Editing Spectrum of a ring (section)

== Global or relative Spec ==
There is a relative version of the functor <math>\operatorname{Spec}</math> called global <math>\operatorname{Spec}</math>, or relative <math>\operatorname{Spec}</math>.  If <math>S</math> is a scheme, then relative <math>\operatorname{Spec}</math> is denoted by <math>\underline{\operatorname{Spec}}_S</math> or <math>\mathbf{Spec}_S</math>. If <math>S</math> is clear from the context, then relative Spec may be denoted by <math>\underline{\operatorname{Spec}}</math> or <math>\mathbf{Spec}</math>.  For a scheme <math>S</math> and a [[quasi-coherent sheaf|quasi-coherent]] [[sheaf of algebras|sheaf of <math>\mathcal{O}_S</math>-algebras]] <math>\mathcal{A}</math>, there is a scheme <math>\underline{\operatorname{Spec}}_S(\mathcal{A})</math> and a morphism <math>f : \underline{\operatorname{Spec}}_S(\mathcal{A}) \to S</math> such that for every open affine <math>U \subseteq S</math>, there is an isomorphism <math>f^{-1}(U) \cong \operatorname{Spec}(\mathcal{A}(U))</math>, and such that for open affines <math>V \subseteq U</math>, the inclusion <math>f^{-1}(V) \to f^{-1}(U)</math> is induced by the restriction map <math>\mathcal{A}(U) \to \mathcal{A}(V)</math>.  That is, as ring homomorphisms induce opposite maps of spectra, the restriction maps of a sheaf of algebras induce the inclusion maps of the spectra that make up the '''Spec''' of the sheaf.

Global Spec has a universal property similar to the universal property for ordinary Spec.  More precisely, just as Spec and the global section functor are contravariant right adjoints between the category of commutative rings and schemes, global Spec and the direct image functor for the structure map are contravariant right adjoints between the category of commutative <math>\mathcal{O}_S</math>-algebras and schemes over <math>S</math>.{{dubious|don't you need quasi-coherent?|date=January 2018}}  In formulas,
:<math>\operatorname{Hom}_{\mathcal{O}_S\text{-alg}}(\mathcal{A}, \pi_*\mathcal{O}_X) \cong \operatorname{Hom}_{\text{Sch}/S}(X, \mathbf{Spec}(\mathcal{A})),</math>
where <math>\pi \colon X \to S</math> is a morphism of schemes.

=== Example of a relative Spec ===
The relative spec is the correct tool for parameterizing the family of lines through the origin of <math>\mathbb{A}^2_\mathbb{C}</math> over <math>X = \mathbb{P}^1_{a,b}.</math> Consider the sheaf of algebras <math>\mathcal{A} = \mathcal{O}_X[x,y],</math> and let <math>\mathcal{I} = (ay-bx)</math> be a sheaf of ideals of <math>\mathcal{A}.</math> Then the relative spec <math>\underline{\operatorname{Spec}}_X(\mathcal{A}/\mathcal{I}) \to \mathbb{P}^1_{a,b}</math> parameterizes the desired family. In fact, the fiber over <math>[\alpha:\beta]</math> is the line through the origin of <math>\mathbb{A}^2</math> containing the point <math>(\alpha,\beta).</math> Assuming <math>\alpha \neq 0,</math> the fiber can be computed by looking at the composition of pullback diagrams
:<math>\begin{matrix}
\operatorname{Spec}\left( \frac{\mathbb{C}[x,y]}{\left(y-\frac{\beta}{\alpha}x\right)} \right) & \to & \operatorname{Spec}\left( \frac{\mathbb{C}\left[\frac{b}{a}\right] [x,y]}{\left(y-\frac{b}{a}x\right)} \right) & \to & \underline{\operatorname{Spec}}_X\left( \frac{\mathcal{O}_X[x,y]}{\left(ay-bx\right)} \right)\\
\downarrow & & \downarrow & &  \downarrow \\
\operatorname{Spec}(\mathbb{C})& \to & \operatorname{Spec}\left(\mathbb{C}\left[\frac{b}{a}\right]\right)=U_a & \to &  \mathbb{P}^1_{a,b}
\end{matrix}</math>
where the composition of the bottom arrows
:<math>\operatorname{Spec}(\mathbb{C})\xrightarrow{[\alpha:\beta]} \mathbb{P}^1_{a,b}</math>
gives the line containing the point <math>(\alpha,\beta)</math> and the origin. This example can be generalized to parameterize the family of lines through the origin of <math>\mathbb{A}^{n+1}_\mathbb{C}</math> over <math>X = \mathbb{P}^n_{a_0,...,a_n}</math> by letting <math>\mathcal{A} = \mathcal{O}_X[x_0,...,x_n]</math> and <math>\mathcal{I} = \left( 2\times 2 \text{ minors of } \begin{pmatrix}a_0 & \cdots & a_n \\
x_0 & \cdots & x_n\end{pmatrix} \right).</math>