The Stacks project

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 21.29.1 to the following situation: $\mathcal{C} = \mathcal{X}$, $\tau = fppf$, $\tau ' = {\acute{e}tale}$, $\mathcal{O} = \mathcal{O}_\mathcal {X}$, $\mathcal{A} = \textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$, 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 103.8.1). Condition (3) holds because every scheme has a Zariski open covering by affines. Condition (4) follows from Descent, Lemma 35.12.4.

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

Proof. The functor $g^* = g^{-1}$ is exact, hence (1)(a), (2)(a), (1)(b), and (2)(b) follow from Cohomology of Stacks, Lemmas 103.16.3 and 103.14.6.

Proof of (1)(c) and (2)(c). The construction of $Lg_!$ in Lemma 104.3.1 (via Cohomology on Sites, Lemma 21.37.2 which in turn uses Derived Categories, Proposition 13.29.2) shows that $Lg_!$ on any object $\mathcal{H}^\bullet $ of $D(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ is computed as

(termwise colimits) where the quasi-isomorphism $\mathop{\mathrm{colim}}\nolimits \mathcal{K}_ n^\bullet \to \mathcal{H}^\bullet $ induces quasi-isomorphisms $\mathcal{K}_ n^\bullet \to \tau _{\leq n} \mathcal{H}^\bullet $. Since the inclusion functors

are compatible with filtered colimits we see that it suffices to prove (c) on bounded above complexes $\mathcal{H}^\bullet $ in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ and in $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$. In this case to show that $H^ n(Lg_!\mathcal{H}^\bullet )$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ we can argue by induction on the integer $m$ such that $\mathcal{H}^ i = 0$ for $i > m$. If $m < n$, then $H^ n(Lg_!\mathcal{H}^\bullet ) = 0$ and the result holds. In general consider the distinguished triangle

(Derived Categories, Remark 13.12.4) and apply the functor $Lg_!$. Since $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ is a weak Serre subcategory of the module category it suffices to prove (c) for two out of three. We have the result for $Lg_!\tau _{\leq m - 1}\mathcal{H}^\bullet $ by induction and we have the result for $Lg_!H^ m(\mathcal{H}^\bullet )[-m]$ by Lemma 104.3.3. Whence (c) holds.

Let us prove (2)(d). By (2)(a) and (2)(b) the functor $g^{-1} = g^*$ induces a functor

see Derived Categories, Lemma 13.6.8. Thus we have the following diagram of triangulated categories

where $q$ is the quotient functor, the inner triangle is commutative, and $g^{-1}Lg_! = \text{id}$. For any object of $E$ of $D_{\textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ the map $a : Lg_!g^{-1}E \to E$ maps to a quasi-isomorphism in $D(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$. Hence the cone on $a$ maps to zero under $g^{-1}$ and by (2)(b) we see that $q(a)$ is an isomorphism. Thus $q \circ Lg_!$ is a quasi-inverse to $c$.

In the case of the lisse-étale site exactly the same argument as above proves that

is equivalent to $D_\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$. Applying the last equivalence of Lemma 104.5.2 finishes the proof. $\square$

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 103.14. Set $E' = Lg_!g^*E$ and let $P$ be the cone on the adjunction map $E' \to E$, see Lemma 104.3.1 part (4). By Lemma 104.5.3 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 104.3.1 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 104.5.3 part (2)(b). Finally, for $P'$ in $D_{\textit{Parasitic} \cap \textit{LQCoh}^{fbc}}(\mathcal{O}_\mathcal {X})$ we have

as $g^*P' = 0$ by Lemma 104.5.3 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. $\square$