Lemma 15.90.15. Let $(R \to R',f)$ be a glueing pair. Then $\text{Tor}^ R_1(R', R/R[f^\infty ]) = 0$.

**Proof.**
We have $R/R[f^\infty ] = \mathop{\mathrm{colim}}\nolimits R/R[f^ n] = \mathop{\mathrm{colim}}\nolimits f^ nR$. As formation of Tor groups commutes with filtered colimits (Algebra, Lemma 10.76.2) we may apply Lemma 15.90.14.
$\square$

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