The relative point of view is that much of algebraic geometry should be developed for a morphism X → Y of schemes (called a scheme X over Y), rather than for an individual scheme. In other words, it is a ringed space which is locally a spectrum of a commutative ring. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem.įormally, 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. Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory was 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).
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety and different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
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