The Stacks project

Lemma 102.9.4. Let $\mathcal{X}$ be an algebraic stack. Let $\alpha : \mathcal{F} \to \mathcal{G}$ and $\beta : \mathcal{G} \to \mathcal{H}$ be maps in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ with $\beta \circ \alpha = 0$. The following are equivalent:

  1. in the abelian category $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ the complex $\mathcal{F} \to \mathcal{G} \to \mathcal{H}$ is exact at $\mathcal{G}$,

  2. $\mathop{\mathrm{Ker}}(\beta )/\mathop{\mathrm{Im}}(\alpha )$ computed in either $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ or $\textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_\mathcal {X})$ is parasitic.

Proof. We have $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \subset \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$, see Section 102.8. Hence $\mathop{\mathrm{Ker}}(\beta )/\mathop{\mathrm{Im}}(\alpha )$ computed in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ or $\textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_\mathcal {X})$ agree, see Proposition 102.8.1. From now on we will use the ├ętale topology on $\mathcal{X}$.

Let $\mathcal{E}$ be the cohomology of $\mathcal{F} \to \mathcal{G} \to \mathcal{H}$ computed in the abelian category $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$. Let $x : U \to \mathcal{X}$ be a flat morphism where $U$ is a scheme. As we are using the ├ętale topology, the restriction functor $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \to \textit{Mod}(U_{\acute{e}tale}, \mathcal{O}_ U)$ is exact. On the other hand, by Lemma 102.4.1 and Sheaves on Stacks, Lemma 95.14.2 the restriction functor

\[ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \xrightarrow {x^*} \mathit{QCoh}((\mathit{Sch}/U)_{\acute{e}tale}, \mathcal{O}) \xrightarrow {{-}|_{U_{\acute{e}tale}}} \mathit{QCoh}(U_{\acute{e}tale}, \mathcal{O}_ U) \]

is exact too. We conclude that $\mathcal{E}|_{U_{\acute{e}tale}} = (\mathop{\mathrm{Ker}}(\beta )/\mathop{\mathrm{Im}}(\alpha ))|_{U_{\acute{e}tale}}$.

If (1) holds, then $\mathcal{E} = 0$ hence $\mathop{\mathrm{Ker}}(\beta )/\mathop{\mathrm{Im}}(\alpha )$ restricts to zero on $U_{\acute{e}tale}$ for all $U$ flat over $\mathcal{X}$ and this is the definition of a parasitic module. If (2) holds, then $\mathop{\mathrm{Ker}}(\beta )/\mathop{\mathrm{Im}}(\alpha )$ restricts to zero on $U_{\acute{e}tale}$ for all $U$ flat over $\mathcal{X}$ hence $\mathcal{E}$ restricts to zero on $U_{\acute{e}tale}$ for all $U$ flat over $\mathcal{X}$. This certainly implies that the quasi-coherent module $\mathcal{E}$ is zero, for example apply Lemma 102.4.2 to the map $0 \to \mathcal{E}$. $\square$

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