The Stacks project

Remark 23.12.2. In the situation above, if $R$ is Noetherian, we can inductively choose a sequence of integers $1 = n_0 < n_1 < n_2 < \ldots $ such that for $i = 1, 2, 3, \ldots $ we have maps $K_{n_ i} \to A_ i \to R/(f_1^{n_ i}, \ldots , f_ r^{n_ i})$ and $A_ i \to K_{n_{i - 1}}$ as in Lemma 23.12.1. Denote $A_{i + 1} \to A_ i$ the composition $A_{i + 1} \to K_{n_ i} \to A_ i$. Then the diagram

\[ \xymatrix{ K_{n_1} \ar[d] & K_{n_2} \ar[d] \ar[l] & K_{n_3} \ar[d] \ar[l] & \ldots \ar[l] \\ A_1 \ar[d] & A_2 \ar[l] \ar[d] & A_3 \ar[l] \ar[d] & \ldots \ar[l] \\ K_1 & K_{n_1} \ar[l] & K_{n_2} \ar[l] & \ldots \ar[l] } \]

commutes. In this way we see that the inverse systems $(K_ n)$ and $(A_ n)$ are pro-isomorphic in the category of differential graded $R$-algebras with compatible divided powers.

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