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 10.123.3. Let $R \to S$ be a ring map, $\mathfrak q$ a prime of $S$ lying over $\mathfrak p$ in $R$. If

  1. $R$ is Noetherian,

  2. $R \to S$ is of finite type, and

  3. $R \to S$ is quasi-finite at $\mathfrak q$,

then $R_\mathfrak p^\wedge \otimes _ R S = S_\mathfrak q^\wedge \times B$ for some $R_\mathfrak p^\wedge $-algebra $B$.

Proof. There exists a finite $R$-algebra $S' \subset S$ and an element $g \in S'$, $g \not\in \mathfrak q' = S' \cap \mathfrak q$ such that $S'_ g = S_ g$ and in particular $S'_{\mathfrak q'} = S_\mathfrak q$, see Lemma 10.122.14. We have

\[ R_\mathfrak p^\wedge \otimes _ R S' = (S'_{\mathfrak q'})^\wedge \times B' \]

by Lemma 10.96.8. Note that we have a commutative diagram

\[ \xymatrix{ R_\mathfrak p^\wedge \otimes _ R S \ar[r] & S_\mathfrak q^\wedge \\ R_\mathfrak p^\wedge \otimes _ R S' \ar[r] \ar[u] & (S'_{\mathfrak q'})^\wedge \ar[u] } \]

where the right vertical is an isomorphism and the lower horizontal arrow is the projection map of the product decomposition above. The lemma follows. $\square$


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