Lemma 15.9.6. Let $A$ be a ring, let $I \subset A$ be an ideal. Let $f \in A[x]$ be a monic polynomial. Let $\overline{f} = \overline{g} \overline{h}$ be a factorization of $f$ in $A/I[x]$ and assume

1. the leading coefficient of $\overline{g}$ is an invertible element of $A/I$, and

2. $\overline{g}$, $\overline{h}$ generate the unit ideal in $A/I[x]$.

Then there exists an étale ring map $A \to A'$ which induces an isomorphism $A/I \to A'/IA'$ and a factorization $f = g' h'$ in $A'[x]$ lifting the given factorization over $A/I$.

Proof. Applying Lemma 15.9.1 we may assume that the leading coefficient of $\overline{g}$ is the reduction of an invertible element $u \in A$. Then we may replace $\overline{g}$ by $\overline{u}^{-1}\overline{g}$ and $\overline{h}$ by $\overline{u}\overline{h}$. Thus we may assume that $\overline{g}$ is monic. Since $f$ is monic we conclude that $\overline{h}$ is monic too. In this case the result follows from Lemma 15.9.5. $\square$

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