The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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


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