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 10.141.22. Let $R \to S$ be a ring map. Let $\mathfrak p \subset R$ be a prime. Assume $R \to S$ finite type. Then there exists

  1. an étale ring map $R \to R'$,

  2. a prime $\mathfrak p' \subset R'$ lying over $\mathfrak p$,

  3. a product decomposition

    \[ R' \otimes _ R S = A_1 \times \ldots \times A_ n \times B \]

with the following properties

  1. we have $\kappa (\mathfrak p) = \kappa (\mathfrak p')$,

  2. each $A_ i$ is finite over $R'$,

  3. each $A_ i$ has exactly one prime $\mathfrak r_ i$ lying over $\mathfrak p'$, and

  4. $R' \to B$ not quasi-finite at any prime lying over $\mathfrak p'$.

Proof. Denote $F = S \otimes _ R \kappa (\mathfrak p)$ the fibre ring of $S/R$ at the prime $\mathfrak p$. As $F$ is of finite type over $\kappa (\mathfrak p)$ it is Noetherian and hence $\mathop{\mathrm{Spec}}(F)$ has finitely many isolated closed points. If there are no isolated closed points, i.e., no primes $\mathfrak q$ of $S$ over $\mathfrak p$ such that $S/R$ is quasi-finite at $\mathfrak q$, then the lemma holds. If there exists at least one such prime $\mathfrak q$, then we may apply Lemma 10.141.21. This gives a diagram

\[ \xymatrix{ S \ar[r] & R'\otimes _ R S \ar@{=}[r] & A_1 \times B' \\ R \ar[r] \ar[u] & R' \ar[u] \ar[ru] } \]

as in said lemma. Since the residue fields at $\mathfrak p$ and $\mathfrak p'$ are the same, the fibre rings of $S/R$ and $(A \times B)/R'$ are the same. Hence, by induction on the number of isolated closed points of the fibre we may assume that the lemma holds for $R' \to B$ and $\mathfrak p'$. Thus we get an étale ring map $R' \to R''$, a prime $\mathfrak p'' \subset R''$ and a decomposition

\[ R'' \otimes _{R'} B' = A_2 \times \ldots \times A_ n \times B \]

We omit the verification that the ring map $R \to R''$, the prime $\mathfrak p''$ and the resulting decomposition

\[ R'' \otimes _ R S = (R'' \otimes _{R'} A_1) \times A_2 \times \ldots \times A_ n \times B \]

is a solution to the problem posed in the lemma. $\square$


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