Editing Spectrum of a ring (section)

=== 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>