Scheme (Mathematics)

In algebraic geometry, a scheme is a structure that enlarges the notion of an algebraic variety by accounting for multiplicities, so that equations such as x = 0 and x-squared = 0 define the same variety but different schemes, and by allowing varieties defined over any commutative ring rather than only over the real or complex numbers. Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise Elements de geometrie algebrique, developed in part to solve deep problems of algebraic geometry such as the Weil conjectures, and it strongly draws on commutative algebra to allow a systematic use of topology and homological algebra; scheme theory also unified algebraic geometry with much of number theory, eventually contributing to Andrew Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a ringed space covered by affine schemes, where an affine scheme is the spectrum of a commutative ring, its points are the ring's prime ideals and its closed points are the ring's maximal ideals.