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

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

  3. the finite field extensions $\kappa (\mathfrak p') \subset \kappa (\mathfrak r_ i)$ are purely inseparable, and

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

Proof. The strategy of the proof is to make two étale ring extensions: first we control the residue fields, then we apply Lemma 10.141.22.

Denote $F = S \otimes _ R \kappa (\mathfrak p)$ the fibre ring of $S/R$ at the prime $\mathfrak p$. As in the proof of Lemma 10.141.22 there are finitely may primes, say $\mathfrak q_1, \ldots , \mathfrak q_ n$ of $S$ lying over $R$ at which the ring map $R \to S$ is quasi-finite. Let $\kappa (\mathfrak p) \subset L_ i \subset \kappa (\mathfrak q_ i)$ be the subfield such that $\kappa (\mathfrak p) \subset L_ i$ is separable, and the field extension $L_ i \subset \kappa (\mathfrak q_ i)$ is purely inseparable. Let $\kappa (\mathfrak p) \subset L$ be a finite Galois extension into which $L_ i$ embeds for $i = 1, \ldots , n$. By Lemma 10.141.15 we can find an étale ring extension $R \to R'$ together with a prime $\mathfrak p'$ lying over $\mathfrak p$ such that the field extension $\kappa (\mathfrak p) \subset \kappa (\mathfrak p')$ is isomorphic to $\kappa (\mathfrak p) \subset L$. Thus the fibre ring of $R' \otimes _ R S$ at $\mathfrak p'$ is isomorphic to $F \otimes _{\kappa (\mathfrak p)} L$. The primes lying over $\mathfrak q_ i$ correspond to primes of $\kappa (\mathfrak q_ i) \otimes _{\kappa (\mathfrak p)} L$ which is a product of fields purely inseparable over $L$ by our choice of $L$ and elementary field theory. These are also the only primes over $\mathfrak p'$ at which $R' \to R' \otimes _ R S$ is quasi-finite, by Lemma 10.121.8. Hence after replacing $R$ by $R'$, $\mathfrak p$ by $\mathfrak p'$, and $S$ by $R' \otimes _ R S$ we may assume that for all primes $\mathfrak q$ lying over $\mathfrak p$ for which $S/R$ is quasi-finite the field extensions $\kappa (\mathfrak p) \subset \kappa (\mathfrak q)$ are purely inseparable.

Next apply Lemma 10.141.22. The result is what we want since the field extensions do not change under this étale ring extension. $\square$


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