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 10.126.11. Suppose $R \to S$ is a local homomorphism of local rings. Assume that $S$ is essentially of finite presentation over $R$. Then there exists a directed set $(\Lambda , \leq )$, and a system of local homomorphism $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 $S_\lambda \otimes _{R_\lambda } R_\mu $ at a prime ideal.

Proof. By assumption we may choose an isomorphism $\Phi : (R[x_1, \ldots , x_ n]/I)_{\mathfrak q} \to S$ where $I \subset R[x_1, \ldots , x_ n]$ is a finitely generated ideal, and $\mathfrak q \subset R[x_1, \ldots , x_ n]/I$ is a prime. (Note that $R \cap \mathfrak q$ is equal to the maximal ideal $\mathfrak m$ of $R$.) We also choose generators $f_1, \ldots , f_ m \in I$ for the ideal $I$. Write $R$ in any way as a colimit $R = \mathop{\mathrm{colim}}\nolimits R_\lambda $ over a directed set $(\Lambda , \leq )$, with each $R_\lambda $ local and essentially of finite type over $\mathbf{Z}$. There exists some $\lambda _0 \in \Lambda $ such that $f_ j$ is the image of some $f_{j, \lambda _0} \in R_{\lambda _0}[x_1, \ldots , x_ n]$. For all $\lambda \geq \lambda _0$ denote $f_{j, \lambda } \in R_{\lambda }[x_1, \ldots , x_ n]$ the image of $f_{j, \lambda _0}$. Thus we obtain a system of ring maps

\[ R_\lambda [x_1, \ldots , x_ n]/(f_{1, \lambda }, \ldots , f_{n, \lambda }) \to R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ n) \to S \]

Set $\mathfrak q_\lambda $ the inverse image of $\mathfrak q$. Set $S_\lambda = (R_\lambda [x_1, \ldots , x_ n]/ (f_{1, \lambda }, \ldots , f_{n, \lambda }))_{\mathfrak q_\lambda }$. We leave it to the reader to see that this works. $\square$


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