$\text{Tor}_i^A(M,N) \cong \text{Tor}_i^{S^{-1}A}(M,N)$ for $S^{-1}A$-Modules $M,N$

Question

Let $A$ be a unitary commutative ring, $S\subset A$ multiplicatively closed and $S^{-1}A$ its localization. Let $M,N$ be $S^{-1}A$-Modules which can be viewed as $A$-Modules via the canonical map $A\to S^{-1}A$. Show that there is an isomorphism of $A$-Modules $$\text{Tor}_i^A(M,N) \cong \text{Tor}_i^{S^{-1}A}(M,N)$$ I know that $S^{-1}\text{Tor}_i^A(M,N) \cong \text{Tor}_i^{S^{-1}A}(S^{-1}M,S^{-1}N)$ for $A$-Modules $M,N$ but I don't really know how to use this to prove the statement above, can someone help me?

Answer 1

Let $X = \mathrm{Tor}^A(M,N)$ and let $Y$ be the Tor group over the localization. You already know that there is a (natural) isomorphism $S^{-1}X \to Y$, so it suffices to show that the canonical map $X\to S^{-1}X$ is an isomorphism. To do this, you can argue that every element of $S$ acts on $X$ by an isomorphism. Can you do this? The point is that $M$ and $N$ are already modules where the elements of $S$ act by isomorphisms, of course, so you have to take it from there.