Lemma 101.4.4. Let $\mathcal{X}$ be an algebraic stack. Let $E$ be an object of $D_{\mathcal{M}_\mathcal {X}}(\mathcal{O}_\mathcal {X})$. There exists a canonical distinguished triangle

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

in $D_{\mathcal{M}_\mathcal {X}}(\mathcal{O}_\mathcal {X})$ such that $P$ is in $D_{\mathcal{P}_\mathcal {X}}(\mathcal{O}_\mathcal {X})$ and

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

for all $P'$ in $D_{\mathcal{P}_\mathcal {X}}(\mathcal{O}_\mathcal {X})$.

Proof. Consider the morphism of ringed topoi $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{flat, fppf}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{X}_{fppf})$. Set $E' = Lg_!g^{-1}E$ and let $P$ be the cone on the adjunction map $E' \to E$. Since $g^{-1}E' \to g^{-1}E$ is an isomorphism we see that $P$ is an object of $D_{\mathcal{P}_\mathcal {X}}(\mathcal{O}_\mathcal {X})$ by Lemma 101.4.3 (2)(b). Finally, $\mathop{\mathrm{Hom}}\nolimits (E', P') = \mathop{\mathrm{Hom}}\nolimits (Lg_!g^{-1}E, P') = \mathop{\mathrm{Hom}}\nolimits (g^{-1}E, g^{-1}P') = 0$ as $g^{-1}P' = 0$.

Uniqueness. Suppose that $E'' \to E \to P'$ is a second distinguished triangle as in the statement of the lemma. Since $\mathop{\mathrm{Hom}}\nolimits (E', P') = 0$ the morphism $E' \to E$ factors as $E' \to E'' \to E$, see Derived Categories, Lemma 13.4.2. Similarly, the morphism $E'' \to E$ factors as $E'' \to E' \to E$. Consider the composition $\varphi : E' \to E'$ of the maps $E' \to E''$ and $E'' \to E'$. Note that $\varphi - 1 : E' \to E'$ fits into the commutative diagram

$\xymatrix{ E' \ar[d]^{\varphi - 1} \ar[r] & E \ar[d]^0 \\ E' \ar[r] & E }$

hence factors through $P[-1] \to E$. Since $\mathop{\mathrm{Hom}}\nolimits (E', P[-1]) = 0$ we see that $\varphi = 1$. Whence the maps $E' \to E''$ and $E'' \to E'$ are inverse to each other. $\square$

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