The Stacks project

Proof. We apply Lemma 16.7.1 to

and the image of $\pi$ in $D$. By Lemma 16.2.7 we see that $\pi$ is strictly standard in $A \otimes _ R D$ over $D$. As our section $\rho : (A \otimes _ R D)/\pi ^4 (A \otimes _ R D) \to D/\pi ^4 D$ we take the map induced by the map in (2). Thus Lemma 16.7.1 applies and we obtain a factorization $A \otimes _ R D \to B \to \Lambda$ with $B$ of finite presentation and $\mathfrak a B \subset H_{B/D}$ where

For any prime $\mathfrak q$ of $D$ such that $D_\mathfrak q$ is flat over $R$ we have $\text{Ann}_{D_\mathfrak q}(\pi ^2)/\text{Ann}_{D_\mathfrak q}(\pi ) = 0$ because annihilators of elements commutes with flat base change and we assumed $\text{Ann}_ R(\pi ) = \text{Ann}_ R(\pi ^2)$. Because $D$ is Noetherian we see that $\text{Ann}_ D(\pi ^2)/\text{Ann}_ D(\pi )$ is a finite $D$-module, hence formation of its annihilator commutes with localization. Thus we see that $\mathfrak a \not\subset \mathfrak q$. Hence we see that $D \to B$ is smooth at any prime of $B$ lying over $\mathfrak q$. Since any prime of $D$ where $R \to D$ is smooth is one where $D_\mathfrak q$ is flat over $R$ we conclude that $H_{D/R}B \subset H_{B/D}$. The final inclusion $H_{D/R}B \subset H_{B/R}$ follows because compositions of smooth ring maps are smooth (Algebra, Lemma 10.137.14). $\square$