Editing Scheme (mathematics) (section)

==Definition==
An '''affine scheme''' is a [[locally ringed space]] isomorphic to the [[spectrum of a ring|spectrum]] <math>\operatorname{Spec}(R)</math> of a commutative ring <math>R</math>. A '''scheme''' is a locally ringed space <math>X</math> admitting a covering by open sets <math>U_i</math>, such that each <math>U_i</math> (as a locally ringed space) is an affine scheme.{{sfn|Hartshorne|1997|loc=section II.2}} In particular, <math>X</math> comes with a sheaf <math>\mathcal{O}_X</math>, which assigns to every open subset <math>U</math> a commutative ring <math>\mathcal{O}_X(U)</math> called the '''ring of regular functions''' on <math>U</math>. One can think of a scheme as being covered by "coordinate charts" that are affine schemes. The definition means exactly that schemes are obtained by gluing together affine schemes using the Zariski topology. 

In the early days, this was called a ''prescheme'', and a scheme was defined to be a [[separated scheme|separated]] prescheme.  The term prescheme has fallen out of use, but can still be found in older books, such as Grothendieck's "Éléments de géométrie algébrique" and [[David Mumford|Mumford]]'s "Red Book".{{sfn|Mumford|1999|loc=Chapter II}} The sheaf properties of <math>\mathcal{O}_X(U)</math> mean that its elements'','' which are not necessarily functions, can neverthess be patched together from their restrictions in the same way as functions.

A basic example of an affine scheme is '''affine <math>n</math>-space''' over a field <math>k</math>, for a [[natural number]] <math>n</math>. By definition, <math>A_k^n</math> is the spectrum of the polynomial ring <math>k[x_1,\dots,x_n]</math>. In the spirit of scheme theory, affine <math>n</math>-space can in fact be defined over any commutative ring <math>R</math>, meaning <math>\operatorname{Spec}(R[x_1,\dots,x_n])</math>.