Editing Spectrum of a ring (section)

== Sheaves and schemes ==

Given the space <math>X = \operatorname{Spec}(R)</math> with the Zariski topology, the '''structure sheaf''' <math>\mathcal O_X</math> is defined on the distinguished open subsets <math>D_f</math> by setting <math>\Gamma(D_f,\mathcal O_X) = R_f,</math> the [[localization of a ring|localization]] of <math>R</math> by the powers of <math>f</math>. It can be shown that this defines a [[B-sheaf]] and therefore that it defines a [[sheaf (mathematics)|sheaf]]. In more detail, the distinguished open subsets are a [[base (topology)|basis]] of the Zariski topology, so for an arbitrary open set <math>U</math>, written as the union of <math display=inline>U=\bigcup_{i\in I}D_{f_i}</math>, we set <math display=inline>\Gamma(U,\mathcal O_X)=\varprojlim_{i\in I}R_{f_i},</math> where <math>\varprojlim</math> denotes the [[inverse limit]] with respect to the natural ring homomorphisms <math>R_f\to R_{fg}.</math> One may check that this [[presheaf]] is a sheaf, so <math>\operatorname{Spec}(R)</math> is a [[ringed space]]. Any ringed space isomorphic to one of this form is called an '''affine scheme'''. General [[scheme (mathematics)|schemes]] are obtained by gluing affine schemes together.

Similarly, for a [[module (mathematics)|module]] <math>M</math> over the ring <math>R</math>, we may define a sheaf <math>\widetilde{M}</math> on <math>\operatorname{Spec}(R)</math>. On the distinguished open subsets set <math>\Gamma(D_f,\widetilde M)=M_f,</math> using the [[localization of a module]]. As above, this construction extends to a presheaf on all open subsets of <math>\operatorname{Spec}(R)</math> and satisfies the [[gluing axiom]]. A sheaf of this form is called a [[quasicoherent sheaf]].

If <math>\mathfrak{p}</math> is a point in <math>\operatorname{Spec}(R)</math>, that is, a prime ideal, then the [[stalk (mathematics)|stalk]] of the structure sheaf at <math>\mathfrak{p}</math> equals the [[localization of a ring|localization]] of <math>R</math> at the ideal <math>\mathfrak{p}</math>, which is generally denoted <math>R_{\mathfrak{p}}</math>, and this is a [[local ring]].  Consequently, <math>\operatorname{Spec}(R)</math> is a [[locally ringed space]].

If <math>R</math> is an [[integral domain]], with [[field of fractions]] <math>K</math>, then we can describe the ring <math>\Gamma(U, \mathcal O_X)</math> more concretely as follows.  We say that an element <math>f</math> in <math>K</math> is regular at a point <math>\mathfrak{p}</math> in <math>X=\operatorname{Spec}{R}</math> if it can be represented as a fraction <math>f=a/b</math> with <math>b\notin\mathfrak{p}</math>.  Note that this agrees with the notion of a [[regular function]] in algebraic geometry.  Using this definition, we can describe <math>\Gamma(U, \mathcal O_X)</math> as precisely the set of elements of <math>K</math> that are regular at every point <math>\mathfrak{p}</math> in <math>U</math>.