Let $M_i$ be a arbitrary colection of $A$-modules and $S$ a multiplicative subset of $A$. I want to show that
$$S^{-1}\left(\bigoplus_i M_i\right)\cong \bigoplus_i S^{-1}M_i$$
as $A$-modules and as $S^{-1}A$-modules. I know how to explicitly write an isomorphism between them but I want to do it using the universal properties to understand better how they work.
Here's what I've done:
Let $M=\bigoplus_i M_i$ with the canonical injections $\iota_i:M_i\to M$. Also let $\Phi:M\to S^{-1}M$ be the canonical morphism associated with the localization. Composing these morphisms we get
$$\Phi\circ\iota_i:M_i\to S^{-1}M.$$
By the universal property of the localization, we obtain a unique morphism
$$\overline{\Phi\circ\iota_i}:S^{-1}M_i\to S^{-1}M$$
such that $\overline{\Phi\circ\iota_i}\circ\Phi_i=\Phi\circ\iota_i$, where $\Phi_i:M_i\to S^{-1}M_i$ is the localization morphism.
Finally, by the universal property of the coproduct we obtain a morphism
$$\bigoplus_i S^{-1}M_i\to S^{-1}M.$$
I think this morphism might be the desired isomorphism but I don't know how to prove it.
(Maybe a good idea would be to find its inverse but for it I need some kind of morphism $M\to M_i$, which is not available when the direct sum is infinite.)
Also, I know there are a couple questions here about similar things but they do either the explicit isomorphism or the finite case.
Edit: after @GreginGre commentaries, I have a morphism $S^{-1}M\to S^{-1}M_i$ but I don't know neither how to obtain a morphism $S^{-1}M\to\bigoplus S^{-1}M_i$ (since this is the wrong side for the universal property of coproducts) nor how to show that these two morphisms are inverses of each other.
Best Answer
Might not be exactly what you want, but you only need to understand the following fact:
The question is then nothing but the fact that tensor product commutes with arbitrary direct sums (see e.g. this wiki page "distributive property").