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
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