In the proposition Atiyah proves the following:
If $M$ is an $A-$module, then the $S^{-1}A-$ modules $S^{-1}M$ and $S^{-1}A\otimes_A M$ are isomorphic.
My problem is that in the proof he proves that the 2 modules are isomorphic as $A-$modules, in fact he makes an $A-$linear isomorphism between them using the universal proprerty of the tensor product $S^{-1}A\otimes_A M$. Do I forget anything?
Noobest, the isomorphism is the unique $A$-module map $f$ such that $m/s\mapsto 1/s\otimes m$. Then $f(a/s \cdot m/r)=f(am/sr)=1/sr\otimes am=a/sr\otimes m=a/s(1/r\otimes m)$.