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 10.126.9. Suppose $R \to S$ is a local homomorphism of local rings. There exists a directed set $(\Lambda , \leq )$, and a system of local homomorphisms $R_\lambda \to S_\lambda $ of local rings such that

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

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

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

Proof. Denote $\varphi : R \to S$ the ring map. Let $\mathfrak m \subset R$ be the maximal ideal of $R$ and let $\mathfrak n \subset S$ be the maximal ideal of $S$. Let

\[ \Lambda = \{ (A, B) \mid A \subset R, B \subset S, \# A < \infty , \# B < \infty , \varphi (A) \subset B \} . \]

As partial ordering we take the inclusion relation. For each $\lambda = (A, B) \in \Lambda $ we let $R'_\lambda $ be the sub $\mathbf{Z}$-algebra generated by $a \in A$, and we let $S'_\lambda $ be the sub $\mathbf{Z}$-algebra generated by $b$, $b \in B$. Let $R_\lambda $ be the localization of $R'_\lambda $ at the prime ideal $R'_\lambda \cap \mathfrak m$ and let $S_\lambda $ be the localization of $S'_\lambda $ at the prime ideal $S'_\lambda \cap \mathfrak n$. In a picture

\[ \xymatrix{ B \ar[r] & S'_\lambda \ar[r] & S_\lambda \ar[r] & S \\ A \ar[r] \ar[u] & R'_\lambda \ar[r] \ar[u] & R_\lambda \ar[r] \ar[u] & R \ar[u] }. \]

The transition maps are clear. We leave the proofs of the other assertions to the reader. $\square$


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