Show that a direct limit of exact sequences is exact. More generally, show that homology commutes with direct limits: If \(\{C_\alpha, f_{\alpha\beta}\}\) is a directed system of chain complexes, with the maps \(f_{\alpha\beta}: C_\alpha \to C_\beta\) chain maps, then \(H_n(\varinjlim C_\alpha) = \varinjlim H_n(C_\alpha)\).
Solution
For a directed system of abelian groups, recall that \[ \varinjlim A_\alpha \] is the disjoint union of the \(A_\alpha\) modulo the relation that \(a_\alpha\) and \(a_\beta\) are equal when their images agree at some later stage.
Consider a directed system of exact sequences \[ A_\alpha\xrightarrow{u_\alpha}B_\alpha\xrightarrow{v_\alpha}C_\alpha. \] Certainly \(\operatorname{im}(\varinjlim u_\alpha)\subseteq\ker(\varinjlim v_\alpha)\). Conversely, let a class \([b_\alpha]\) map to zero in \(\varinjlim C_\alpha\). Then for some \(\beta\ge\alpha\), the image \(b_\beta\) maps to zero in \(C_\beta\). Exactness at stage \(\beta\) gives \(a_\beta\in A_\beta\) with \[ u_\beta(a_\beta)=b_\beta. \] Thus \([b_\alpha]\) lies in the image of \(\varinjlim A_\alpha\). Hence filtered direct limits preserve exact sequences.
Now let \(C_\alpha\) be a directed system of chain complexes. Since direct limits are exact, they commute with kernels, images, and cokernels occurring in the short exact sequences \[ 0\to Z_n(C_\alpha)\to C_{\alpha,n}\to B_{n-1}(C_\alpha)\to0 \] and \[ 0\to B_n(C_\alpha)\to Z_n(C_\alpha)\to H_n(C_\alpha)\to0. \] Therefore \[ Z_n(\varinjlim C_\alpha)\cong\varinjlim Z_n(C_\alpha), \qquad B_n(\varinjlim C_\alpha)\cong\varinjlim B_n(C_\alpha), \] and taking the quotient gives \[ \boxed{H_n(\varinjlim C_\alpha)\cong\varinjlim H_n(C_\alpha).} \]