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\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 10.148.12. Let $\varphi : R \to S$ be a local homomorphism of strictly henselian local rings. Let $P_1, \ldots , P_ n \in R[x_1, \ldots , x_ n]$ be polynomials such that $R[x_1, \ldots , x_ n]/(P_1, \ldots , P_ n)$ is ├ętale over $R$. Then the map

\[ R^ n \longrightarrow S^ n, \quad (h_1, \ldots , h_ n) \longmapsto (\varphi (h_1), \ldots , \varphi (h_ n)) \]

induces a bijection between

\[ \{ (r_1, \ldots , r_ n) \in R^ n \mid P_ i(r_1, \ldots , r_ n) = 0, \ i = 1, \ldots , n \} \]

and

\[ \{ (s_1, \ldots , s_ n) \in S^ n \mid P'_ i(s_1, \ldots , s_ n) = 0, \ i = 1, \ldots , n \} \]

where $P'_ i \in S[x_1, \ldots , x_ n]$ are the images of the $P_ i$ under $\varphi $.

Proof. The first solution set is canonically isomorphic to the set

\[ \mathop{\mathrm{Hom}}\nolimits _ R(R[x_1, \ldots , x_ n]/(P_1, \ldots , P_ n), R). \]

As $R$ is henselian the map $R \to R/\mathfrak m_ R$ induces a bijection between this set and the set of solutions in the residue field $R/\mathfrak m_ R$, see Lemma 10.148.3. The same is true for $S$. Now since $R[x_1, \ldots , x_ n]/(P_1, \ldots , P_ n)$ is ├ętale over $R$ and $R/\mathfrak m_ R$ is separably algebraically closed we see that $R/\mathfrak m_ R[x_1, \ldots , x_ n]/(\overline{P_1}, \ldots , \overline{P_ n})$ is a finite product of copies of $R/\mathfrak m_ R$. Hence the tensor product

\[ R/\mathfrak m_ R[x_1, \ldots , x_ n]/(\overline{P_1}, \ldots , \overline{P_ n}) \otimes _{R/\mathfrak m_ R} S/\mathfrak m_ S = S/\mathfrak m_ S[x_1, \ldots , x_ n]/(\overline{P_1'}, \ldots , \overline{P_ n'}) \]

is also a finite product of copies of $S/\mathfrak m_ S$ with the same index set. This proves the lemma. $\square$


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