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I read that $\lim\limits_{\longleftarrow}\mathrm{Hom}(N_j,M)\cong\mathrm{Hom}(\lim\limits_{\longrightarrow}N_j,M)$. I was wondering if we can write $\lim\limits_{\longrightarrow}\mathrm{Hom}(N_j,M)$ as $\mathrm{Hom}(X,M)$ for some $X$. Do you know if we can? I would be happy also if you can give me only a reference.

(Here I'm talking of modules over commutative rings, we can also suppose that the $N_j$'s and $M$ are finitely generated, we can also add some other conditions if you wish, maybe noetherianity).

I also read that $\mathrm{Hom}(M,\lim\limits_{\longleftarrow}N_j)\cong\lim\limits_{\longleftarrow}\mathrm{Hom}(M,N_j)$. I was wondering if under some hypothesis $\mathrm{Hom}(M,\lim\limits_{\longrightarrow}N_j)\cong \lim\limits_{\longrightarrow}\mathrm{Hom}(M,N_j)$. I would be happy if you can help me even in only one of those questions.

EDIT: what about if $M$ is not even finitely generated?

EDIT: Suppose that the transition maps in $\{N_j\}$ are injective. We can see $M$ as a direct limit of finitely generated modules. Is it true that $\lim\limits_{\longleftarrow_n}\lim\limits_{\longrightarrow_j}Hom(M_n,N_j)$=$\lim\limits_{\longrightarrow_j}\lim\limits_{\longleftarrow_n}Hom(M_n,N_j)$? If this is true then we can drop the hypothesis $M$ finitely generated in the answer of Matt E.

EDIT: what about $\mathrm{Hom}(\lim\limits_{\longleftarrow}\;M_n,N)$?

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Chris, note LaTeX has already a command for the limit operator, namely \lim. Compare, however the cases \lim\limits_{x\to a} which renders $\lim\limits_{x\to a}$ versus \lim_{x\to a} which renders $\lim_{x\to a}$. Using double dollar signs gives the same for both, but will center the output and force a line break: $$\lim_{x\to a}$$ $$\lim\limits_{x\to a}$$ – Pedro Tamaroff Nov 1 '12 at 2:45
You can also obtain nicer renders with \longrightarrow and \longleftarrow which give $\lim\limits_{\longleftarrow}$ and $\lim\limits_{\longrightarrow}$. The code \rm works the same as \mathrm and can shorten things up. Also, using \left( and \right) will give "elastic" parenthesis (or any other delimiter). Compare \left(1+\frac 1 n \right)^n:$\left(1+\frac 1 n \right)^n$ with (1+\frac 1 n )^n: $(1+\frac 1 n )^n$ – Pedro Tamaroff Nov 1 '12 at 2:47
Dear Chris, The limits you ask about don't commute, and the conditions in my answer are if and only if. Regards, – Matt E Nov 1 '12 at 11:03
Just to make a link about the first point you have read about:… – user39158 Jun 29 '15 at 16:30
up vote 6 down vote accepted

There is a natural isomorphism $\varinjlim Hom(M,N_j) \cong Hom(M,\varinjlim N_j)$ for all filtered direct systems $N_j$ if and only if $M$ is finitely presented. If you assume that the transition maps in the system $\{N_j\}$ are injective, then finitely generated is enough. (See this answer.)

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what about if $M$ is not even finitely generated? There are any chances to answer at least the other question? – Chris Nov 1 '12 at 2:58
I wanted to leave a comment here but I had some problems with Tex, I posted that comment as an edit in the question – Chris Nov 1 '12 at 3:49
if you look to the answer of Pierre-Yves Gaillard to the question you linked it seems that inverse limits and direct limits as written in my question commute (or maybe I'm misunderstanding his answer) – Chris Nov 1 '12 at 3:59

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