 Scheme (mathematics)

In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern algebraic geometry. Technically, 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.
Contents
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 nonmaximal 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 had 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, JeanPierre 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
See also: spectrum of a ringAlexander Grothendieck then gave the decisive definition, bringing to a conclusion a generation of experimental suggestions and partial developments. He defined 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 Zariskiopen set he assigns 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 general varieties can be obtained by gluing together affine varieties.
The generality of the scheme concept was initially criticized: some schemes are removed from having straightforward geometrical interpretation, which made the concept difficult to grasp. However, admitting arbitrary schemes makes the whole category of schemes betterbehaved. Moreover, natural considerations regarding, for example, moduli spaces, lead to schemes which are "nonclassical". The occurrence of these schemes that are not varieties (nor built up simply from varieties) in problems that could be posed in classical terms made for the gradual acceptance of the new foundations of the subject.
Subsequent work on algebraic spaces and algebraic stacks by Deligne, Mumford, and Michael Artin, originally in the context of moduli problems, has further enhanced the geometric flexibility of modern algebraic geometry. Grothendieck advocated certain types of ringed toposes as generalisations of schemes, and following his proposals relative schemes over ringed toposes were developed by M. Hakim. Recent ideas about higher algebraic stacks and homotopical or derived algebraic geometry have regard to further expanding the algebraic reach of geometric intuition, bringing algebraic geometry closer in spirit to homotopy theory.
Definitions
An affine scheme is a locally ringed space isomorphic to Spec(A) for some commutative ring 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.
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
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.
Types of schemes
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 detail on the development of scheme theory, which quickly becomes technically demanding, see first glossary of scheme theory.
O_{X} modules
Just as the Rmodules 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.
References
 David Eisenbud; Joe Harris (1998). The Geometry of Schemes. SpringerVerlag. ISBN 0387986375.
 Robin Hartshorne (1997). Algebraic Geometry. SpringerVerlag. ISBN 0387902449.
 David Mumford (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians (2nd ed. ed.). SpringerVerlag. doi:10.1007/b62130. ISBN 354063293X.
 Qing Liu (2002). Algebraic Geometry and Arithmetic Curves. Oxford University Press. ISBN 0198502842.
Categories: Scheme theory
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