The Stacks project

Lemma 16.7.3. Let $R$ be a Noetherian ring. Let $\Lambda $ be an $R$-algebra. Let $\pi \in R$ and assume that $\text{Ann}_ R(\pi ) = \text{Ann}_ R(\pi ^2)$ and $\text{Ann}_\Lambda (\pi ) = \text{Ann}_\Lambda (\pi ^2)$. Let $A \to \Lambda $ be an $R$-algebra map with $A$ of finite presentation and assume $\pi $ is strictly standard in $A$ over $R$. Let

\[ A/\pi ^8A \to \bar C \to \Lambda /\pi ^8\Lambda \]

be a factorization with $\bar C$ of finite presentation. Then we can find a factorization $A \to B \to \Lambda $ with $B$ of finite presentation such that $R_\pi \to B_\pi $ is smooth and such that

\[ H_{\bar C/(R/\pi ^8 R)} \cdot \Lambda /\pi ^8\Lambda \subset \sqrt{H_{B/R} \Lambda } \bmod \pi ^8\Lambda . \]

Proof. Apply Lemma 16.6.1 to get $R \to D \to \Lambda $ with a factorization $\bar C/\pi ^4\bar C \to D/\pi ^4 D \to \Lambda /\pi ^4\Lambda $ such that $R \to D$ is smooth at any prime not containing $\pi $ and at any prime lying over a prime of $\bar C/\pi ^4\bar C$ where $R/\pi ^8 R \to \bar C$ is smooth. By Lemma 16.7.2 we can find a finitely presented $R$-algebra $B$ and factorizations $A \to B \to \Lambda $ and $D \to B \to \Lambda $ such that $H_{D/R}B \subset H_{B/R}$. We omit the verification that this is a solution to the problem posed by the lemma. $\square$

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