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The algebraic geometers of the Italian school had often used the somewhat foggy concept of "generic point" when proving statements about algebraic varieties. What is true for the generic point is true for all points of the variety except a small number of special points. In the 1920s, Emmy Noether had first suggested a way to clarify the concept: start with the coordinate ring of the variety (the ring of all polynomial functions defined on the variety); the maximal ideals of this ring will correspond to ordinary points of the variety (under suitable conditions), and the non-maximal prime ideals will correspond to the various generic points. By taking all prime ideals, one thus gets the whole collection of ordinary and generic points. Noether did not pursue this approach.
In the 1930s, Wolfgang Krull turned things around and took a radical step: start with any commutative ring, consider the set of its prime ideals, turn it into a topological space by introducing the Zariski topologyIn mathematics, the Zariski topology is a structure basic to algebraic geometry, especially since 1950. It is named after its originator, Oscar Zariski. The Zariski topology is defined by defining the closed sets to be the sets consisting of the mutual ze, and study the algebraic geometry of these quite general objects. Others did not see the point of this generality and Krull abandoned it.
Andre Weil was especially interested in algebraic geometry over finite fieldIn abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theos and other rings. In the 1940Events January-February January 5 FM radio is demonstrated to the FCC for the first time. January 6 World War II: Mass execution of Poles, committed by Germans in the Poznan, Warthegau. January 12 World War II: Russia bombs cities in Finland. February 2 Fs he returned to the prime ideal approach; he needed an abstract variety (outside projective spaceProjective geometry In mathematics, a projective space is a fundamental construction from any vector space. It generalises the projective plane that may be constructed from a three-dimensional vector space, over any field. While the theory of projective p) for foundational reasons, particularly for the existence in an algebraic setting of the Jacobian variety. In Weil's main foundational book, generic points are constructed by taking points in a very large algebraically closed field, called a universal domain.Around 1942 Oscar Zariski had defined an abstract Zariski space from the function field of an algebraic variety, for the needs of birational geometry: this is like a direct limit of ordinary varieties (under 'blowing up'), and the construction, reminiscent of locale theory , used valuation rings as points.
In the 1950s, Jean-Pierre Serre and Chevalley-Nagata, motivated by the Weil conjectures relating number theory and algebraic geometry, pursued similar approaches with prime ideals as points. According to Pierre Cartier , the word scheme was first used in the 1956 Chevalley Seminar, in which Chevalley was pursuing Zariski's ideas; and it was Martineau who suggested to Serre the move to the current spectrum of a ring in general.
Alexander Grothendieck then gave the decisive definition. He defines the spectrum of a commutative ring as the space of prime ideals with Zariski topology, but augments it with a sheaf of rings: to every Zariski-open set he defines a commutative ring, thought of as the ring of "polynomial functions" defined on that set. These objects are the "affine schemes"; a general scheme is then obtained by "gluing together" several such affine schemes, in analogy to the fact that projective varieties can be obtained by gluing together affine varieties.See also the article on spectrum of a ring for a motivation of the paradigm "points are prime ideals".
The generality of the scheme concept was initially criticized: some schemes are extremely far removed from having any geometrical interpretation. Grothendieck and Dieudonné studied the category of all schemes, and Grothendieck's student Pierre Deligne later wrote that admitting bizarre schemes made the whole category of schemes much nicer.
The evolution of the scheme concept was not the end of the road. Subsequent work on algebraic space s and algebraic stacks by Deligne, Mumford, and Michael Artin , originally in the context of moduli problems, have significantly enhanced the geometric flexibility of modern algebraic geometry. Recent ideas about higher algebraic stacks and derived algebraic geometry promise to further expand the algebraic reach of geometric intuition, bringing algebraic geometry closer in spirit to algebraic topology and homotopy theory.