Lemma 10.8.7. Let $(M_ i, \mu _{ij})$, $(N_ i, \nu _{ij})$ be systems of $R$-modules over the same preordered set. A morphism of systems $\Phi = (\phi _ i)$ from $(M_ i, \mu _{ij})$ to $(N_ i, \nu _{ij})$ induces a unique homomorphism

$\mathop{\mathrm{colim}}\nolimits \phi _ i : \mathop{\mathrm{colim}}\nolimits M_ i \longrightarrow \mathop{\mathrm{colim}}\nolimits N_ i$

such that

$\xymatrix{ M_ i \ar[r] \ar[d]_{\phi _ i} & \mathop{\mathrm{colim}}\nolimits M_ i \ar[d]^{\mathop{\mathrm{colim}}\nolimits \phi _ i} \\ N_ i \ar[r] & \mathop{\mathrm{colim}}\nolimits N_ i }$

commutes for all $i \in I$.

Proof. Write $M = \mathop{\mathrm{colim}}\nolimits M_ i$ and $N = \mathop{\mathrm{colim}}\nolimits N_ i$ and $\phi = \mathop{\mathrm{colim}}\nolimits \phi _ i$ (as yet to be constructed). We will use the explicit description of $M$ and $N$ in Lemma 10.8.2 without further mention. The condition of the lemma is equivalent to the condition that

$\xymatrix{ \bigoplus _{i\in I} M_ i \ar[r] \ar[d]_{\bigoplus \phi _ i} & M \ar[d]^\phi \\ \bigoplus _{i\in I} N_ i \ar[r] & N }$

commutes. Hence it is clear that if $\phi$ exists, then it is unique. To see that $\phi$ exists, it suffices to show that the kernel of the upper horizontal arrow is mapped by $\bigoplus \phi _ i$ to the kernel of the lower horizontal arrow. To see this, let $j \leq k$ and $x_ j \in M_ j$. Then

$(\bigoplus \phi _ i)(x_ j - \mu _{jk}(x_ j)) = \phi _ j(x_ j) - \phi _ k(\mu _{jk}(x_ j)) = \phi _ j(x_ j) - \nu _{jk}(\phi _ j(x_ j))$

which is in the kernel of the lower horizontal arrow as required. $\square$

## Comments (2)

Comment #1834 by Patrizio on

Would it not be easier just to use the universal property? In fact, We can take the composition M_i \rightarrow N_i \rightarrow lim N_i and then we know that there exists a unique morphism which factorizes by lim M_i, just by definition. Is that idea wrong?

Comment #1871 by on

Yes, I agree that works and I agree that it is better. If you feel so inclined, please edit the latex file and send it to me. Thanks, Johan

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00DA. Beware of the difference between the letter 'O' and the digit '0'.