Any module is the colimit of its finitely generated submodules.

Question

Fact A : If $E_i$ is a directed system of $R$-modules, and $F$ is another $R$-module. Then

$$\varinjlim(F\otimes E_{i})=F\otimes(\varinjlim E_{i}).$$

Fact B : "Every module is a direct limit of its finitely generated submodules."

How does Fact A implies Fact B?

Serge Lang claims this in his Algebra book.

possible duplicate of Tensor products commute with direct limits — rschwieb, Aug 5 '14 at 13:25
@rschwieb: I am not asking for the proof of the expression I just wrote. I am asking how does it follow from that "Any module is a direct limit of its finitely generated submodule." — Babai, Aug 5 '14 at 13:27
Yes, that's the reduction. My thought right now is to further reduce to the case of $R=k$, a field, but I'm not having so much luck managing it. — Cass, Aug 5 '14 at 14:39
Just as a note, by the way, I don't really see why any of this is necessary. The map $f:lim E_i \rightarrow E$ is injective because direct limit preserves injectivity. $f$ is also surjective because every element $e$ of $E$ can be put into a finitely generated submodule, namely $Re$. So $f$ is an isomorphism. Maybe there's some other way with this Fact, but the proof I just gave is about as straight ahead as anyone could hope. — Cass, Aug 5 '14 at 14:56
Take the trivial directed system $F_i = E$ for all $i$ over the same index set as $E_i$. Then, we have injective maps $E_i \rightarrow F_i$ for all $i$ and these are compatible with directed system maps. Take direct limit on both sides. lim is left (and right) exact, so you get an injective map $lim E_i \rightarrow lim F_i$. But $lim F_i$ is clearly $E$. — Cass, Aug 5 '14 at 15:14

Martin Brandenburg · Accepted Answer · 2014-08-06 13:11:06Z

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I am 100% sure that Fact A and Fact B cannot be proven from another.

[So in some sense, I don't give an answer here, but rather the meta-answer that there is no answer.]

answered Aug 5 '14 at 20:38

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$\begingroup$ You've misread his question. He is not trying to prove the tensor product fact (call it Fact A) using the Fact B = "Modules Are Limits of F.G. Submodules". He is trying to prove Fact B using Fact A. He already believes Fact A is true. But in any case, as I said in a comment, I agree that it is not necessary or even useful to appeal to Fact A in proving Fact B. $\endgroup$ – Cass Aug 6 '14 at 1:13
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$\begingroup$ You are right, I've misread the question. But as for the original version, is was quite easy to misunderstand it ... the new version is more precise. I've edited accordingly. $\endgroup$ – Martin Brandenburg Aug 6 '14 at 13:10

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