Lemma 10.90.3. Let $R$ be a ring.

1. A finite submodule of a coherent module is coherent.

2. Let $\varphi : N \to M$ be a homomorphism from a finite module to a coherent module. Then $\mathop{\mathrm{Ker}}(\varphi )$ is finite.

3. Let $\varphi : N \to M$ be a homomorphism of coherent modules. Then $\mathop{\mathrm{Ker}}(\varphi )$ and $\mathop{\mathrm{Coker}}(\varphi )$ are coherent modules.

4. Given a short exact sequence of $R$-modules $0 \to M_1 \to M_2 \to M_3 \to 0$ if two out of three are coherent so is the third.

Proof. The first statement is immediate from the definition. During the rest of the proof we will use the results of Lemma 10.5.3 without further mention.

Let $\varphi : N \to M$ satisfy the assumptions of (2). Suppose that $N$ is generated by $x_1, \ldots , x_ n$. By Definition 10.90.1 the kernel $K$ of the induced map $R^{\oplus n} \to M$, $e_ i \mapsto \varphi (x_ i)$ is of finite type. Hence $\mathop{\mathrm{Ker}}(\varphi )$ which is the image of the composition $K \to R^{\oplus n} \to N$ is of finite type. This proves (2).

Let $\varphi : N \to M$ satisfy the assumptions of (3). By (2) the kernel of $\varphi$ is of finite type and hence by (1) it is coherent.

With the same hypotheses let us show that $\mathop{\mathrm{Coker}}(\varphi )$ is coherent. Since $M$ is finite so is $\mathop{\mathrm{Coker}}(\varphi )$. Let $\overline{x}_ i \in \mathop{\mathrm{Coker}}(\varphi )$. We have to show that the kernel of the associated morphism $\overline{\Psi } : R^{\oplus n} \to \mathop{\mathrm{Coker}}(\varphi )$ is finite. Choose $x_ i \in M$ lifting $\overline{x}_ i$. Choose additionally generators $y_1, \ldots , y_ m$ of $\mathop{\mathrm{Im}}(\varphi )$. Let $\Phi : R^{\oplus m} \to \mathop{\mathrm{Im}}(\varphi )$ using $y_ j$ and $\Psi : R^{\oplus m} \oplus R^{\oplus n} \to M$ using $y_ j$ and $x_ i$ be the corresponding maps. Consider the following commutative diagram

$\xymatrix{ 0 \ar[r] & R^{\oplus m} \ar[d]_\Phi \ar[r] & R^{\oplus m} \oplus R^{\oplus n} \ar[d]_\Psi \ar[r] & R^{\oplus n} \ar[d]_{\overline{\Psi }} \ar[r] & 0 \\ 0 \ar[r] & \mathop{\mathrm{Im}}(\varphi ) \ar[r] & M \ar[r] & \mathop{\mathrm{Coker}}(\varphi ) \ar[r] & 0 }$

with exact rows. By Lemma 10.4.1 we get an exact sequence $\mathop{\mathrm{Ker}}(\Psi ) \to \mathop{\mathrm{Ker}}(\overline{\Psi }) \to 0$. Since $\mathop{\mathrm{Ker}}(\Psi )$ is a finite $R$-module, we see that $\mathop{\mathrm{Ker}}(\overline{\Psi })$ is finite.

Statement (4) follows from (3).

Let $0 \to M_1 \to M_2 \to M_3 \to 0$ be a short exact sequence of $R$-modules. It suffices to prove that if $M_1$ and $M_3$ are coherent so is $M_2$. By Lemma 10.5.3 we see that $M_2$ is finite. Let $x_1, \ldots , x_ n$ be finitely many elements of $M_2$. We have to show that the module of relations $K$ between them is finite. Consider the following commutative diagram

$\xymatrix{ 0 \ar[r] & 0 \ar[r] \ar[d] & \bigoplus _{i = 1}^{n} R \ar[r] \ar[d] & \bigoplus _{i = 1}^{n} R \ar[r] \ar[d] & 0 \\ 0 \ar[r] & M_1 \ar[r] & M_2 \ar[r] & M_3 \ar[r] & 0 }$

with obvious notation. By the snake lemma we get an exact sequence $0 \to K \to K_3 \to M_1$ where $K_3$ is the module of relations among the images of the $x_ i$ in $M_3$. Since $M_3$ is coherent we see that $K_3$ is a finite module. Since $M_1$ is coherent we see that the image $I$ of $K_3 \to M_1$ is coherent. Hence $K$ is the kernel of the map $K_3 \to I$ between a finite module and a coherent module and hence finite by (2). $\square$

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