Spectrum of A Ring - Sheaves and Schemes

Sheaves and Schemes

Given the space X=Spec(R) with the Zariski topology, the structure sheaf O_X is defined on the D_f by setting Γ(D_f, O_X) = R_f, the localization of R at the multiplicative system {1,f,f2,f3,...}. It can be shown that this satisfies the necessary axioms to be a B-Sheaf. Next, if U is the union of {D_fi}_i∈I, we let Γ(U,O_X) = lim_i∈I R_fi, and this produces a sheaf; see the Gluing axiom article for more detail.

If R is an integral domain, with field of fractions K, then we can describe the ring Γ(U,O_X) more concretely as follows. We say that an element f in K is regular at a point P in X if it can be represented as a fraction f = a/b with b not in P. Note that this agrees with the notion of a regular function in algebraic geometry. Using this definition, we can describe Γ(U,O_X) as precisely the set of elements of K which are regular at every point P in U.

If P is a point in Spec(R), that is, a prime ideal, then the stalk at P equals the localization of R at P, and this is a local ring. Consequently, Spec(R) is a locally ringed space.

Every locally ringed space isomorphic to one of this form is called an affine scheme. General schemes are obtained by "gluing together" several affine schemes.