Lemma 9

Let $A$ be a ring and $E$ a finitely presented $A$-module. For every exact sequence

where $G$ is finitely generated, the module $F$ is finitely generated.

Let $L, \xrightarrow{r} L_0 \xrightarrow{s} E \to 0$ be a finite presentation; if $(e_i)$ is a basis of $L_0$, there exists for each $i$ an element $g_i \in G$ such that $p(g_i) = s(e_i)$; the homomorphism $u : L_0 \to G$ such that $u(e_i) = g_i$ for all $i$ then satisfies $s = p \circ u$. As $s \circ r = 0$, we have $u(r(L_1)) \subset \mathrm{Ker}\, p$, and as $\mathrm{Ker}\, p$ is isomorphic to $F$, it can be seen that there is a homomorphism $v : L, \to F$ such that the diagram

is commutative. As $j$ is injective and $s$ surjective, we can apply the snake diagram ($\S$ 1, no. 4, Proposition 4), in other words there is an exact sequence

This shows that $\mathrm{Coker}\, v$ is isomorphic to $G / u(L_0)$, which is finitely generated by hypothesis. Moreover we have the exact sequence

and as $v(L_1)$ and $\mathrm{Coker}\, v$ are finitely generated, so is $F$ (Algebra, Chapter II, § 1, no. 7, Corollary 5 to Proposition 9).