The Stacks project

Lemma 46.4.9. Let $A$ be a ring. Let $M$ be an $A$-module. Let $L$ be a linearly adequate functor on $\textit{Alg}_ A$. Then $\mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {P}(L, \underline{M}) = 0$ for $i > 0$ where $\mathcal{P}$ is the category of module-valued functors on $\textit{Alg}_ A$.

Proof. Since $L$ is linearly adequate there exists an exact sequence

\[ 0 \to L \to \underline{A^{\oplus m}} \to \underline{A^{\oplus n}} \to \underline{P} \to 0 \]

Here $P = \mathop{\mathrm{Coker}}(A^{\oplus m} \to A^{\oplus n})$ is the cokernel of the map of finite free $A$-modules which is given by the definition of linearly adequate functors. By Lemma 46.4.8 we have the vanishing of $\mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {P}(\underline{P}, \underline{M})$ and $\mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {P}(\underline{A}, \underline{M})$ for $i > 0$. Let $K = \mathop{\mathrm{Ker}}(\underline{A^{\oplus n}} \to \underline{P})$. By the long exact sequence of Ext groups associated to the exact sequence $0 \to K \to \underline{A^{\oplus n}} \to \underline{P} \to 0$ we conclude that $\mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {P}(K, \underline{M}) = 0$ for $i > 0$. Repeating with the sequence $0 \to L \to \underline{A^{\oplus m}} \to K \to 0$ we win. $\square$

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