Proposition 6

Tag 0DPE, permanent. Cite it as [Bourbaki, Tag 0DPE].

Algebra, chapter X, § 1. Compléments d’algèbre linéaire

Let A be a ring, E an a-module of finite presentation. For every exact sequence

0 \to F \xrightarrow{j} G \xrightarrow{p} E \to 0

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

Let L_1 \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 is therefore such that s = p \circ u. Since s \circ r = 0, we have u(r(L_1)) \subset \mathrm{Ker}\ p, and since Ker p is isomorphic to F, we see that there is a homomorphism v : L_1 \to F such that the diagram

\begin{array}{ccccccc} L_1 & \xrightarrow{r} & L_0 & \xrightarrow{s} & E & \to & 0 \\ v \downarrow & & u \downarrow & & 1_E \downarrow & & \\ F & \xrightarrow{j} & G & \xrightarrow{p} & E & \to & 0 \end{array}

is commutative. Since j is injective and s surjective, we can apply Proposition 2 of X, p. 4, in other words there is an exact sequence

\mathrm{Ker}\ 1_E \xrightarrow{d} \mathrm{Coker}\ v \to \mathrm{Coker}\ u \to \mathrm{Coker}\ 1_E.

This shows that Coker v is isomorphic to G/u(L_0), which is finitely generated by assumption. We have in addition the exact sequence

0 \to v(L_1) \to F \to \mathrm{Coker}\ v \to 0

and since v(L_1) and Coker v are finitely generated, the same is true of F (II, p. 17, cor. 5).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Cites

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