If \(\{C_*^\alpha\}_{\alpha \in I}\) is a directed system of chain complexes, show that “taking homology commutes with direct limits”: that is, for any \(n\), \[ \varinjlim H_n(C_*^\alpha) = H_n\left(\varinjlim C_*^\alpha\right). \]
Solution
1. For each \(\alpha\), write\[ Z_n^\alpha=\ker(d_n:C_n^\alpha\to C_{n-1}^\alpha), \qquad B_n^\alpha=\operatorname{im}(d_{n+1}:C_{n+1}^\alpha\to C_n^\alpha). \]The structure maps of the directed system preserve cycles and boundaries.
Every structure map is a chain map, so it commutes with differentials. Hence it sends kernels to kernels and images to images.
2. Directed colimits preserve the exact sequences\[ 0\to Z_n^\alpha\to C_n^\alpha\xrightarrow{d_n}B_{n-1}^\alpha\to0 \]and\[ 0\to B_n^\alpha\to Z_n^\alpha\to H_n(C_*^\alpha)\to0. \]
Directed colimit is exact by the preceding result on direct limits.
3. In the colimit complex \(C_*:=\varinjlim C_*^\alpha\),\[ Z_n(C_*)\cong\varinjlim Z_n^\alpha, \qquad B_n(C_*)\cong\varinjlim B_n^\alpha. \]
From the colimit of the first short exact sequence, \[ 0\to\varinjlim Z_n^\alpha\to\varinjlim C_n^\alpha \to\varinjlim B_{n-1}^\alpha\to0. \] The last module embeds in \(\varinjlim C_{n-1}^\alpha\) by exactness applied to \(0\to B_{n-1}^\alpha\to C_{n-1}^\alpha\). Therefore the kernel of the induced differential on \(C_n\) is exactly \(\varinjlim Z_n^\alpha\). The image statement follows similarly from the factorization of the differential through \(B_n^\alpha\) and surjectivity onto the colimit of these image modules.
4. Consequently\[ \boxed{\varinjlim H_n(C_*^\alpha)\cong H_n\!\left(\varinjlim C_*^\alpha\right).} \]
Taking the colimit of \[ 0\to B_n^\alpha\to Z_n^\alpha\to H_n(C_*^\alpha)\to0 \] gives \[ 0\to\varinjlim B_n^\alpha\to\varinjlim Z_n^\alpha \to\varinjlim H_n(C_*^\alpha)\to0. \] Using 3, its cokernel is \[ Z_n(C_*)/B_n(C_*)=H_n(C_*). \]