The Stacks project

There exist $\delta _1, \ldots , \delta _ d \in \Lambda $, $\delta _ i \not\in \mathfrak q$ and a factorization $D \to D' \to \Lambda _\mathfrak q/\mathfrak q^ n\Lambda _\mathfrak q$ with $D'$ local Artinian, $D \to D'$ essentially smooth, the map $D' \to \Lambda _\mathfrak q/\mathfrak q^ n\Lambda _\mathfrak q$ flat such that, with $\pi _ i' = \delta _ i \pi _ i$, we have for $i = 1, \ldots , d$

1. $(\pi _ i')^{2N} = \sum a_ j\lambda _{ij}$ in $\Lambda $ where $\lambda _{ij} \bmod \mathfrak q^ n\Lambda _\mathfrak q$ is an element of $D'$,

2. $\text{Ann}_{\Lambda /({\pi '}_1^ e, \ldots , {\pi '}_{i - 1}^ e)}({\pi '}_ i) = \text{Ann}_{\Lambda /({\pi '}_1^ e, \ldots , {\pi '}_{i - 1}^ e)}({\pi '}_ i^2)$,

3. $\delta _ i \bmod \mathfrak q^ n\Lambda _\mathfrak q$ is an element of $D'$.