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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