Filtered colimits are exact. Directed colimits are exact.

Lemma 10.8.8. Let $I$ be a directed set. Let $(L_ i, \lambda _{ij})$, $(M_ i, \mu _{ij})$, and $(N_ i, \nu _{ij})$ be systems of $R$-modules over $I$. Let $\varphi _ i : L_ i \to M_ i$ and $\psi _ i : M_ i \to N_ i$ be morphisms of systems over $I$. Assume that for all $i \in I$ the sequence of $R$-modules

$\xymatrix{ L_ i \ar[r]^{\varphi _ i} & M_ i \ar[r]^{\psi _ i} & N_ i }$

is a complex with homology $H_ i$. Then the $R$-modules $H_ i$ form a system over $I$, the sequence of $R$-modules

$\xymatrix{ \mathop{\mathrm{colim}}\nolimits _ i L_ i \ar[r]^\varphi & \mathop{\mathrm{colim}}\nolimits _ i M_ i \ar[r]^\psi & \mathop{\mathrm{colim}}\nolimits _ i N_ i }$

is a complex as well, and denoting $H$ its homology we have

$H = \mathop{\mathrm{colim}}\nolimits _ i H_ i.$

Proof. It is clear that $\xymatrix{ \mathop{\mathrm{colim}}\nolimits _ i L_ i \ar[r]^\varphi & \mathop{\mathrm{colim}}\nolimits _ i M_ i \ar[r]^\psi & \mathop{\mathrm{colim}}\nolimits _ i N_ i }$ is a complex. For each $i \in I$, there is a canonical $R$-module morphism $H_ i \to H$ (sending each $[m] \in H_ i = \mathop{\mathrm{Ker}}(\psi _ i) / \mathop{\mathrm{Im}}(\varphi _ i)$ to the residue class in $H = \mathop{\mathrm{Ker}}(\psi ) / \mathop{\mathrm{Im}}(\varphi )$ of the image of $m$ in $\mathop{\mathrm{colim}}\nolimits _ i M_ i$). These give rise to a morphism $\mathop{\mathrm{colim}}\nolimits _ i H_ i \to H$. It remains to show that this morphism is surjective and injective.

We are going to repeatedly use the description of colimits over $I$ as in Lemma 10.8.3 without further mention. Let $h \in H$. Since $H = \mathop{\mathrm{Ker}}(\psi )/\mathop{\mathrm{Im}}(\varphi )$ we see that $h$ is the class mod $\mathop{\mathrm{Im}}(\varphi )$ of an element $[m]$ in $\mathop{\mathrm{Ker}}(\psi ) \subset \mathop{\mathrm{colim}}\nolimits _ i M_ i$. Choose an $i$ such that $[m]$ comes from an element $m \in M_ i$. Choose a $j \geq i$ such that $\nu _{ij}(\psi _ i(m)) = 0$ which is possible since $[m] \in \mathop{\mathrm{Ker}}(\psi )$. After replacing $i$ by $j$ and $m$ by $\mu _{ij}(m)$ we see that we may assume $m \in \mathop{\mathrm{Ker}}(\psi _ i)$. This shows that the map $\mathop{\mathrm{colim}}\nolimits _ i H_ i \to H$ is surjective.

Suppose that $h_ i \in H_ i$ has image zero on $H$. Since $H_ i = \mathop{\mathrm{Ker}}(\psi _ i)/\mathop{\mathrm{Im}}(\varphi _ i)$ we may represent $h_ i$ by an element $m \in \mathop{\mathrm{Ker}}(\psi _ i) \subset M_ i$. The assumption on the vanishing of $h_ i$ in $H$ means that the class of $m$ in $\mathop{\mathrm{colim}}\nolimits _ i M_ i$ lies in the image of $\varphi$. Hence there exists a $j \geq i$ and an $l \in L_ j$ such that $\varphi _ j(l) = \mu _{ij}(m)$. Clearly this shows that the image of $h_ i$ in $H_ j$ is zero. This proves the injectivity of $\mathop{\mathrm{colim}}\nolimits _ i H_ i \to H$. $\square$

Comment #895 by Kestutis Cesnavicius on

Suggested slogan: Formation of homology of complexes of modules commutes with filtered direct limits

Comment #907 by on

My slogan for this is: Filtered colimits are exact. This is mathematically imprecise, but it is really kind of a slogan and gets used over and over again and this lemma is a good example of it.

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