Lemma 56.2.3. Let $A$, $\mathcal{B}$, $F$ be as in Lemma 56.2.1. Assume $\mathcal{B}$ is additive, has cokernels, and $F$ is right exact. Then $F'$ is additive, right exact, and commutes with arbitrary direct sums.

Proof. Since $F$ is right exact, $F$ commutes with coproducts 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 right exact themselves instead of reading the proof below.

To show that $F'$ is right exact, it suffices to show that $F'$ commutes with coequalizers, see Categories, Lemma 4.23.3. Now, if $a, b : K \to L$ are maps of $A$-modules, then the coequalizer of $a$ and $b$ is the cokernel of $a - b : K \to L$. Thus let $K \to L \to M \to 0$ be an exact sequence of $A$-modules. We have to show that in

$F'(K) \to F'(L) \to F'(M) \to 0$

the second arrow is a cokernel for the first arrow 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 $K \to L_ i \to M_ i \to 0$ over $I$. Since colimits commute with colimits by Categories, Lemma 4.14.10 and since cokernels are a type of coequalizer, it suffices to show that $F'(L_ i) \to F(M_ i)$ is a cokernel of $F'(K) \to F'(L_ i)$ 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 with the property that each $L_ i$ surjects onto $M$. Let $K_ i = K \times _ L L_ i$. We obtain a system of short exact sequences $K_ i \to L_ i \to M \to 0$ over $I$. Repeating the argument already given, we reduce to showing $F(L_ i) \to F(M_ i)$ is a cokernel of $F'(K) \to F(L_ i)$ in $\mathcal{B}$ for all $i \in I$. In other words, we may assume both $L$ and $M$ are finitely presented $A$-modules. In this case the module $\mathop{\mathrm{Ker}}(L \to M)$ is finite (Algebra, Lemma 10.5.3). Thus we can write $K = \mathop{\mathrm{colim}}\nolimits _{i \in I} K_ i$ as a filtered colimit of finitely presented $A$-modules each surjecting onto $\mathop{\mathrm{Ker}}(L \to M)$. We obtain a system of short exact sequences $K_ i \to L \to M \to 0$ over $I$. Repeating the argument already given, we reduce to showing $F(L) \to F(M)$ is a cokernel of $F(K_ i) \to F(L)$ in $\mathcal{B}$ for all $i \in I$. In other words, we may assume $K$, $L$, and $M$ are finitely presented $A$-modules. This final case follows from the assumption that $F$ is right exact. $\square$

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