The Stacks project

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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