Definitions
An affine scheme is a locally ringed space isomorphic to the spectrum of a commutative ring. We denote the spectrum of a commutative ring A by Spec(A). A scheme is a locally ringed space X admitting a covering by open sets Ui, such that the restriction of the structure sheaf OX to each Ui is an affine scheme. Therefore one may think of a scheme as being covered by "coordinate charts" of affine schemes. The above formal definition means exactly that schemes are obtained by glueing together affine schemes for the Zariski topology.
In the early days, this was called a prescheme, and a scheme was defined to be a 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 Mumford's Red Book.
Read more about this topic: Scheme (mathematics)
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