Choose an integer $m$ and a $k$-algebra map $\varphi : k[y_1, \ldots , y_ m] \to \Lambda $ and a factorization by local Artinian rings
\[ k[y_1, \ldots , y_ m]_\mathfrak p/\mathfrak p^ n k[y_1, \ldots , y_ m]_\mathfrak p \to D \to \Lambda _\mathfrak q/\mathfrak q^ n\Lambda _\mathfrak q \]such that the first arrow is essentially smooth, the second is flat, $E$ is contained in $D$, with $\mathfrak p = \varphi ^{-1}(\mathfrak q)$ the map $k[y_1, \ldots , y_ m]_\mathfrak p \to \Lambda _\mathfrak q$ is flat, and $\mathfrak p \Lambda _\mathfrak q = \mathfrak q \Lambda _\mathfrak q$.
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