The Stacks project

Lemma 15.90.13. Let $(R \to R', f)$ be a glueing pair. Then $\text{Tor}^ R_1(R', f^ n R) = 0$ for each $n > 0$.

Proof. From the exact sequence $0 \to R[f^ n] \to R \to f^ n R \to 0$ we see that it suffices to check that $R[f^ n] \otimes _ R R' \to R'$ is injective. By Lemma 15.88.8 we have $R[f^ n] \otimes _ R R' = R[f^ n]$ and by Lemma 15.90.6 we see that $R[f^ n] \to R'$ is injective as $(R \to R', f)$ is a glueing pair. $\square$

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