Scheme (mathematics)

In mathematics, schemes connect the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck in 1960 in his treatise Éléments de géométrie algébrique, with the aim of 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).^[1] Schemes enlarge the notion of algebraic variety to include nilpotent elements (the equations x = 0 and x² = 0 define the same points, but different schemes), and "varieties" defined over any commutative ring.

To be technically precise, a scheme is a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra (spaces of prime ideals) of commutative rings along their open subsets. In other words, it is a locally ringed space which is locally a spectrum of a commutative ring. There are many ways one can qualify a scheme. According to a basic idea of Grothendieck, conditions should be applied to a morphism of schemes. Any scheme S has a unique morphism to Spec(Z), so this attitude, part of the relative point of view, doesn't lose anything. For details on the development of scheme theory, which quickly becomes technically demanding, see first glossary of scheme theory.

History and motivation

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, one for each subvariety. 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 topology, and study the algebraic geometry of these quite general objects. Others did not see the point of this generality and Krull abandoned it.

André Weil was especially interested in algebraic geometry over finite fields and other rings. In the 1940s he returned to the prime ideal approach; he needed an abstract variety (outside projective space) for foundational reasons, particularly for the existence in an algebraic setting of the Jacobian variety. In Weil's main foundational book (1946), generic points are constructed by taking points in a very large algebraically closed field, called a universal domain.

In 1944 Oscar Zariski defined an abstract Zariski–Riemann 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, Claude Chevalley and Masayoshi Nagata, motivated largely 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 André Martineau who suggested to Serre the move to the current spectrum of a ring in general.

Modern definitions of the objects of algebraic geometry

Definitions

An affine scheme is a locally ringed space isomorphic to the spectrum of a commutative ring. We denote the spectrum of a commutative ring A by Spec(A). A scheme is a locally ringed space X admitting a covering by open sets U_i, such that the restriction of the structure sheaf O_X to each U_i is an affine scheme. Therefore one may think of a scheme as being covered by "coordinate charts" of affine schemes. The above formal definition means exactly that schemes are obtained by glueing together affine schemes for 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 category of schemes

Schemes form a category if we take as morphisms the morphisms of locally ringed spaces. (See also: morphism of schemes.)

Morphisms from schemes to affine schemes are completely understood in terms of ring homomorphisms by the following contravariant adjoint pair: For every scheme X and every commutative ring A we have a natural equivalence

\operatorname{Hom}_{\rm Schemes}(X, \operatorname{Spec}(A)) \cong \operatorname{Hom}_{\rm CRing}(A, {\mathcal O}_X(X)).

Since Z is an initial object in the category of rings, the category of schemes has Spec(Z) as a final object.

The category of schemes has finite products, but one has to be careful: the underlying topological space of the product scheme of (X, O_X) and (Y, O_Y) is normally not equal to the product of the topological spaces X and Y. In fact, the underlying topological space of the product scheme often has more points than the product of the underlying topological spaces. For example, if K is the field with nine elements, then Spec K × Spec K ≈ Spec (K ⊗_Z K) ≈ Spec (K ⊗_Z/3Z K) ≈ Spec (K × K), a set with two elements, though Spec K has only a single element.

For a scheme $S$ , the category of schemes over $S$ has also fibre products, and since it has a final object $S$ , it follows that it has finite limits.

O_X modules

Main article: Sheaf of modules

Just as the R-modules are central in commutative algebra when studying the commutative ring R, so are the O_X-modules central in the study of the scheme X with structure sheaf O_X. (See locally ringed space for a definition of O_X-modules.) The category of O_X-modules is abelian. Of particular importance are the coherent sheaves on X, which arise from finitely generated (ordinary) modules on the affine parts of X. The category of coherent sheaves on X is also abelian.

The sections of the structure sheaf O_X of X are called regular functions, which are defined on each open subsets U in X. The invertible subsheaf of O_X, denoted O^*_X, consists only of the invertible germs of regular functions under the multiplication. In most situations, the sheaf K_X is defined on an open affine subset Spec A of X as the total quotient rings Q(A) (though there are cases where the definition is more complicated).^[2] The sections of K_X are called rational functions on X. The invertible subsheaf of K_X is denoted by K^*_X. The equivalent class of this invertible sheaf turns to be an abelian group with tensor products and isomorphic to H¹(X, O^*_X), which is called Picard group. On projective varieties the sections of the structure sheaf O_X defined on each open subsets U of X are also called regular functions though there are no global sections except for constants.

Generalizations

A commonly used generalization of schemes are the algebraic stacks. All schemes are algebraic stacks, but the category of algebraic stacks is richer in that it contains many quotient objects and moduli spaces that cannot be constructed as schemes; stacks can also have negative dimension. Standard constructions of scheme theory, such as sheaves and étale cohomology, can be extended to algebraic stacks.

Notes

↑ Introduction of the first edition of Éléments de géométrie algébrique
↑ Kleiman, Misconceptions about K_X, L'Enseignement Mathematique.

References

David Eisenbud; Joe Harris (1998). The Geometry of Schemes. Springer-Verlag. ISBN 0-387-98637-5.
Robin Hartshorne (1997). Algebraic Geometry. Springer-Verlag. ISBN 0-387-90244-9.
David Mumford (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians (2nd ed.). Springer-Verlag. doi:10.1007/b62130. ISBN 3-540-63293-X.
Qing Liu (2002). Algebraic Geometry and Arithmetic Curves. Oxford University Press. ISBN 0-19-850284-2.