Filtered colimits of abelian groups are exact

By [F1] the categories $\mathbf{A}\mathbf{b}$ and $\mathbb{Z}\text{-}\mathbf{M}\mathbf{o}\mathbf{d}$ have the same objects and morphisms, so every statement about the categorical structure of one holds for the other. By [F2] $\mathbb{Z}\text{-}\mathbf{M}\mathbf{o}\mathbf{d}$ is a Grothendieck category, hence by [F3] an abelian category satisfying AB5 and having a generator, and satisfying AB5 includes AB3 by [F4], so $\mathbb{Z}\text{-}\mathbf{M}\mathbf{o}\mathbf{d}$ is a cocomplete abelian category satisfying AB5. By [F1] the same holds for $\mathbf{A}\mathbf{b}$, and [F5] then gives that for every small filtered category $\mathcal{J}$ the filtered colimit functor ${\mathrm{colim}}_{\mathcal{J}}:{\mathbf{A}\mathbf{b}}^{\mathcal{J}}\to \mathbf{A}\mathbf{b}$ is exact.

Let $j\mapsto (0\to A_j\to B_j\to C_j\to 0)$ be a diagram of short exact sequences of abelian groups. Viewing it as an object of ${\mathbf{A}\mathbf{b}}^{\mathcal{J}}$ concentrated in cohomological degrees $0,1,2$, it is a diagram of complexes that is exact in each degree. By [F6] the exact functor ${\mathrm{colim}}_{\mathcal{J}}$ preserves kernels and cokernels, hence also images; applied to the diagrams $j\mapsto A_j$, $j\mapsto B_j$, $j\mapsto C_j$ with their structure maps it therefore yields the short exact sequence $0\to {\mathrm{colim}}_j{A}_j\to {\mathrm{colim}}_j{B}_j\to {\mathrm{colim}}_j{C}_j\to 0$. Concretely, injectivity on the left is preservation of the kernel of $A_j\to B_j$, surjectivity on the right is preservation of the cokernel of $B_j\to C_j$, and exactness at the middle term follows because the image of a morphism is the kernel of its cokernel: $\mathrm{im}({\mathrm{colim}}_j{A}_j\to {\mathrm{colim}}_j{B}_j)={\mathrm{colim}}_j\mathrm{im}(A_j\to B_j)={\mathrm{colim}}_j\ker (B_j\to C_j)=\ker ({\mathrm{colim}}_j{B}_j\to {\mathrm{colim}}_j{C}_j)$.

Let $j\mapsto K_j^{\bullet }$ be a diagram of cochain complexes of abelian groups with differentials $d_j^p$, and put $Z_j^p:=\ker d_j^p$ and $B_j^p:=\mathrm{im}{d}_j^{p-1}$, so that $0\to Z_j^p\to K_j^p\to B_j^{p+1}\to 0$ and $0\to B_j^p\to Z_j^p\to H^p(K_j^{\bullet })\to 0$ are pointwise exact sequences of diagrams of abelian groups. By [step 2.1] the colimits of these two diagrams of short exact sequences are short exact: $0\to {\mathrm{colim}}_j{Z}_j^p\to {\mathrm{colim}}_j{K}_j^p\to {\mathrm{colim}}_j{B}_j^{p+1}\to 0$ and $0\to {\mathrm{colim}}_j{B}_j^p\to {\mathrm{colim}}_j{Z}_j^p\to {\mathrm{colim}}_j{H}^p(K_j^{\bullet })\to 0$. Since ${\mathrm{colim}}_{\mathcal{J}}$ is exact it preserves kernels and images [F6], so ${\mathrm{colim}}_j{Z}_j^p=\ker ({\mathrm{colim}}_j{K}_j^p\to {\mathrm{colim}}_j{K}_j^{p+1})=Z^p({\mathrm{colim}}_j{K}_j^{\bullet })$ and ${\mathrm{colim}}_j{B}_j^{p+1}=\mathrm{im}({\mathrm{colim}}_j{K}_j^p\to {\mathrm{colim}}_j{K}_j^{p+1})=B^{p+1}({\mathrm{colim}}_j{K}_j^{\bullet })$. It also preserves cokernels, so ${\mathrm{colim}}_j(Z_j^p\mathrm{/}{B}_j^p)\cong ({\mathrm{colim}}_j{Z}_j^p)\mathrm{/}({\mathrm{colim}}_j{B}_j^p)$; combining the identifications gives $H^p({\mathrm{colim}}_j{K}_j^{\bullet })=({\mathrm{colim}}_j{Z}_j^p)\mathrm{/}({\mathrm{colim}}_j{B}_j^p)={\mathrm{colim}}_j(Z_j^p\mathrm{/}{B}_j^p)={\mathrm{colim}}_j{H}^p(K_j^{\bullet })$, which is assertion 2.

If every complex $K_j^{\bullet }$ is exact, then $H^p(K_j^{\bullet })=0$ for all $j$ and all $p$, so assertion 2 gives $H^p({\mathrm{colim}}_j{K}_j^{\bullet })=0$ for every $p$ and the colimit complex is exact in every degree. Assertion 1 is [step 2.1] and the exactness of the filtered colimit functor itself is [step 1.1]; no choice principle is used anywhere, the proof resting only on the identification of abelian groups with $\mathbb{Z}$-modules and on AB5 for module categories. ∎