The Stacks project

Lemma 21.29.1. With $\epsilon : (\mathcal{C}_\tau , \mathcal{O}_\tau ) \to (\mathcal{C}_{\tau '}, \mathcal{O}_{\tau '})$ as above. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Let $\mathcal{A} \subset \textit{PMod}(\mathcal{O})$ be a full subcategory. Assume

  1. every object of $\mathcal{A}$ is a sheaf for the $\tau $-topology,

  2. $\mathcal{A}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_\tau )$,

  3. every object of $\mathcal{C}$ has a $\tau '$-covering whose members are elements of $\mathcal{B}$, and

  4. for every $U \in \mathcal{B}$ we have $H^ p_\tau (U, \mathcal{F}) = 0$, $p > 0$ for all $\mathcal{F} \in \mathcal{A}$.

Then $\mathcal{A}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_{\tau '})$ and there is an equivalence of triangulated categories $D_\mathcal {A}(\mathcal{O}_\tau ) = D_\mathcal {A}(\mathcal{O}_{\tau '})$ given by $\epsilon ^*$ and $R\epsilon _*$.

Proof. Since $\epsilon ^{-1}\mathcal{O}_{\tau '} = \mathcal{O}_\tau $ we see that $\epsilon $ is a flat morphism of ringed sites and that in fact $\epsilon ^{-1} = \epsilon ^*$ on sheaves of modules. By property (1) we can think of every object of $\mathcal{A}$ as a sheaf of $\mathcal{O}_\tau $-modules and as a sheaf of $\mathcal{O}_{\tau '}$-modules. In other words, we have fully faithful inclusion functors

\[ \mathcal{A} \to \textit{Mod}(\mathcal{O}_\tau ) \to \textit{Mod}(\mathcal{O}_{\tau '}) \]

To avoid confusion we will denote $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}_{\tau '})$ the image of $\mathcal{A}$. Then it is clear that $\epsilon _* : \mathcal{A} \to \mathcal{A}'$ and $\epsilon ^* : \mathcal{A}' \to \mathcal{A}$ are quasi-inverse equivalences (see discussion preceding the lemma and use that objects of $\mathcal{A}'$ are sheaves in the $\tau $ topology).

Conditions (3) and (4) imply that $R^ p\epsilon _*\mathcal{F} = 0$ for $p > 0$ and $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. This is true because $R^ p\epsilon _*$ is the sheaf associated to the presheave $U \mapsto H^ p_\tau (U, \mathcal{F})$, see Lemma 21.7.4. Thus any exact complex in $\mathcal{A}$ (which is the same thing as an exact complex in $\textit{Mod}(\mathcal{O}_\tau )$ whose terms are in $\mathcal{A}$, see Homology, Lemma 12.10.3) remains exact upon applying the functor $\epsilon _*$.

Consider an exact sequence

\[ \mathcal{F}'_0 \to \mathcal{F}'_1 \to \mathcal{F}'_2 \to \mathcal{F}'_3 \to \mathcal{F}'_4 \]

in $\textit{Mod}(\mathcal{O}_{\tau '})$ with $\mathcal{F}'_0, \mathcal{F}'_1, \mathcal{F}'_3, \mathcal{F}'_4$ in $\mathcal{A}'$. Apply the exact functor $\epsilon ^*$ to get an exact sequence

\[ \epsilon ^*\mathcal{F}'_0 \to \epsilon ^*\mathcal{F}'_1 \to \epsilon ^*\mathcal{F}'_2 \to \epsilon ^*\mathcal{F}'_3 \to \epsilon ^*\mathcal{F}'_4 \]

in $\textit{Mod}(\mathcal{O}_\tau )$. Since $\mathcal{A}$ is a weak Serre subcategory and since $\epsilon ^*\mathcal{F}'_0, \epsilon ^*\mathcal{F}'_1, \epsilon ^*\mathcal{F}'_3, \epsilon ^*\mathcal{F}'_4$ are in $\mathcal{A}$, we conclude that $\epsilon ^*\mathcal{F}_2$ is in $\mathcal{A}$ by Homology, Definition 12.10.1. Consider the map of sequences

\[ \xymatrix{ \mathcal{F}'_0 \ar[r] \ar[d] & \mathcal{F}'_1 \ar[r] \ar[d] & \mathcal{F}'_2 \ar[r] \ar[d] & \mathcal{F}'_3 \ar[r] \ar[d] & \mathcal{F}'_4 \ar[d] \\ \epsilon _*\epsilon ^*\mathcal{F}'_0 \ar[r] & \epsilon _*\epsilon ^*\mathcal{F}'_1 \ar[r] & \epsilon _*\epsilon ^*\mathcal{F}'_2 \ar[r] & \epsilon _*\epsilon ^*\mathcal{F}'_3 \ar[r] & \epsilon _*\epsilon ^*\mathcal{F}'_4 } \]

The lower row is exact by the discussion in the preceding paragraph. The vertical arrows with index $0$, $1$, $3$, $4$ are isomorphisms by the discussion in the first paragraph. By the $5$ lemma (Homology, Lemma 12.5.20) we find that $\mathcal{F}'_2 \cong \epsilon _*\epsilon ^*\mathcal{F}'_2$ and hence $\mathcal{F}'_2$ is in $\mathcal{A}'$. In this way we see that $\mathcal{A}'$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_{\tau '})$, see Homology, Definition 12.10.1.

At this point it makes sense to talk about the derived categories $D_\mathcal {A}(\mathcal{O}_\tau )$ and $D_{\mathcal{A}'}(\mathcal{O}_{\tau '})$, see Derived Categories, Section 13.17. To finish the proof we show that conditions (1) – (5) of Lemma 21.28.7 apply. We have already seen (1), (2), (3) above. Note that since every object has a $\tau '$-covering by objects of $\mathcal{B}$, a fortiori every object has a $\tau $-covering by objects of $\mathcal{B}$. Hence condition (4) of Lemma 21.28.7 is satisfied. Similarly, condition (5) is satisfied as well. $\square$

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