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Scheme (mathematics)
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise Éléments de géométrie algébrique (EGA); 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. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem.
Schemes elaborate the fundamental idea that an algebraic variety is best analyzed through the coordinate ring of regular algebraic functions defined on it (or on its subsets), and each subvariety corresponds to the ideal of functions which vanish on the subvariety. Intuitively, a scheme is a topological space consisting of closed points which correspond to geometric points, together with non-closed points which are generic points of irreducible subvarieties. The space is covered by an atlas of open sets, each endowed with a coordinate ring of regular functions, with specified coordinate changes between the functions over intersecting open sets. Such a structure is called a ringed space or a sheaf of rings. The cases of main interest are the Noetherian schemes, in which the coordinate rings are Noetherian rings.
Formally, a scheme is a ringed space covered by affine schemes. An affine scheme is the spectrum of a commutative ring; its points are the prime ideals of the ring, and its closed points are maximal ideals. The coordinate ring of an affine scheme is the ring itself, and the coordinate rings of open subsets are rings of fractions.
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 the base Y ), rather than for an individual scheme. For example, in studying algebraic surfaces, it can be useful to consider families of algebraic surfaces over any scheme Y. In many cases, the family of all varieties of a given type can itself be viewed as a variety or scheme, known as a moduli space.
For some of the detailed definitions in the theory of schemes, see the glossary of scheme theory.
The origins of algebraic geometry mostly lie in the study of polynomial equations over the real numbers. 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 of complex numbers, which has the advantage of being algebraically closed. 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, and more generally over 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 k : the maximal ideals in the polynomial ring k[x1, ... , xn] are in one-to-one correspondence with the set kn of n-tuples of elements of k, and the prime ideals correspond to the irreducible algebraic sets in kn, known as affine varieties. Motivated by these ideas, Emmy Noether and Wolfgang Krull developed commutative algebra in the 1920s and 1930s. 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 dimension of a commutative ring in terms of prime ideals and, at least when the ring is Noetherian, he proved that this definition satisfies many of the intuitive properties of geometric dimension.
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Scheme (mathematics)
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise Éléments de géométrie algébrique (EGA); 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. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem.
Schemes elaborate the fundamental idea that an algebraic variety is best analyzed through the coordinate ring of regular algebraic functions defined on it (or on its subsets), and each subvariety corresponds to the ideal of functions which vanish on the subvariety. Intuitively, a scheme is a topological space consisting of closed points which correspond to geometric points, together with non-closed points which are generic points of irreducible subvarieties. The space is covered by an atlas of open sets, each endowed with a coordinate ring of regular functions, with specified coordinate changes between the functions over intersecting open sets. Such a structure is called a ringed space or a sheaf of rings. The cases of main interest are the Noetherian schemes, in which the coordinate rings are Noetherian rings.
Formally, a scheme is a ringed space covered by affine schemes. An affine scheme is the spectrum of a commutative ring; its points are the prime ideals of the ring, and its closed points are maximal ideals. The coordinate ring of an affine scheme is the ring itself, and the coordinate rings of open subsets are rings of fractions.
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 the base Y ), rather than for an individual scheme. For example, in studying algebraic surfaces, it can be useful to consider families of algebraic surfaces over any scheme Y. In many cases, the family of all varieties of a given type can itself be viewed as a variety or scheme, known as a moduli space.
For some of the detailed definitions in the theory of schemes, see the glossary of scheme theory.
The origins of algebraic geometry mostly lie in the study of polynomial equations over the real numbers. 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 of complex numbers, which has the advantage of being algebraically closed. 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, and more generally over 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 k : the maximal ideals in the polynomial ring k[x1, ... , xn] are in one-to-one correspondence with the set kn of n-tuples of elements of k, and the prime ideals correspond to the irreducible algebraic sets in kn, known as affine varieties. Motivated by these ideas, Emmy Noether and Wolfgang Krull developed commutative algebra in the 1920s and 1930s. 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 dimension of a commutative ring in terms of prime ideals and, at least when the ring is Noetherian, he proved that this definition satisfies many of the intuitive properties of geometric dimension.