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.2 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.3 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.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 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.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.5 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.6
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.7 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.8 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".9 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.10
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.11 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 .
Examples
Here and below, all the rings considered are commutative.
Affine space
Let k be an algebraically closed field. The affine space is the algebraic variety of all points with coordinates in k; its coordinate ring is the polynomial ring . The corresponding scheme is a topological space with the Zariski topology, whose closed points are the maximal ideals , the set of polynomials vanishing at . (This topology is compact and T1-separated, but not Hausdorff.) The scheme also contains a non-closed point for each non-maximal prime ideal , whose vanishing defines an irreducible subvariety ; the topological closure of the scheme point is the subscheme , which includes all the closed points of the subvariety, i.e. with , or equivalently . The closed subscheme has coordinate ring : that is , since prime ideals are in natural bijection with .
The scheme has a basis of open subsets given by the complements of hypersurfaces,
for irreducible polynomials . This set is endowed with its coordinate ring of regular functions
The sheaf structure on this open cover induces a unique sheaf which gives the usual ring of rational functions regular on a given open set .
Each ring element , a polynomial function on , also defines a function on the points of the scheme whose value at lies in the quotient ring , the residue ring. We define as the image of under the natural map . A maximal ideal gives the residue field , with the natural isomorphism , so that corresponds to the original value .
The vanishing locus of a polynomial is a hypersurface subvariety , also called a divisor, corresponding to the principal ideal with corresponding closed subscheme . For example, taking k to be the complex or real numbers, the equation defines a nodal cubic curve in the affine plane , corresponding to the scheme . Its singularity at can be analyzed via its , the pair of diagonal lines defined by , the lowest-degree terms of ; this has coordinate ring given by the of at the ideal :
Motivation for schemes
Here are some of the ways in which schemes go beyond older notions of algebraic varieties, and their significance.
- Field extensions. Given some polynomial equations in n variables over a field k, one can study the set X(k) of solutions of the equations in the product set kn. If the field k is algebraically closed (for example the complex numbers), then one can base algebraic geometry on sets such as X(k): define the Zariski topology on X(k), consider polynomial mappings between different sets of this type, and so on. But if k is not algebraically closed, then the set X(k) is not rich enough. Indeed, one can study the solutions X(E) of the given equations in any field extension E of k, but these sets are not determined by X(k) in any reasonable sense. For example, the plane curve X over the real numbers defined by x2 + y2 = −1 has X(R) empty, but X(C) not empty. (In fact, X(C) can be identified with C − 0.) By contrast, a scheme X over a field k has enough information to determine the set X(E) of E-rational points for every extension field E of k. (In particular, the closed subscheme of A2
R defined by x2 + y2 = −1 is a nonempty topological space.) - Generic point. The points of the affine line A1
C, as a scheme, are its complex points (one for each complex number) together with one generic point (whose closure is the whole scheme). The generic point is the image of a natural morphism Spec(C(x)) → A1
C, where C(x) is the field of in one variable. To see why it is useful to have an actual "generic point" in the scheme, consider the following example.
Coherent sheaves
Main article: Coherent sheaf
A central part of scheme theory is the notion of coherent sheaves, generalizing the notion of (algebraic) vector bundles. For a scheme X, one starts by considering the abelian category of OX-modules, which are sheaves of abelian groups on X that form a module over the sheaf of regular functions OX. In particular, a module M over a commutative ring R determines an associated OX-module ~M on X = Spec(R). A quasi-coherent sheaf on a scheme X means an OX-module that is the sheaf associated to a module on each affine open subset of X. Finally, a coherent sheaf (on a Noetherian scheme X, say) is an OX-module that is the sheaf associated to a finitely generated module on each affine open subset of X.
Coherent sheaves include the important class of vector bundles, which are the sheaves that locally come from finitely generated free modules. An example is the tangent bundle of a smooth variety over a field. However, coherent sheaves are richer; for example, a vector bundle on a closed subscheme of can be viewed as a coherent sheaf on that is zero outside (by the construction). In this way, coherent sheaves on a scheme include information about all closed subschemes of . Moreover, has good properties for coherent (and quasi-coherent) sheaves. The resulting theory of is perhaps the main technical tool in algebraic geometry.
Generalizations
Considered as its functor of points, a scheme is a functor that is a sheaf of sets for the Zariski topology on the category of commutative rings, and that, locally in the Zariski topology, is an affine scheme. This can be generalized in several ways. One is to use the étale topology. Michael Artin defined an algebraic space as a functor that is a sheaf in the étale topology and that, locally in the étale topology, is an affine scheme. Equivalently, an algebraic space is the quotient of a scheme by an étale equivalence relation. A powerful result, the Artin representability theorem, gives simple conditions for a functor to be represented by an algebraic space.20
A further generalization is the idea of a stack. Crudely speaking, algebraic stacks generalize algebraic spaces by having an algebraic group attached to each point, which is viewed as the automorphism group of that point. For example, any action of an algebraic group G on an algebraic variety X determines a quotient stack [X/G], which remembers the stabilizer subgroups for the action of G. More generally, moduli spaces in algebraic geometry are often best viewed as stacks, thereby keeping track of the automorphism groups of the objects being classified.
Grothendieck originally introduced stacks as a tool for the theory of descent. In that formulation, stacks are (informally speaking) sheaves of categories. From this general notion, Artin defined the narrower class of algebraic stacks (or "Artin stacks"), which can be considered geometric objects. These include (similar to in topology), for which the stabilizer groups are finite, and algebraic spaces, for which the stabilizer groups are trivial. The says that an algebraic stack with finite stabilizer groups has a that is an algebraic space.
References
- Arapura, Donu (2011), "Frobenius amplitude, ultraproducts, and vanishing on singular spaces", Illinois Journal of Mathematics, 55 (4): 1367–1384, arXiv:0806.1033, doi:10.1215/ijm/1373636688, MR 3082873
- Cartier, Pierre (2001), "A mad day's work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry", Bulletin of the American Mathematical Society, 38 (4): 389–408, doi:10.1090/S0273-0979-01-00913-2, MR 1848254
- Dieudonné, Jean (1985), History of Algebraic Geometry, Wadsworth, ISBN 978-0-534-03723-9,
the spectrum of the Gaussian integers , and the quadratic extension corresponds to a branched double cover , . The fiber over consists of -values which are roots of modulo , which may split, ramify, or remain inert. For example, at , the fiber splits into two points , since in ; or equivalently, the rational prime 5 factors into the Gaussian primes . At , we get one ramified double-point , since or with a unit. At , the fiber is inert, meaning is a prime ideal corresponding to in an extension field ; since we cannot distinguish between these values (they are symmetric under the Galois group), we should picture as two fused points. Overall, is a curve of degree 2, having in some sense a double fiber over each . Not all finite extensions of the integers appear in this way as quotients of , only those which are generated by a single element.
![Spec Z[x]](https://thumb.wikimedia.org/wikipedia/commons/thumb/1/11/SpecZx.png/500px-SpecZx.png?utm_source=en.wikipedia.org&utm_campaign=parser&utm_content=thumbnail)
