Proposition 7

Let M be an a-module. There exists an ordered set I inductive to the right and an inductive system of a-modules of finite presentation $(M_\alpha, \varphi_{\beta\alpha})$ relative to I such that M is isomorphic to $\lim_\alpha M_\alpha$. If M possesses a generating system of n elements, we can suppose that the same is true of the $M_\alpha$.

Consider a presentation

let I be the set of pairs $\alpha = (K', L')$, where K' (resp. L') is a finite subset of K (resp.L), such that u induces a mapping u_\alpha from the submodule $A_s^{K'}$ of $A_s^{(K)}$ into the submodule $A_s^{L'}$ of $A_s^{(L)}$ ; for $\alpha \in I$, let $M_\alpha$ be the cokernel of $u_\alpha$ and $v_\alpha : A_s^{L'} \to M_\alpha$ the canonical mapping, so that we have a commutative diagram with exact rows:

where $i_\alpha$ and $j_\alpha$ are the canonical injections, and where $f_\alpha$ is deduced from $j_\alpha$ by passing to quotients. We order the set I by the relation

for $\alpha\leq\beta$, let $\varphi_{\beta\alpha}:M_\alpha\longrightarrow M_\beta$ be the homomorphism deduced by passing to quotients from the inclusion of $A_s^{L'}$ in $A_s^{L''}$. One verifies immediately that the ordered set I is inductive, that $(M_\alpha,\varphi_{\beta\alpha})$ is an inductive system of a-modules and that $(\varphi_\alpha)$ is an inductive system of a-homomorphisms. By passing to the inductive limit, we obtain a commutative diagram

the rows of this diagram are exact (II, p. 91, prop. 3); since $i$ and $j$ are bijective, $\varphi$ is so also (X, p. 7, cor. 3), whence the proposition.