The Stacks project

Lemma 10.150.5. Let $R \to S$ be a ring map. Let $J \subset S$ be an ideal such that $R \to S/J$ is surjective; let $I \subset R$ be the kernel. If $R \to S$ is formally étale, then $\bigoplus I^ n/I^{n + 1} \to \bigoplus J^ n/J^{n + 1}$ is an isomorphism of graded rings.

Proof. Using the lifting property inductively we find dotted arrows

\[ \xymatrix{ S \ar[r] \ar@{-->}[rd] & S/J = R/I \\ R \ar[r] \ar[u] & R/I^2 \ar[u] } \quad \xymatrix{ S \ar[r] \ar@{-->}[rd] & R/I^2 \\ R \ar[r] \ar[u] & R/I^3 \ar[u] } \quad \xymatrix{ S \ar[r] \ar@{-->}[rd] & R/I^3 \\ R \ar[r] \ar[u] & R/I^4 \ar[u] } \]

The corresponding maps $S/J^ n \to R/I^ n$ are isomorphisms since the compositions $S/J^ n \to R/I^ n \to S/J^ n$ are (inductively) the identity by the uniqueness in the lifting property of formally étale ring maps. $\square$

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