Editing Scheme (mathematics) (section)

==Origin of schemes==
The theory took its definitive form in Grothendieck's ''Éléments de géométrie algébrique'' (EGA) and the later ''Séminaire de géométrie algébrique'' (SGA), bringing to a conclusion a generation of experimental suggestions and partial developments.{{sfn|Dieudonné|1985|loc=sections VII.4, VIII.2, VIII.3}} Grothendieck defined the [[spectrum of a ring|spectrum]] <math>X</math> of a [[commutative ring]] <math>R</math> as the space of [[prime ideal]]s of <math>R</math> with a natural topology (known as the Zariski topology), but augmented it with a [[sheaf (mathematics)|sheaf]] of rings: to every open subset <math>U</math> he assigned a commutative ring <math>\mathcal{O}_X(U)</math>, which may be thought of as the coordinate ring of regular functions on <math>U</math>. These objects <math>\operatorname{Spec}(R)</math> are the affine schemes; a general scheme is then obtained by "gluing together" affine schemes.

Much of algebraic geometry focuses on projective or quasi-projective varieties over a field <math>k</math>, most often over the complex numbers. Grothendieck developed a large body of theory for arbitrary schemes extending much of the geometric intuition for varieties. For example, it is common to construct a moduli space first as a scheme, and only later study whether it is a more concrete object such as a projective variety. Applying Grothendieck's theory to schemes over the integers and other number fields led to powerful new perspectives in number theory.