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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07BA. Beware of the difference between the letter 'O' and the digit '0'.