Scheme theory
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety to include, among other things multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety and different schemes) and "varieties" defined over rings (for example Fermat curves are defined over the integers).
Schemes were introduced by Alexander Grothendieck in 1960 in his treatise Éléments de géométrie algébrique; 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). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. By including rationality questions inside the formalism, scheme theory introduces a strong connection between algebraic geometry and number theory, which eventually allowed Wiles' proof of Fermat's Last Theorem.
To be technically precise, a scheme is a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra (spaces of prime ideals) of commutative rings along their open subsets. In other words, it is a locally ringed space which is locally a spectrum of a commutative ring.
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