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 chain complexes, show that “direct limit commutes with taking the associated long exact sequence”.
Solution
1. Taking directed colimits degreewise gives a short exact sequence of chain complexes\[ 0\to\varinjlim A_*^\alpha\to\varinjlim B_*^\alpha\to\varinjlim C_*^\alpha\to0. \]
For each degree \(n\), the sequence \[ 0\to A_n^\alpha\to B_n^\alpha\to C_n^\alpha\to0 \] is a directed system of short exact sequences. Exactness of directed colimits gives a short exact sequence in each degree, and naturality of the differentials makes these into chain maps.
2. The sequence in 1 has its usual long exact homology sequence\[ \cdots\to H_n(\varinjlim A_*^\alpha)\to H_n(\varinjlim B_*^\alpha) \to H_n(\varinjlim C_*^\alpha)\xrightarrow{\partial}\cdots. \]
Every short exact sequence of chain complexes induces a natural long exact sequence in homology.
3. For each of the three systems,\[ H_n(\varinjlim A_*^\alpha)\cong\varinjlim H_n(A_*^\alpha), \]and similarly for \(B\) and \(C\).
Taking homology commutes with directed colimits by the preceding result.
4. Under the identifications in 3, the connecting homomorphism \(\partial\) in 2 is the direct limit of the stagewise connecting homomorphisms \(\partial_\alpha\).
The connecting map is constructed by choosing a cycle in \(C_n\), lifting it to \(B_n\), applying the differential, and then viewing the result in \(A_{n-1}\). This construction is natural with respect to morphisms of short exact sequences of chain complexes. Hence the structure maps commute with every \(\partial_\alpha\), so these maps induce a colimit map, and the representative construction is exactly the connecting map for the colimit sequence.
5. Thus the direct limit of the stagewise long exact sequences is canonically the long exact sequence associated to the direct-limit short exact sequence.
Combine <1>2–<1>4. In particular, corresponding terms and all maps, including connecting maps, agree under the canonical colimit/homology isomorphisms.