Affine scheme

A generalization of the concept of an affine variety, which plays the role of a local object in the theory of schemes. Let be a commutative ring with a unit. An affine scheme consists of a topological space and a sheaf of rings on . Here, is the set of all prime ideals of (called the points of the affine scheme), provided with the Zariski topology (or, which is the same, with the spectral topology), in which the basis of open sets is constituted by the subsets , where runs through the elements of . The sheaf of local rings is defined by the condition , where is the localization of the ring with respect to the multiplicative system , cf. Localization in a commutative algebra.

Affine schemes were first introduced by A. Grothendieck [1], who created the theory of schemes. A scheme is a ringed space which is locally isomorphic to an affine scheme.

An affine scheme is called Noetherian (integral, reduced, normal, or regular, respectively) if the ring is Noetherian (integral, without nilpotents, integrally closed, or regular, respectively). An affine scheme is called connected (irreducible, discrete, or quasi-compact, respectively) if the topological space also has these properties. The space of an affine scheme is always compact (and usually not Hausdorff).

The affine schemes form a category if the morphisms of these schemes, considered as locally ringed spaces, are considered as morphisms of affine schemes. Each homomorphism of rings defines a morphism of affine schemes: , consisting of the continuous mapping ( for ), and a homomorphism of sheaves of rings , which transforms the section of the sheaf over the set into the section . The morphisms of an arbitrary scheme into an affine scheme () (which are also called -valued points of ) are in a one-to-one correspondence with the homomorphisms of rings ; thus, the correspondence is a contravariant functor from the category of commutative rings with a unit into the category of affine schemes, which establishes an anti-equivalence of these categories. In particular, in the category of affine schemes there are finite direct sums and fibre products, dual to the constructions of the direct sum and the tensor product of rings. The morphisms of affine schemes which correspond to surjective homomorphisms of rings are called closed imbeddings of affine schemes.

The most important examples of affine schemes are affine varieties; other examples are affine group schemes (cf. Group scheme).

In a manner similar to the construction of the sheaf it is possible to construct, for any -module , a sheaf of -modules on for which

Such sheaves are called quasi-coherent. The category of -modules is equivalent to the category of quasi-coherent sheaves of -modules on ; projective modules correspond to locally free sheaves. The cohomology spaces of quasi-coherent sheaves over an affine scheme are described by Serre's theorem:

The converse of this theorem (Serre's criterion for affinity) states that if is a compact separable scheme, and if for any quasi-coherent sheaf of -modules , then is an affine scheme. Other criteria for affinity also exist [1], [4].

References

[1]	A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" Publ. Math. IHES , 4 (1960)
[2]	J. Dieudonné, "Algebraic geometry" Adv. in Math. , 1 (1969) pp. 233–321
[3]	Yu.I. Manin, "Lectures on algebraic geometry" , 1 , Moscow (1970) (In Russian)
[4]	J. Goodman, R. Hartshorne, "Schemes with finite-dimensional cohomology groups" Amer. J. Math. , 91 (1969) pp. 258–266

Comments

Reference [a1] is, of course, standard. It replaces [3]. An alternative to [1] is [a2].

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977)
[a2]	A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" , I. Le langage des schémes , Springer (1971)