The Stacks project

Lemma 46.3.16. Let $A$ be a ring. Let $0 \to F \to G \to H \to 0$ be a short exact sequence of module-valued functors on $\textit{Alg}_ A$. If $F$ and $H$ are adequate, so is $G$.

Proof. Choose an exact sequence $0 \to F \to \underline{M} \to \underline{N}$. If we can show that $(\underline{M} \oplus G)/F$ is adequate, then $G$ is the kernel of the map of adequate functors $(\underline{M} \oplus G)/F \to \underline{N}$, hence adequate by Lemma 46.3.11. Thus we may assume $F = \underline{M}$.

We can choose a surjection $L \to H$ where $L$ is a direct sum of linearly adequate functors, see Lemma 46.3.6. If we can show that the pullback $G \times _ H L$ is adequate, then $G$ is the cokernel of the map $\mathop{\mathrm{Ker}}(L \to H) \to G \times _ H L$ hence adequate by Lemma 46.3.10. Thus we may assume that $H = \bigoplus L_ i$ is a direct sum of linearly adequate functors. By Lemma 46.3.14 each of the pullbacks $G \times _ H L_ i$ is adequate. By Lemma 46.3.12 we see that $\bigoplus G \times _ H L_ i$ is adequate. Then $G$ is the cokernel of

\[ \bigoplus \nolimits _{i \not= i'} F \longrightarrow \bigoplus G \times _ H L_ i \]

where $\xi $ in the summand $(i, i')$ maps to $(0, \ldots , 0, \xi , 0, \ldots , 0, -\xi , 0, \ldots , 0)$ with nonzero entries in the summands $i$ and $i'$. Thus $G$ is adequate by Lemma 46.3.10. $\square$

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