Remark 15.90.9. If $R \to R^\wedge$ is flat, then for each positive integer $n$ tensoring the sequence $0 \to R[f^ n] \to R \to R$ with $R^\wedge$ gives the sequence $0 \to R[f^ n] \otimes _ R R^\wedge \to R^\wedge \to R^\wedge$. Combined with Lemma 15.90.2 we conclude that $R[f^ n] \to R^\wedge [f^ n]$ is an isomorphism. Thus $(R, f)$ is a glueing pair. This holds in particular if $R$ is Noetherian, see Algebra, Lemma 10.97.2.

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