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 10.126.10. Suppose $R \to S$ is a local homomorphism of local rings. Assume that $S$ is essentially of finite type over $R$. Then 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 $.

  4. For each $\lambda \leq \mu $ the map $S_\lambda \otimes _{R_\lambda } R_\mu \to S_\mu $ presents $S_\mu $ as the localization of a quotient of $S_\lambda \otimes _{R_\lambda } R_\mu $.

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 $x_1, \ldots , x_ n \in S$ be elements such that $S$ is a localization of the sub $R$-algebra of $S$ generated by $x_1, \ldots , x_ n$. In other words, $S$ is a quotient of a localization of the polynomial ring $R[x_1, \ldots , x_ n]$.

Let $\Lambda = \{ A \subset R \mid \# A < \infty \} $ be the set of finite subsets of $R$. As partial ordering we take the inclusion relation. For each $\lambda = A \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 $\varphi (a)$, $a \in A$ and the elements $x_1, \ldots , x_ n$. 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

\[ \xymatrix{ \varphi (A) \amalg \{ x_ i\} \ar[r] & S'_\lambda \ar[r] & S_\lambda \ar[r] & S \\ A \ar[r] \ar[u] & R'_\lambda \ar[r] \ar[u] & R_\lambda \ar[r] \ar[u] & R \ar[u] } \]

It is clear that if $A \subset B$ corresponds to $\lambda \leq \mu $ in $\Lambda $, then there are canonical maps $R_\lambda \to R_\mu $, and $S_\lambda \to S_\mu $ and we obtain a system over the directed set $\Lambda $.

The assertion that $R = \mathop{\mathrm{colim}}\nolimits R_\lambda $ is clear because all the maps $R_\lambda \to R$ are injective and any element of $R$ eventually is in the image. The same argument works for $S = \mathop{\mathrm{colim}}\nolimits S_\lambda $. Assertions (2), (3) are true by construction. The final assertion holds because clearly the maps $S'_\lambda \otimes _{R'_\lambda } R'_\mu \to S'_\mu $ are surjective. $\square$


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