Show that direct limit is an exact functor: that is, if \[ \{0 \to A_\alpha \to B_\alpha \to C_\alpha \to 0\}_{\alpha \in I} \] is a directed system of short exact sequences of modules, then the corresponding sequence of direct limits is also exact. Use this to prove the corresponding result for long exact sequences.
Solution
1. Let\[ 0\to A_\alpha\xrightarrow{i_\alpha}B_\alpha\xrightarrow{p_\alpha}C_\alpha\to0 \]be a directed system of short exact sequences. The induced map\[ \varinjlim A_\alpha\longrightarrow\varinjlim B_\alpha \]is injective.
Suppose a class represented by \(a\in A_\alpha\) maps to zero in \(\varinjlim B_\alpha\). By the definition of a directed colimit, there is some \(\beta\ge\alpha\) such that the image of \(i_\alpha(a)\) in \(B_\beta\) is zero. Commutativity of the system says this image equals \(i_\beta(a_\beta)\). Since \(i_\beta\) is injective, \(a_\beta=0\), so \([a]=0\) already in \(\varinjlim A_\alpha\).
2. The induced sequence is exact at \(\varinjlim B_\alpha\).
Let \([b]\), represented by \(b\in B_\alpha\), map to zero in \(\varinjlim C_\alpha\). Then for some \(\beta\ge\alpha\), the image \(b_\beta\in B_\beta\) maps to zero in \(C_\beta\). Exactness at \(B_\beta\) gives \(a_\beta\in A_\beta\) with \(i_\beta(a_\beta)=b_\beta\). Hence \([b]\) lies in the image of \([a_\beta]\in\varinjlim A_\alpha\). The reverse inclusion follows from \(p_\alpha i_\alpha=0\) at every stage.
3. The map\[ \varinjlim B_\alpha\longrightarrow\varinjlim C_\alpha \]is surjective.
A class in the target is represented by \(c\in C_\alpha\). Since \(p_\alpha\) is surjective, choose \(b\in B_\alpha\) with \(p_\alpha(b)=c\). Then \([b]\) maps to \([c]\).
4. Therefore directed colimit is an exact functor on modules:\[ \boxed{0\to\varinjlim A_\alpha\to\varinjlim B_\alpha\to\varinjlim C_\alpha\to0.} \]
Combine <1>1–<1>3.
5. The directed colimit of a directed system of long exact sequences is again exact.
Exactness at any chosen term only involves the two adjacent maps. The same representative argument as in 2 applies verbatim: if a class maps to zero in the next colimit, it maps to zero at some sufficiently large stage, where exactness produces a preimage from the previous term. Since this works at every term, the colimit sequence is long exact.