Lemma 60.17.1. Let D and D(n) be as in (60.17.0.1) and (60.17.0.4). The coprojection P \to P \otimes _ A \ldots \otimes _ A P, f \mapsto f \otimes 1 \otimes \ldots \otimes 1 induces an isomorphism
60.17.1.1
\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 + 1st 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 60.2.5.
\square
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