The Stacks project

Lemma 101.4.2. Let $\mathcal{X}$ be an algebraic stack. The comparison morphism $\epsilon : \mathcal{X}_{fppf} \to \mathcal{X}_{\acute{e}tale}$ induces a commutative diagram

\[ \xymatrix{ D_{\mathcal{P}_\mathcal {X}}(\mathcal{O}_\mathcal {X}) \ar[r] & D_{\mathcal{M}_\mathcal {X}}(\mathcal{O}_\mathcal {X}) \ar[r] & D(\mathcal{O}_\mathcal {X}) \\ D_{\mathcal{P}_\mathcal {X}}( \mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \ar[r] \ar[u]^{\epsilon ^*} & D_{\mathcal{M}_\mathcal {X}}( \mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \ar[r] \ar[u]^{\epsilon ^*} & D(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \ar[u]^{\epsilon ^*} } \]

Moreover, the left two vertical arrows are equivalences of triangulated categories, hence we also obtain an equivalence

\[ D_{\mathcal{M}_\mathcal {X}}( \mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) / D_{\mathcal{P}_\mathcal {X}}( \mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_\mathcal {X}) \]

Proof. Since $\epsilon ^*$ is exact it is clear that we obtain a diagram as in the statement of the lemma. We will show the middle vertical arrow is an equivalence by applying Cohomology on Sites, Lemma 21.28.1 to the following situation: $\mathcal{C} = \mathcal{X}$, $\tau = fppf$, $\tau ' = {\acute{e}tale}$, $\mathcal{O} = \mathcal{O}_\mathcal {X}$, $\mathcal{A} = \mathcal{M}_\mathcal {X}$, and $\mathcal{B}$ is the set of objects of $\mathcal{X}$ lying over affine schemes. To see the lemma applies we have to check conditions (1), (2), (3), (4). Conditions (1) and (2) are clear from the discussion above (explicitly this follows from Cohomology of Stacks, Proposition 100.7.4). Condition (3) holds because every scheme has a Zariski open covering by affines. Condition (4) follows from Descent, Lemma 35.9.4.

We omit the verification that the equivalence of categories $\epsilon ^* : D_{\mathcal{M}_\mathcal {X}}( \mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X}) \to D_{\mathcal{M}_\mathcal {X}}(\mathcal{O}_\mathcal {X})$ induces an equivalence of the subcategories of complexes with parasitic cohomology sheaves. $\square$


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