Lemma 84.24.4 (Unbounded BBD glueing lemma). In Situation 84.24.1. Assume

1. $\mathcal{C}$ has equalizers and fibre products,

2. there is a morphism of sites $f : \mathcal{C} \to \mathcal{D}$ given by a continuous functor $u : \mathcal{D} \to \mathcal{C}$ such that

1. $\mathcal{D}$ has equalizers and fibre products and $u$ commutes with them,

2. $\mathcal{B}$ is a full subcategory of $\mathcal{D}$ and $u : \mathcal{B} \to \mathcal{C}$ is the restriction of $u$,

3. every object of $\mathcal{D}$ has a covering whose members are objects of $\mathcal{B}$,

3. all negative self-exts of $E_ U$ in $D(\mathcal{O}_{u(U)})$ are zero, and

4. there exist weak Serre subcategories $\mathcal{A}_ U \subset \textit{Mod}(\mathcal{O}_ U)$ for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ satisfying conditions (1), (2), and (3),

5. $E_ U \in D_{\mathcal{A}_ U}(\mathcal{O}_ U)$.

Then there exists a solution unique up to unique isomorphism.

Proof. The proof is exactly the same as the proof of Lemma 84.24.3. The only change is that $E$ is an object of $D_{\mathcal{A}_{total}}(\mathcal{O})$ and hence we use Lemma 84.23.3 to obtain $F$ with $E = a^*F$ instead of Lemma 84.18.4. $\square$

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