Editing Scheme (mathematics) (section)

==Development==
The origins of algebraic geometry mostly lie in the study of [[polynomial]] equations over the [[real number]]s. By the 19th century, it became clear (notably in the work of [[Jean-Victor Poncelet]] and [[Bernhard Riemann]]) that algebraic geometry over the real numbers is simplified by working over the [[field (mathematics)|field]] of [[complex number]]s, which has the advantage of being [[algebraically closed field|algebraically closed]].{{sfn|Dieudonné|1985|loc=Chapters IV and V}} The early 20th century saw analogies between algebraic geometry and number theory, suggesting the question: can algebraic geometry be developed over other fields, such as those with positive [[characteristic (algebra)|characteristic]], and more generally over [[Number ring|number rings]] like the integers, where the tools of topology and [[complex analysis]] used to study complex varieties do not seem to apply?

[[Hilbert's Nullstellensatz]] suggests an approach to algebraic geometry over any algebraically closed field {{math|''k''}} : the [[maximal ideal]]s in the [[polynomial ring]] {{math|''k''[''x''<sub>1</sub>, ... , ''x''<sub>''n''</sub>]}} are in one-to-one correspondence with the set {{math|''k''<sup>''n''</sup>}} of {{math|''n''}}-tuples of elements of {{math|''k''}}, and the [[prime ideal]]s correspond to the irreducible algebraic sets in {{math|''k''<sup>''n''</sup>}}, known as affine varieties. Motivated by these ideas, [[Emmy Noether]] and [[Wolfgang Krull]] developed commutative algebra in the 1920s and 1930s.{{sfn|Dieudonné|1985|loc=sections VII.2 and VII.5}} Their work generalizes algebraic geometry in a purely algebraic direction, generalizing the study of points (maximal ideals in a polynomial ring) to the study of prime ideals in any commutative ring. For example, Krull defined the [[Krull dimension|dimension]] of a commutative ring in terms of prime ideals and, at least when the ring is [[Noetherian ring|Noetherian]], he proved that this definition satisfies many of the intuitive properties of geometric dimension.

Noether and Krull's commutative algebra can be viewed as an algebraic approach to ''affine'' algebraic varieties. However, many arguments in algebraic geometry work better for [[projective varieties]], essentially because they are [[compact space|compact]]. From the 1920s to the 1940s, [[Bartel Leendert van der Waerden|B. L. van der Waerden]], [[André Weil]] and [[Oscar Zariski]] applied commutative algebra as a new foundation for algebraic geometry in the richer setting of projective (or [[quasi-projective]]) varieties.{{sfn|Dieudonné|1985|loc=section VII.4}} In particular, the [[Zariski topology]] is a useful topology on a variety over any algebraically closed field, replacing to some extent the classical topology on a complex variety (based on the [[metric topology]] of the complex numbers).

For applications to number theory, van der Waerden and Weil formulated algebraic geometry over any field, not necessarily algebraically closed. Weil was the first to define an ''abstract variety'' (not embedded in [[projective space]]), by gluing affine varieties along open subsets, on the model of abstract [[manifold]]s in topology. He needed this generality for his construction of the [[Jacobian variety]] of a curve over any field. (Later, Jacobians were shown to be projective varieties by Weil, [[Wei-Liang Chow|Chow]] and [[Teruhisa Matsusaka|Matsusaka]].)

The algebraic geometers of the [[Italian school of algebraic geometry|Italian school]] had often used the somewhat foggy concept of the [[generic point]] of an algebraic variety. What is true for the generic point is true for "most" points of the variety. In Weil's ''Foundations of Algebraic Geometry'' (1946), generic points are constructed by taking points in a very large algebraically closed field, called a ''universal domain''.{{sfn|Dieudonné|1985|loc=section VII.4}} This worked awkwardly: there were many different generic points for the same variety. (In the later theory of schemes, each algebraic variety has a single generic point.)

In the 1950s, [[Claude Chevalley]], [[Masayoshi Nagata]] and [[Jean-Pierre Serre]], motivated in part by the [[Weil conjectures]] relating number theory and algebraic geometry, further extended the objects of algebraic geometry, for example by generalizing the base rings allowed. The word ''scheme'' was first used in the 1956 Chevalley Seminar, in which Chevalley pursued Zariski's ideas.<ref>{{citation|last=Chevalley|first= C. |title=Les schémas|series= Séminaire Henri Cartan|volume= 8 |year=1955–1956|issue= 5|pages= 1–6 |url= http://www.numdam.org/item?id=SHC_1955-1956__8__A5_0}}</ref> According to [[Pierre Cartier (mathematician)|Pierre Cartier]], it was [[André Martineau]] who suggested to Serre the possibility of using the spectrum of an arbitrary commutative ring as a foundation for algebraic geometry.{{sfn|Cartier|2001|loc=note 29}}