Lemma 13.17.4. Let \mathcal{A} be an abelian category. Let \mathcal{B} \subset \mathcal{A} be a Serre subcategory. Assume that for every surjection X \to Y with X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) and Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B}) there exists X' \subset X, X' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B}) which surjects onto Y. Then the functor D^-(\mathcal{B}) \to D^-_\mathcal {B}(\mathcal{A}) of (13.17.1.1) is an equivalence.
Proof. Let X^\bullet be a bounded above complex of \mathcal{A} such that H^ i(X^\bullet ) \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B}) for all i \in \mathbf{Z}. Moreover, suppose we are given B^ i \subset X^ i, B^ i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B}) for all i \in \mathbf{Z}. Claim: there exists a subcomplex Y^\bullet \subset X^\bullet such that
Y^\bullet \to X^\bullet is a quasi-isomorphism,
Y^ i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B}) for all i \in \mathbf{Z}, and
B^ i \subset Y^ i for all i \in \mathbf{Z}.
To prove the claim, using the assumption of the lemma we first choose C^ i \subset \mathop{\mathrm{Ker}}(d^ i : X^ i \to X^{i + 1}), C^ i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B}) surjecting onto H^ i(X^\bullet ). Setting D^ i = C^ i + d^{i - 1}(B^{i - 1}) + B^ i we find a subcomplex D^\bullet satisfying (2) and (3) such that H^ i(D^\bullet ) \to H^ i(X^\bullet ) is surjective for all i \in \mathbf{Z}. For any choice of E^ i \subset X^ i with E^ i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B}) and d^ i(E^ i) \subset D^{i + 1} + E^{i + 1} we see that setting Y^ i = D^ i + E^ i gives a subcomplex whose terms are in \mathcal{B} and whose cohomology surjects onto the cohomology of X^\bullet . Clearly, if d^ i(E^ i) = (D^{i + 1} + E^{i + 1}) \cap \mathop{\mathrm{Im}}(d^ i) then we see that the map on cohomology is also injective. For n \gg 0 we can take E^ n equal to 0. By descending induction we can choose E^ i for all i with the desired property. Namely, given E^{i + 1}, E^{i + 2}, \ldots we choose E^ i \subset X^ i such that d^ i(E^ i) = (D^{i + 1} + E^{i + 1}) \cap \mathop{\mathrm{Im}}(d^ i). This is possible by our assumption in the lemma combined with the fact that (D^{i + 1} + E^{i + 1}) \cap \mathop{\mathrm{Im}}(d^ i) is in \mathcal{B} as \mathcal{B} is a Serre subcategory of \mathcal{A}.
The claim above implies the lemma. Essential surjectivity is immediate from the claim. Let us prove faithfulness. Namely, suppose we have a morphism f : U^\bullet \to V^\bullet of bounded above complexes of \mathcal{B} whose image in D(\mathcal{A}) is zero. Then there exists a quasi-isomorphism s : V^\bullet \to X^\bullet into a bounded above complex of \mathcal{A} such that s \circ f is homotopic to zero. Choose a homotopy h^ i : U^ i \to X^{i - 1} between 0 and s \circ f. Apply the claim with B^ i = h^{i + 1}(U^{i + 1}) + s^ i(V^ i). The resulting map s' : V^\bullet \to Y^\bullet is a quasi-isomorphism as well and s' \circ f is homotopic to zero as is clear from the fact that h^ i factors through Y^{i - 1}. This proves faithfulness. Fully faithfulness is proved in the exact same manner. \square
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