In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of an algebraic variety in several ways, such as taking account of multiplicities (the equations and 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).Introduction of the first edition of "Éléments de géométrie algébrique". 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 of schemes (called a scheme over the base ), 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 . 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.
Development 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 : the maximal ideals in the polynomial ring are in one-to-one correspondence with the set of -tuples of elements of , and the prime ideals correspond to the irreducible algebraic sets in , 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.
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. From the 1920s to the 1940s, 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. 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 manifolds 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, Chow and Matsusaka.)
The algebraic geometers of the 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. 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. According to 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.
Origin of schemes The theory took its definitive form in Grothendieck's Éléments de géométrie algébrique (EGA) and the later Séminaire de géométrie algébrique (SGA), bringing to a conclusion a generation of experimental suggestions and partial developments. Grothendieck defined the spectrum of a commutative ring as the space of prime ideals of with a natural topology (known as the Zariski topology), but augmented it with a sheaf of rings: to every open subset he assigned a commutative ring , which may be thought of as the coordinate ring of regular functions on . These objects are the affine schemes; a general scheme is then obtained by "gluing together" affine schemes.
Much of algebraic geometry focuses on projective or quasi-projective varieties over a field , most often over the complex numbers. Grothendieck developed a large body of theory for arbitrary schemes extending much of the geometric intuition for varieties. For example, it is common to construct a moduli space first as a scheme, and only later study whether it is a more concrete object such as a projective variety. Applying Grothendieck's theory to schemes over the integers and other number fields led to powerful new perspectives in number theory.
Definition An affine scheme is a locally ringed space isomorphic to the spectrum of a commutative ring . A scheme is a locally ringed space admitting a covering by open sets , such that each (as a locally ringed space) is an affine scheme. In particular, comes with a sheaf , which assigns to every open subset a commutative ring called the ring of regular functions on . One can think of a scheme as being covered by "coordinate charts" that are affine schemes. The definition means exactly that schemes are obtained by gluing together affine schemes using the Zariski topology.
In the early days, this was called a prescheme, and a scheme was defined to be a separated prescheme. The term prescheme has fallen out of use, but can still be found in older books, such as Grothendieck's "Éléments de géométrie algébrique" and Mumford's "Red Book". The sheaf properties of mean that its elements, which are not necessarily functions, can nevertheless be patched together from their restrictions in the same way as functions.
A basic example of an affine scheme is affine -space over a field , for a natural number . By definition, is the spectrum of the polynomial ring . In the spirit of scheme theory, affine -space can in fact be defined over any commutative ring , meaning .
The category of schemes Schemes form a category, with morphisms defined as morphisms of locally ringed spaces. (See also: morphism of schemes.) For a scheme Y, a scheme X over Y (or a Y-scheme) means a morphism X → Y of schemes. A scheme X over a commutative ring R means a morphism X → Spec(R).
An algebraic variety over a field k can be defined as a scheme over k with certain properties. There are different conventions about exactly which schemes should be called varieties. One standard choice is that a variety over k means an integral separated scheme of finite type over k..
A morphism f: X → Y of schemes determines a pullback homomorphism on the rings of regular functions, f*: O(Y) → O(X). In the case of affine schemes, this construction gives a one-to-one correspondence between morphisms Spec(A) → Spec(B) of schemes and ring homomorphisms B → A. In this sense, scheme theory completely subsumes the theory of commutative rings.
Since Z is an initial object in the category of commutative rings, the category of schemes has Spec(Z) as a terminal object.
For a scheme X over a commutative ring R, an R-point of X means a section of the morphism X → Spec(R). One writes X(R) for the set of R-points of X. In examples, this definition reconstructs the old notion of the set of solutions of the defining equations of X with values in R. When R is a field k, X(k) is also called the set of k-rational points of X.
More generally, for a scheme X over a commutative ring R and any commutative R-algebra S, an S-point of X means a morphism Spec(S) → X over R. One writes X(S) for the set of S-points of X. (This generalizes the old observation that given some equations over a field k, one can consider the set of solutions of the equations in any field extension E of k.) For a scheme X over R, the assignment S ↦ X(S) is a functor from commutative R-algebras to sets. It is an important observation that a scheme X over R is determined by this functor of points.
The fiber product of schemes always exists. That is, for any schemes X and Z with morphisms to a scheme Y, the categorical fiber product exists in the category of schemes. If X and Z are schemes over a field k, their fiber product over Spec(k) may be called the product X × Z in the category of k-schemes. For example, the product of affine spaces and over k is affine space over k.
Since the category of schemes has fiber products and also a terminal object Spec(Z), it has all finite limits.
Examples Here and below, all the rings considered are commutative.
Affine space