The Stacks project

Lemma 56.2.4. Let $A$, $\mathcal{B}$, $F$ be as in Lemma 56.2.1. Assume $A$ is a coherent ring (Algebra, Definition 10.90.1), $\mathcal{B}$ is additive, has kernels, filtered colimits commute with taking kernels, and $F$ is left exact. Then $F'$ is additive, left exact, and commutes with arbitrary direct sums.

Proof. Since $A$ is coherent, the category $\text{Mod}^{fp}_ A$ is abelian with same kernels and cokernels as in $\text{Mod}_ A$, see Algebra, Lemmas 10.90.4 and 10.90.3. Hence all finite limits exist in $\text{Mod}^{fp}_ A$ and Categories, Definition 4.23.1 applies. Since $F$ is left exact, $F$ commutes with products of pairs, which are represented by direct sums. Hence $F$ is additive by Homology, Lemma 12.7.1. Hence $F'$ is additive and commutes with direct sums by Lemma 56.2.2. We urge the reader to prove that $F'$ is left exact themselves instead of reading the proof below.

To show that $F'$ is left exact, it suffices to show that $F'$ commutes with equalizers, see Categories, Lemma 4.23.2. Now, if $a, b : L \to M$ are maps of $A$-modules, then the equalizer of $a$ and $b$ is the kernel of $a - b : L \to M$. Thus let $0 \to K \to L \to M$ be an exact sequence of $A$-modules. We have to show that in

\[ 0 \to F'(K) \to F'(L) \to F'(M) \]

the arrow $F'(K) \to F'(L)$ is a kernel for $F'(L) \to F'(M)$ in $\mathcal{B}$ (if $\mathcal{B}$ were abelian we would say that the displayed sequence is exact). Write $M = \mathop{\mathrm{colim}}\nolimits _{i \in I} M_ i$ as a filtered colimit of finitely presented $A$-modules, see Algebra, Lemma 10.11.3. Let $L_ i = L \times _ M M_ i$. We obtain a system of exact sequences $0 \to K \to L_ i \to M_ i$ over $I$. Since filtered colimits commute with taking kernels in $\mathcal{B}$ by assumption, it suffices to show that $F'(K) \to F'(L_ i)$ is a kernel of $F'(L_ i) \to F(M_ i)$ in $\mathcal{B}$ for all $i \in I$. In other words, we may assume $M$ is finitely presented. Write $L = \mathop{\mathrm{colim}}\nolimits _{i \in I} L_ i$ as a filtered colimit of finitely presented $A$-modules. Let $K_ i = K \times _ L L_ i$. We obtain a system of short exact sequences $0 \to K_ i \to L_ i \to M$ over $I$. Repeating the argument already given, we reduce to showing $F'(K_ i) \to F(L_ i)$ is a kernel of $F(L_ i) \to F(M)$ in $\mathcal{B}$ for all $i \in I$. In other words, we may assume both $L$ and $M$ are finitely presented $A$-modules. Since $A$ is coherent, the $A$-module $K = \mathop{\mathrm{Ker}}(L \to M)$ is of finite presentation as the category of finitely presented $A$-modules is abelian (see references given above). In other words, all three modules $K$, $L$, and $M$ are finitely presented $A$-modules. This final case follows from the assumption that $F$ is left exact. $\square$

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