Lemma 59.17.1. Let $D$ and $D(n)$ be as in (59.17.0.1) and (59.17.0.4). The coprojection $P \to P \otimes _ A \ldots \otimes _ A P$, $f \mapsto f \otimes 1 \otimes \ldots \otimes 1$ induces an isomorphism

59.17.1.1
\begin{equation} \label{crystalline-equation-structure-Dn} D(n) = \mathop{\mathrm{lim}}\nolimits _ e D\langle \xi _ i(j) \rangle /p^ eD\langle \xi _ i(j) \rangle \end{equation}

of algebras over $D$ with

\[ \xi _ i(j) = x_ i \otimes 1 \otimes \ldots \otimes 1 - 1 \otimes \ldots \otimes 1 \otimes x_ i \otimes 1 \otimes \ldots \otimes 1 \]

for $j = 1, \ldots , n$ where the second $x_ i$ is placed in the $j + 1$st slot; recall that $D(n)$ is constructed starting with the $n + 1$-fold tensor product of $P$ over $A$.

**Proof.**
We have

\[ P \otimes _ A \ldots \otimes _ A P = P[\xi _ i(j)] \]

and $J(n)$ is generated by $J$ and the elements $\xi _ i(j)$. Hence the lemma follows from Lemma 59.2.5.
$\square$

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