## 10.90 Coherent rings

We use the discussion on interchanging $\prod $ and $\otimes $ to determine for which rings products of flat modules are flat. It turns out that these are the so-called coherent rings. You may be more familiar with the notion of a coherent $\mathcal{O}_ X$-module on a ringed space, see Modules, Section 17.12.

Definition 10.90.1. Let $R$ be a ring. Let $M$ be an $R$-module.

We say $M$ is a *coherent module* if it is finitely generated and every finitely generated submodule of $M$ is finitely presented over $R$.

We say $R$ is a *coherent ring* if it is coherent as a module over itself.

Thus a ring is coherent if and only if every finitely generated ideal is finitely presented as a module.

Example 10.90.2. A valuation ring is a coherent ring. Namely, every nonzero finitely generated ideal is principal (Lemma 10.50.15), hence free as a valuation ring is a domain, hence finitely presented.

The category of coherent modules is abelian.

Lemma 10.90.3. Let $R$ be a ring.

A finite submodule of a coherent module is coherent.

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

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

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.

Let $\varphi : N \to M$ satisfy the assumptions of (2). First, $\mathop{\mathrm{Im}}(\varphi )$ is finite, hence coherent by (1). In particular $\mathop{\mathrm{Im}}(\varphi )$ is finitely presented, so applying Lemma 10.5.3 to the exact sequence $0\to \mathop{\mathrm{Ker}}(\varphi )\to N\to \mathop{\mathrm{Im}}(\varphi )\to 0$ we see that $\mathop{\mathrm{Ker}}(\varphi )$ is finite. To prove that $\mathop{\mathrm{Coker}}(\varphi )$ is coherent, let $E \subset \mathop{\mathrm{Coker}}(\varphi )$ be a finite subomdule, and let $E'$ be its inverse image in $M$. From the exact sequence $0 \to \mathop{\mathrm{Ker}}(\varphi ) \to E' \to E \to 0$ and since $\mathop{\mathrm{Ker}}(\varphi )$ is finite we conclude by Lemma 10.5.3 that $E'\subset M$ is finite, hence finitely presented because $M$ is coherent. The same exact sequence then shows that $E$ is finitely presented, whence our claim.

Part (3) follows immediately from (1) and (2).

Let $0 \to M_1 \xrightarrow {i} M_2 \xrightarrow {p} M_3 \to 0$ be a short exact sequence of $R$-modules as in (4). It remains 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 $N_2 \subset M_2$ be a finite submodule. Put $N_3 = p(N_2) \subset M_3$ and $N_1 = i^{-1}(N_2) \subset M_1$. We have an exact sequence $0 \to N_1 \to N_2 \to N_3 \to 0$. Clearly $N_3$ is finite (as a quotient of $N_2$), hence finitely presented (as a finite submodule of $M_3$). It follows by Lemma 10.5.3 (5) that $N_1$ is finite, hence finitely presented (as a finite submodule of $M_1$). We conclude by Lemma 10.5.3 (2) that $M_2$ is finitely presented.
$\square$

Lemma 10.90.4. Let $R$ be a ring. If $R$ is coherent, then a module is coherent if and only if it is finitely presented.

**Proof.**
It is clear that a coherent module is finitely presented (over any ring). Conversely, if $R$ is coherent, then $R^{\oplus n}$ is coherent and so is the cokernel of any map $R^{\oplus m} \to R^{\oplus n}$, see Lemma 10.90.3.
$\square$

Lemma 10.90.5. A Noetherian ring is a coherent ring.

**Proof.**
By Lemma 10.31.4 any finite $R$-module is finitely presented. In particular any ideal of $R$ is finitely presented.
$\square$

reference
Proposition 10.90.6. Let $R$ be a ring. The following are equivalent

$R$ is coherent,

any product of flat $R$-modules is flat, and

for every set $A$ the module $R^ A$ is flat.

**Proof.**
Assume $R$ coherent, and let $Q_\alpha $, $\alpha \in A$ be a set of flat $R$-modules. We have to show that $I \otimes _ R \prod _\alpha Q_\alpha \to \prod Q_\alpha $ is injective for every finitely generated ideal $I$ of $R$, see Lemma 10.39.5. Since $R$ is coherent $I$ is an $R$-module of finite presentation. Hence $I \otimes _ R \prod _\alpha Q_\alpha = \prod I \otimes _ R Q_\alpha $ by Proposition 10.89.3. The desired injectivity follows as $I \otimes _ R Q_\alpha \to Q_\alpha $ is injective by flatness of $Q_\alpha $.

The implication (2) $\Rightarrow $ (3) is trivial.

Assume that the $R$-module $R^ A$ is flat for every set $A$. Let $I$ be a finitely generated ideal in $R$. Then $I \otimes _ R R^ A \to R^ A$ is injective by assumption. By Proposition 10.89.2 and the finiteness of $I$ the image is equal to $I^ A$. Hence $I \otimes _ R R^ A = I^ A$ for every set $A$ and we conclude that $I$ is finitely presented by Proposition 10.89.3.
$\square$

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