The Stacks project

Lemma 103.5.4. Let $\mathcal{X}$ be an algebraic stack. Let $E$ be an object of $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$. There exists a canonical distinguished triangle

\[ E' \to E \to P \to E'[1] \]

in $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ such that $P$ is in $D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}} (\mathcal{O}_\mathcal {X})$ and

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_\mathcal {X})}(E', P') = 0 \]

for all $P'$ in $D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$.

Proof. Consider the morphism of ringed topoi $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{flat, fppf}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf})$ studied in Cohomology of Stacks, Section 102.14. Set $E' = Lg_!g^*E$ and let $P$ be the cone on the adjunction map $E' \to E$, see Lemma 103.3.1 part (4). By Lemma 103.5.3 parts (2)(a) and (2)(c) we have that $E'$ is in $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$. Hence also $P$ is in $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$. The map $g^*E' \to g^*E$ is an isomorphism as $g^*Lg_! = \text{id}$ by Lemma 103.3.1 part (4). Hence $g^*P = 0$ and whence $P$ is an object of $D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ by Lemma 103.5.3 part (2)(b). Finally, for $P'$ in $D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ we have

\[ \mathop{\mathrm{Hom}}\nolimits (E', P') = \mathop{\mathrm{Hom}}\nolimits (Lg_!g^*E, P') = \mathop{\mathrm{Hom}}\nolimits (g^*E, g^*P') = 0 \]

as $g^*P' = 0$ by Lemma 103.5.3 part (2)(b). The distinguished triangle $E' \to E \to P \to E'[1]$ is canonical (more precisely unique up to isomorphism of triangles induces the identity on $E$) by the discussion in Derived Categories, Section 13.40. $\square$


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