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*}

Example 10.8.5. Consider the partially ordered set $I = \{ a, b, c\} $ with $a < b$ and $a < c$ and no other strict inequalities. A system $(M_ a, M_ b, M_ c, \mu _{ab}, \mu _{ac})$ over $I$ consists of three $R$-modules $M_ a, M_ b, M_ c$ and two $R$-module homomorphisms $\mu _{ab} : M_ a \to M_ b$ and $\mu _{ac} : M_ a \to M_ c$. The colimit of the system is just

\[ M := \mathop{\mathrm{colim}}\nolimits _{i \in I} M_ i = \mathop{\mathrm{Coker}}(M_ a \to M_ b \oplus M_ c) \]

where the map is $\mu _{ab} \oplus -\mu _{ac}$. Thus the kernel of the canonical map $M_ a \to M$ is $\mathop{\mathrm{Ker}}(\mu _{ab}) + \mathop{\mathrm{Ker}}(\mu _{ac})$. And the kernel of the canonical map $M_ b \to M$ is the image of $\mathop{\mathrm{Ker}}(\mu _{ac})$ under the map $\mu _{ab}$. Hence clearly the result of Lemma 10.8.4 is false for general systems.


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