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 10.126.15. Suppose $R \to S$ is a ring map. Assume that $S$ is integral over $R$. Then there exists a directed set $(\Lambda , \leq )$, and a system of ring maps $R_\lambda \to S_\lambda $ such that

  1. The colimit of the system $R_\lambda \to S_\lambda $ is equal to $R \to S$.

  2. Each $R_\lambda $ is of finite type over $\mathbf{Z}$.

  3. Each $S_\lambda $ is of finite over $R_\lambda $.

Proof. Consider the set $\Lambda $ of pairs $(E, F)$ where $E \subset R$ is a finite subset, $F \subset S$ is a finite subset, and every element $f \in F$ is the root of a monic $P(X) \in R[X]$ whose coefficients are in $E$. Say $(E, F) \leq (E', F')$ if $E \subset E'$ and $F \subset F'$. Given $\lambda = (E, F) \in \Lambda $ set $R_\lambda \subset R$ equal to the $\mathbf{Z}$-subalgebra of $R$ generated by $E$ and $S_\lambda \subset S$ equal to the $\mathbf{Z}$-subalgebra generated by $F$ and the image of $E$ in $S$. It is clear that $R = \mathop{\mathrm{colim}}\nolimits R_\lambda $. We have $S = \mathop{\mathrm{colim}}\nolimits S_\lambda $ as every element of $S$ is integral over $S$. The ring maps $R_\lambda \to S_\lambda $ are finite by Lemma 10.35.5 and the fact that $S_\lambda $ is generated over $R_\lambda $ by the elements of $F$ which are integral over $R_\lambda $ by our condition on the pairs $(E, F)$. The lemma follows. $\square$


Comments (2)

Comment #3269 by Dennis Keeler on

Part (3) is missing the word "type" in "finite type."

Comment #3270 by Dario WeiƟmann on

No, and are constructed sucht that is finite and not just of finite type. (We take the algebra generated by finitely many integral elements)


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