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

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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)