The Stacks project

Lemma 46.4.10. With notation as in Lemma 46.4.2 we have $R^ pQ(F) = 0$ for all $p > 0$ and any adequate functor $F$.

Proof. Choose an exact sequence $0 \to F \to \underline{M^0} \to \underline{M^1}$. Set $M^2 = \mathop{\mathrm{Coker}}(M^0 \to M^1)$ so that $0 \to F \to \underline{M^0} \to \underline{M^1} \to \underline{M^2} \to 0$ is a resolution. By Derived Categories, Lemma 13.21.3 we obtain a spectral sequence

\[ R^ pQ(\underline{M^ q}) \Rightarrow R^{p + q}Q(F) \]

Since $Q(\underline{M^ q}) = \underline{M^ q}$ it suffices to prove $R^ pQ(\underline{M}) = 0$, $p > 0$ for any $A$-module $M$.

Choose an injective resolution $\underline{M} \to I^\bullet $ in the category $\mathcal{P}$. Suppose that $R^ iQ(\underline{M})$ is nonzero. Then $\mathop{\mathrm{Ker}}(Q(I^ i) \to Q(I^{i + 1}))$ is strictly bigger than the image of $Q(I^{i - 1}) \to Q(I^ i)$. Hence by Lemma 46.3.6 there exists a linearly adequate functor $L$ and a map $\varphi : L \to Q(I^ i)$ mapping into the kernel of $Q(I^ i) \to Q(I^{i + 1})$ which does not factor through the image of $Q(I^{i - 1}) \to Q(I^ i)$. Because $Q$ is a left adjoint to the inclusion functor the map $\varphi $ corresponds to a map $\varphi ' : L \to I^ i$ with the same properties. Thus $\varphi '$ gives a nonzero element of $\mathop{\mathrm{Ext}}\nolimits ^ i_\mathcal {P}(L, \underline{M})$ contradicting Lemma 46.4.9. $\square$

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