally finite presentation, flat, and cohomologically flat in dimension 0 are schemes. Not every singular algebraic surface is a scheme. Hironaka's example can be used to give a non singular 3 dimensional proper algebraic space that is not a scheme, given by the quotient of a scheme by a group of order 2 acting freely. 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). Every quasi separated algebraic space contains a dense open affine subscheme, and the complement of such a subscheme always has codimension ≥ 1. Thus algebraic spaces are in a sense "close" to affine schemes. The quotient of the complex numbers by a lattice is an algebraic space, but is not an elliptic curve, even though the corresponding analytic space is an elliptic curve (or more precisely is the image of an elliptic curve under the functor from complex algebraic spaces to analytic spaces). In fact this algebraic space quotient is not a scheme, is not complete, and is not even quasi separated. This shows that although the quotient of an algebraic space by an infinite discrete group is an algebraic space, it can have strange properties and might not be the algebraic space one was "expecting". Similar examples are given by the quotient of the complex affine line by the integers, or the quotient of the complex affine line minus the origin by the powers of some number: again the corresponding analytic space is a variety, but the algebraic space is not.

Algebraic spaces over the complex numbers are closely related to analytic spaces and Moishezon manifolds. Roughly speaking, the difference between complex algebraic spaces and analytic spaces is that complex algebraic spaces are formed by gluing affine pieces together using the étale topology, while analytic spaces are formed by gluing with the classical topology. In particular there is a functor from complex algebraic spaces of finite type to analytic spaces. Hopf manifolds give examples of analytic surfaces that do not come from a proper algebraic space (though one can construct non proper and non separated algebraic spaces whose analytic space is the Hopf surface). It is also possible for different algebraic spaces to correspond to the same analytic space: for example, an elliptic curve and the quotient of C by the corresponding lattice are not isomorphic as algebraic spaces, but the corresponding analytic spaces are isomorphic. Artin showed that proper algebraic spaces over the complex numbers are more or less the same as Moishezon spaces.

A far reaching generalization of algebraic spaces is given by algebraic stacks. In the category of stacks we can form even more quotients by group actions than in the category of algebraic spaces (the resulting quotient is called a quotient stack).