Lemma 21.19.8. Let $\mathcal{C}$ be a site. Let $\mathcal{A} \subset \textit{Ab}(\mathcal{C})$ denote the Serre subcategory consisting of torsion abelian sheaves. Then the functor $D(\mathcal{A}) \to D_\mathcal {A}(\mathcal{C})$ is an equivalence.
Proof. A key observation is that an injective abelian sheaf $\mathcal{I}$ is divisible. Namely, if $s \in \mathcal{I}(U)$ is a local section, then we interpret $s$ as a map $s : j_{U!}\mathbf{Z} \to \mathcal{I}$ and we apply the defining property of an injective object to the injective map of sheaves $n : j_{U!}\mathbf{Z} \to j_{U!}\mathbf{Z}$ to see that there exists an $s' \in \mathcal{I}(U)$ with $ns' = s$.
For a sheaf $\mathcal{F}$ denote $\mathcal{F}_{tor}$ its torsion subsheaf. We claim that if $\mathcal{I}^\bullet $ is a complex of injective abelian sheaves whose cohomology sheaves are torsion, then
\[ \mathcal{I}^\bullet _{tor} \to \mathcal{I}^\bullet \]
is a quasi-isomorphism. Namely, by flatness of $\mathbf{Q}$ over $\mathbf{Z}$ we have
\[ H^ p(\mathcal{I}^\bullet ) \otimes _\mathbf {Z} \mathbf{Q} = H^ p(\mathcal{I}^\bullet \otimes _\mathbf {Z} \mathbf{Q}) \]
which is zero because the cohomology sheaves are torsion. By divisibility (shown above) we see that $\mathcal{I}^\bullet \to \mathcal{I}^\bullet \otimes _\mathbf {Z} \mathbf{Q}$ is surjective with kernel $\mathcal{I}^\bullet _{tor}$. The claim follows from the long exact sequence of cohomology sheaves associated to the short exact sequence you get.
To prove the lemma we will construct right adjoint $T : D(\mathcal{C}) \to D(\mathcal{A})$. Namely, given $K$ in $D(\mathcal{C})$ we can represent $K$ by a K-injective complex $\mathcal{I}^\bullet $ whose cohomology sheaves are injective, see Injectives, Theorem 19.12.6. Then we set $T(K) = \mathcal{I}^\bullet _{tor}$, in other words, $T$ is the right derived functor of taking torsion. The functor $T$ is a right adjoint to $i : D(\mathcal{A}) \to D_\mathcal {A}(\mathcal{C})$. This readily follows from the observation that if $\mathcal{F}^\bullet $ is a complex of torsion sheaves, then
\[ \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(\mathcal{F}^\bullet , I^\bullet _{tor}) = \mathop{\mathrm{Hom}}\nolimits _{K(\textit{Ab}(\mathcal{C}))}(\mathcal{F}^\bullet , I^\bullet ) \]
in particular $\mathcal{I}^\bullet _{tor}$ is a K-injective complex of $\mathcal{A}$. Some details omitted; in case of doubt, it also follows from the more general Derived Categories, Lemma 13.30.3. Our claim above gives that $L = T(i(L))$ for $L$ in $D(\mathcal{A})$ and $i(T(K)) = K$ if $K$ is in $D_\mathcal {A}(\mathcal{C})$. Using Categories, Lemma 4.24.4 the result follows. $\square$
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