The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 25.6.7. The category of affine schemes has finite products, and fibre products. In other words, it has finite limits. Moreover, the products and fibre products in the category of affine schemes are the same as in the category of locally ringed spaces. In a formula, we have (in the category of locally ringed spaces)

\[ \mathop{\mathrm{Spec}}(R) \times \mathop{\mathrm{Spec}}(S) = \mathop{\mathrm{Spec}}(R \otimes _{\mathbf{Z}} S) \]

and given ring maps $R \to A$, $R \to B$ we have

\[ \mathop{\mathrm{Spec}}(A) \times _{\mathop{\mathrm{Spec}}(R)} \mathop{\mathrm{Spec}}(B) = \mathop{\mathrm{Spec}}(A \otimes _ R B). \]

Proof. This is just an application of Lemma 25.6.4. First of all, by that lemma, the affine scheme $\mathop{\mathrm{Spec}}(\mathbf{Z})$ is the final object in the category of locally ringed spaces. Thus the first displayed formula follows from the second. To prove the second note that for any locally ringed space $X$ we have

\begin{eqnarray*} \mathop{Mor}\nolimits (X, \mathop{\mathrm{Spec}}(A \otimes _ R B)) & = & \mathop{\mathrm{Hom}}\nolimits (A \otimes _ R B, \mathcal{O}_ X(X)) \\ & = & \mathop{\mathrm{Hom}}\nolimits (A, \mathcal{O}_ X(X)) \times _{\mathop{\mathrm{Hom}}\nolimits (R, \mathcal{O}_ X(X))} \mathop{\mathrm{Hom}}\nolimits (B, \mathcal{O}_ X(X)) \\ & = & \mathop{Mor}\nolimits (X, \mathop{\mathrm{Spec}}(A)) \times _{\mathop{Mor}\nolimits (X, \mathop{\mathrm{Spec}}(R))} \mathop{Mor}\nolimits (X, \mathop{\mathrm{Spec}}(B)) \end{eqnarray*}

which proves the formula. See Categories, Section 4.6 for the relevant definitions. $\square$


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