Editing Scheme (mathematics) (section)

{{Short description|Generalization of algebraic variety}}
In [[mathematics]], specifically [[algebraic geometry]], a '''scheme''' is a [[mathematical structure|structure]] that enlarges the notion of [[algebraic variety]] in several ways, such as taking account of [[multiplicity (mathematics)|multiplicities]] (the equations {{math|1=''x'' = 0}} and {{math|1=''x''<sup>2</sup> = 0}} define the same algebraic variety but different schemes) and allowing "varieties" defined over any [[commutative ring]] (for example, [[Fermat curve]]s are defined over the [[integer]]s). 

'''Scheme theory''' was introduced by [[Alexander Grothendieck]] in 1960 in his treatise ''[[Éléments de géométrie algébrique]]'' (EGA); one of its aims was developing the formalism needed to solve deep problems of [[algebraic geometry]], such as the [[Weil conjectures]] (the last of which was proved by [[Pierre Deligne]]).<ref>Introduction of the first edition of "[[Éléments de géométrie algébrique]]".</ref> Strongly based on [[commutative algebra]], scheme theory allows a systematic use of methods of [[topology]] and [[homological algebra]]. Scheme theory also unifies algebraic geometry with much of [[number theory]], which eventually led to [[Wiles's proof of Fermat's Last Theorem]]. 

Schemes elaborate the fundamental idea that an algebraic variety is best analyzed through the [[coordinate ring]] of regular algebraic functions defined on it (or on its subsets), and each subvariety corresponds to the [[Ideal (ring theory)|ideal]] of functions which vanish on the subvariety. Intuitively, a scheme is a [[topological space]] consisting of closed points which correspond to geometric points, together with non-closed points which are [[Generic point|generic points]] of irreducible subvarieties. The space is covered by an [[Atlas (topology)|atlas]] of open sets, each endowed with a coordinate ring of regular functions, with specified coordinate changes between the functions over intersecting open sets. Such a structure is called a [[ringed space]] or a [[Sheaf (mathematics)|sheaf]] of rings. The cases of main interest are the [[Noetherian scheme|Noetherian schemes]], in which the coordinate rings are [[Noetherian ring|Noetherian rings]].

Formally, a scheme is a ringed space covered by affine schemes. An affine scheme is the [[Spectrum of a ring|spectrum]] of a commutative ring; its points are the [[Prime ideal|prime ideals]] of the ring, and its closed points are [[Maximal ideal|maximal ideals]]. The coordinate ring of an affine scheme is the ring itself, and the coordinate rings of open subsets are [[Localization (commutative algebra)|rings of fractions]].

The [[Grothendieck's relative point of view|relative point of view]] is that much of algebraic geometry should be developed for a morphism {{math|1=''X'' → ''Y''}} of schemes (called a scheme {{math|''X''}} '''over the base''' {{math|''Y''}} ), rather than for an individual scheme. For example, in studying [[algebraic surface]]s, it can be useful to consider families of algebraic surfaces over any scheme {{math|''Y''}}. In many cases, the family of all varieties of a given type can itself be viewed as a variety or scheme, known as a [[moduli space]].

For some of the detailed definitions in the theory of schemes, see the [[glossary of scheme theory]].