This is from Cartan-Eilenberg's book "Homological Algebra".
Lemma. If the $R$-module $M$ is a direct limit of of the direct system $(u_{ij}\colon M_i\to M_j)_{i \leq j}$ of $R$-modules together with canonical injections $u_i\colon M_i\to M$, then there exists projective resolutions of $M_i$ for all $i \in I$ such that their direct limit is a projective resolution of $M$.
The proof provided there is quite short and lacks details. The idea is to construct a projective resolution by recursion.
First, let $F_i = R^{M_i}, F = R^M, \pi_i\colon F_i\to M_i$ is a surjective homomorphism which maps $\sum_{m \in M_i} r_me_{i,m}$ to $\sum_{m \in M_i} r_m m$ and $\pi\colon F\to M$ is, similarly, a surjective homomorphism which maps $\sum_{m \in M} r_me_m$ to $\sum_{m \in M} r_mm$. More generally, a free $R$-module functor $F\colon\mathsf{Set}\to R\text{-}\mathsf{Mod}$, I think, restricts to the free functor $F{\restriction}_{R\text{-}\mathsf{Mod}}\colon R\text{-}\mathsf{Mod}\to R\text{-}\mathsf{Mod}$ (defined by the universal property of a free object in a category). By abstract nonsense, free functors are left adjoint, hence they preserve colimits. Now $F(N) = R^N$ and $F(\phi\colon N\to P)$ is a unique $R$-module homomorphism $F(\phi)\colon R^N\to R^P$ which maps $\sum_{n \in N} r_n e_n$ to $\sum_{n \in N} r_n e_{\phi(n)}$. In particular, we have $\pi_j\circ F(u_{ij}) = u_{ij}\circ \pi_i$ and $\pi\circ F(u_i) = u_i\circ \pi_i$. Now since $F{\restriction}_{R\text{-}\mathsf{Mod}}$ preserves colimits, $R^M$ is a direct limit of the direct system $(F(u_{ij}))_{i \leq j}$ together with canonical injections $F(u_i)$.
Next the authors claim that $\ker(\pi)$ is a direct limits of $\ker(\pi_i)$. Concretely, $v_{ij}\colon \ker(\pi_i)\to \ker(\pi_j)$ and $v_i\colon \ker(\pi_i)\to \ker(\pi)$ are restrictions of $F(u_{ij})$ and $F(u_i)$, respectively, defined by the universal property of a kernel. What I don't see is why $(v_i\colon \ker(\pi_i)\to \ker(\pi))_{i \in I}$ form a direct limit of the direct systyem $(v_{ij}\colon \ker(\pi_i)\to \ker(\pi_j))_{i \leq j}$. Does it have something to do with $(F(u_i))_{i \in I}$ being a direct limit of $(F(u_{ij}))_{i \leq j}$?
Best Answer
Maybe the point you're missing is the following fact (be careful, though, that it does not generalize to arbitrary abelian categories, the ones where it is true are something like Grothendieck categories):
In particular, here, you have $0\to \ker(\pi_i)\to R[M_i]\to M_i\to 0$ which is a direct system of short exact sequences, so its colimit is also exact.
But $\mathrm{colim}_i M_i = M$ by definition, $\mathrm{colim}_i R[M_i] = R[M]$ because $R[-]$ commutes with filtered colimits (again, following the comments : not all colimits, but it is true for filtered ones).
Therefore the colimit short exact sequence is $0\to \mathrm{colim}_i \ker(\pi_i)\to R[M]\to M\to 0$. But the kernel of $R[M]\to M$ is, by definition, $\ker(\pi)$, so $\mathrm{colim}_i \ker(\pi_i) = \ker(\pi)$ (you may check easily that the maps $\ker(\pi_i)\to \ker(\pi)$ are the correct ones)