The Stacks project

Lemma 10.76.2. Let $R$ be a ring. Let $M = \mathop{\mathrm{colim}}\nolimits M_ i$ be a filtered colimit of $R$-modules. Let $N$ be an $R$-module. Then $\text{Tor}_ n^ R(M, N) = \mathop{\mathrm{colim}}\nolimits \text{Tor}_ n^ R(M_ i, N)$ for all $n$.

Proof. Choose a free resolution $F_\bullet $ of $N$. Then $F_\bullet \otimes _ R M = \mathop{\mathrm{colim}}\nolimits F_\bullet \otimes _ R M_ i$ as complexes by Lemma 10.12.9. Thus the result by Lemma 10.8.8. $\square$

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