The Stacks project

Proof. Part (1) is Sheaves on Stacks, Lemma 96.23.1.

Part (2) for the embedding $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \subset \textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ we have seen in the proof of Lemma 103.7.3. Let us prove (2) for the embedding $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X}) \subset \textit{Mod}(\mathcal{O}_\mathcal {X})$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism between objects of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. Since $\mathop{\mathrm{Ker}}(\varphi )$ is the same whether computed in the étale or the fppf topology, we see that $\mathop{\mathrm{Ker}}(\varphi )$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ by the étale case. On the other hand, the cokernel computed in the fppf topology is the fppf sheafification of the cokernel computed in the étale topology. However, this étale cokernel is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ hence an fppf sheaf by (1) and we see that the cokernel is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. Finally, suppose that

is an exact sequence in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ (i.e., using the fppf topology) with $\mathcal{F}_1$, $\mathcal{F}_2$ in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. In order to show that $\mathcal{F}_2$ is an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ it suffices to show that the sequence is also exact in the étale topology. To do this it suffices to show that any element of $H^1_{fppf}(x, \mathcal{F}_1)$ becomes zero on the members of an étale covering of $x$ (for any object $x$ of $\mathcal{X}$). This is true because $H^1_{fppf}(x, \mathcal{F}_1) = H^1_{\acute{e}tale}(x, \mathcal{F}_1)$ by Sheaves on Stacks, Lemma 96.23.2 and because of locality of cohomology, see Cohomology on Sites, Lemma 21.7.3. This proves (2).

Part (3) follows from Lemma 103.7.2 and Sheaves on Stacks, Lemma 96.12.3.

Part (4)(b) for $R^ if_*\mathcal{F}$ computed in the étale cohomology follows from Lemma 103.7.3. Whereupon part (4)(a) follows from Sheaves on Stacks, Lemma 96.23.2 combined with (1) above.

Part (5) for the étale topology follows from Sheaves on Stacks, Lemma 96.12.4 and Lemma 103.7.2. The fppf version then follows as the colimit in the étale topology is already an fppf sheaf by part (1).

Parts (6) and (7) follow from the corresponding parts of Lemma 103.7.2 and Sheaves on Stacks, Lemma 96.12.4. $\square$

Proof. Part (1) follows from a combination of Lemmas 103.6.1 and 103.7.2. The proof of (2) is analogous to the proof of Lemma 103.6.3. Let $\mathcal{F}$ of a sheaf of $\mathcal{O}_\mathcal {X}$-modules on $\mathcal{X}_{fppf}$.

First, suppose there is a morphism $a : \mathcal{U} \to \mathcal{X}$ which is surjective, flat, locally of finite presentation, quasi-compact, and quasi-separated such that $a^*\mathcal{F}$ is locally quasi-coherent and has the flat base change property. Then there is an exact sequence

where $b$ is the morphism $b : \mathcal{U} \times _\mathcal {X} \mathcal{U} \to \mathcal{X}$, see Sheaves on Stacks, Proposition 96.19.7 and Lemma 96.19.10. Moreover, the pullback $b^*\mathcal{F}$ is the pullback of $a^*\mathcal{F}$ via one of the projection morphisms, hence is locally quasi-coherent and has the flat base change property, see Proposition 103.8.1. The modules $a_*a^*\mathcal{F}$ and $b_*b^*\mathcal{F}$ are locally quasi-coherent and have the flat base change property by Proposition 103.8.1. We conclude that $\mathcal{F}$ is locally quasi-coherent and has the flat base change property by Proposition 103.8.1.

Choose a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$. By part (1) it suffices to show that $x^*\mathcal{F}$ is locally quasi-coherent and has the flat base change property. Again by part (1) it suffices to do this (Zariski) locally on $U$, hence we may assume that $U$ is affine. By Morphisms of Stacks, Lemma 101.27.14 there exists an fppf covering $\{ a_ i : U_ i \to U\} $ such that each $x \circ a_ i$ factors through some $f_ j$. Hence the module $a_ i^*\mathcal{F}$ on $(\mathit{Sch}/U_ i)_{fppf}$ is locally quasi-coherent and has the flat base change property. After refining the covering we may assume $\{ U_ i \to U\} _{i = 1, \ldots , n}$ is a standard fppf covering. Then $x^*\mathcal{F}$ is an fppf module on $(\mathit{Sch}/U)_{fppf}$ whose pullback by the morphism $a : U_1 \amalg \ldots \amalg U_ n \to U$ is locally quasi-coherent and has the flat base change property. Hence by the previous paragraph we see that $x^*\mathcal{F}$ is locally quasi-coherent and has the flat base change property as desired. $\square$

Proof. By Sheaves on Stacks, Lemma 96.21.3 we can reduce to the case where $\mathcal{X}$ is represented by $U$ and $\mathcal{Y}$ is represented by $V$. Of course this also uses that the pullback of $\mathcal{F}$ to $U$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_ U)$ by Proposition 103.8.1. Then the result follows from Sheaves on Stacks, Lemma 96.22.2 and the fact that $R^ if_*$ may be computed in the étale topology by Proposition 103.8.1. $\square$

Proof. The functors exist by Proposition 103.8.1. By Lemma 103.8.3 this reduces to the case of an affine morphism of algebraic spaces taking higher direct images in the setting of quasi-coherent modules on algebraic spaces. By the discussion in Cohomology of Spaces, Section 69.3 we reduce to the case of an affine morphism of schemes. For affine morphisms of schemes we have the vanishing of higher direct images on quasi-coherent modules by Cohomology of Schemes, Lemma 30.2.3. The vanishing for $R^1f_*$ implies exactness of $f_*$. Commuting with direct sums follows from Morphisms, Lemma 29.11.6 for example. $\square$