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).

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.

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.

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 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.

[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] 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 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.

An affine scheme is a locally ringed space isomorphic to the spectrum

For a scheme X over a commutative ring R, an R-point of X means a section of the morphism X → Spec(R).

over k. Since the category of schemes has fiber products and also a terminal object Spec(Z), it has all finite limits.

Arakelov theory overcomes this obstacle by compactifying affine arithmetic schemes, adding points at infinity corresponding to valuations.

In this way, coherent sheaves on a scheme X include information about all closed subschemes of X.

Moreover, sheaf cohomology has good properties for coherent (and quasi-coherent) sheaves.

The resulting theory of coherent sheaf cohomology is perhaps the main technical tool in algebraic geometry.

[18][19] 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.

A powerful result, the Artin representability theorem, gives simple conditions for a functor to be represented by an algebraic space.

Another type of generalization is to enrich the structure sheaf, bringing algebraic geometry closer to homotopy theory.

These sheaves admit algebraic operations that are associative and commutative only up to an equivalence relation.

Taking the quotient by this equivalence relation yields the structure sheaf of an ordinary scheme.

Not taking the quotient, however, leads to a theory that can remember higher information, in the same way that derived functors in homological algebra yield higher information about operations such as tensor product and the Hom functor on modules.