The Stacks project

Lemma 21.30.9. In Situation 21.30.1. For any $X$ in $\mathcal{C}$ the category $\mathcal{A}_ X \subset \textit{Ab}(\mathcal{C}_\tau /X)$ is a weak Serre subcategory and the functor

\[ R\epsilon _{X, *} : D^+_{\mathcal{A}_ X}(\mathcal{C}_\tau /X) \longrightarrow D^+_{\mathcal{A}'_ X}(\mathcal{C}_{\tau '}/X) \]

is an equivalence with quasi-inverse given by $\epsilon _ X^{-1}$.

Proof. We need to check the conditions listed in Homology, Lemma 12.10.3 for $\mathcal{A}_ X$. If $\varphi : \mathcal{F} \to \mathcal{G}$ is a map in $\mathcal{A}_ X$, then $\epsilon _{X, *}\varphi : \epsilon _{X, *}\mathcal{F} \to \epsilon _{X, *}\mathcal{G}$ is a map in $\mathcal{A}'_ X$. Hence $\mathop{\mathrm{Ker}}(\epsilon _{X, *}\varphi )$ and $\mathop{\mathrm{Coker}}(\epsilon _{X, *}\varphi )$ are objects of $\mathcal{A}'_ X$ as this is a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_{\tau '}/X)$. Applying $\epsilon _ X^{-1}$ we obtain an exact sequence

\[ 0 \to \epsilon _ X^{-1}\mathop{\mathrm{Ker}}(\epsilon _{X, *}\varphi ) \to \mathcal{F} \to \mathcal{G} \to \epsilon _ X^{-1}\mathop{\mathrm{Coker}}(\epsilon _{X, *}\varphi ) \to 0 \]

and we see that $\mathop{\mathrm{Ker}}(\varphi )$ and $\mathop{\mathrm{Coker}}(\varphi )$ are in $\mathcal{A}_ X$. Finally, suppose that

\[ 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 \]

is a short exact sequence in $\textit{Ab}(\mathcal{C}_\tau /X)$ with $\mathcal{F}_1$ and $\mathcal{F}_3$ in $\mathcal{A}_ X$. Then applying $\epsilon _{X, *}$ we obtain an exact sequence

\[ 0 \to \epsilon _{X, *}\mathcal{F}_1 \to \epsilon _{X, *}\mathcal{F}_2 \to \epsilon _{X, *}\mathcal{F}_3 \to R^1\epsilon _{X, *}\mathcal{F}_1 = 0 \]

Vanishing by Lemma 21.30.8. Hence $\epsilon _{X, *}\mathcal{F}_2$ is in $\mathcal{A}'_ X$ as this is a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_{\tau '}/X)$. Pulling back by $\epsilon _ X$ we conclude that $\mathcal{F}_2$ is in $\mathcal{A}_ X$.

Thus $\mathcal{A}_ X$ is a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_\tau /X)$ and it makes sense to consider the category $D^+_{\mathcal{A}_ X}(\mathcal{C}_\tau /X)$. Observe that $\epsilon _ X^{-1} : \mathcal{A}'_ X \to \mathcal{A}_ X$ is an equivalence and that $\mathcal{F}' \to R\epsilon _{X, *}\epsilon _ X^{-1}\mathcal{F}'$ is an isomorphism for $\mathcal{F}'$ in $\mathcal{A}'_ X$ since we have $(V_ n)$ for all $n$ by Lemma 21.30.8. Thus we conclude by Lemma 21.28.5. $\square$

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