The Stacks project

Proof. Part (1) is a consequence of fppf descent of quasi-coherent modules. Namely, suppose that $\{ f_ i : U_ i \to U\} $ is an fpqc covering in $(\textit{Spaces}/X)_{\acute{e}tale}$. Denote $g : U \to X$ the structure morphism. Suppose that we have a family of sections $s_ i \in \Gamma (U_ i , f_ i^*g^*\mathcal{F})$ such that $s_ i|_{U_ i \times _ U U_ j} = s_ j|_{U_ i \times _ U U_ j}$. We have to find the correspond section $s \in \Gamma (U, g^*\mathcal{F})$. We can reinterpret the $s_ i$ as a family of maps $\varphi _ i : f_ i^*\mathcal{O}_ U = \mathcal{O}_{U_ i} \to f_ i^*g^*\mathcal{F}$ compatible with the canonical descent data associated to the quasi-coherent sheaves $\mathcal{O}_ U$ and $g^*\mathcal{F}$ on $U$. Hence by Descent on Spaces, Proposition 74.4.1 we see that we may (uniquely) descend these to a map $\mathcal{O}_ U \to g^*\mathcal{F}$ which gives us our section $s$.

We will deduce (2) – (7) from the corresponding statement for schemes. Choose an étale covering $\{ X_ i \to X\} _{i \in I}$ where each $X_ i$ is a scheme. Observe that $X_ i \times _ X X_ j$ is a scheme too. This covering induces a covering of the final object in each of the three sites $(\textit{Spaces}/X)_{fppf}$, $(\textit{Spaces}/X)_{\acute{e}tale}$, and $X_{\acute{e}tale}$. Hence we see that the category of sheaves on these sites are equivalent to descent data for these coverings, see Sites, Lemma 7.26.5. Parts (2), (3) are local (because we have the glueing statement). Being quasi-coherent is a local property, hence parts (4), (5) are local. Clearly (6) and (7) are local. It follows that it suffices to prove parts (2) – (7) of the lemma when $X$ is a scheme.

Assume $X$ is a scheme. The embeddings $(\mathit{Sch}/X)_{\acute{e}tale}\subset (\textit{Spaces}/X)_{\acute{e}tale}$ and $(\mathit{Sch}/X)_{fppf} \subset (\textit{Spaces}/X)_{fppf}$ determine equivalences of ringed topoi by Lemma 84.3.1. We conclude that (2) – (7) follows from the case of schemes. Étale Cohomology, Lemma 59.101.1. To transport the property of being quasi-coherent via this equivalence use that being quasi-coherent is an intrinsic property of modules as explained in Modules on Sites, Section 18.23. Some minor details omitted. $\square$

Proof. This is an immediate consequence of parts (6) and (7) of Lemma 84.7.1. $\square$

Proof. For any object $f : U \to X$ of $(\textit{Spaces}/X)_{\acute{e}tale}$ consider the restriction $\mathcal{H}_{\acute{e}tale}|_{U_{\acute{e}tale}}$ of $\mathcal{H}_{\acute{e}tale}$ to $U_{\acute{e}tale}$ via the functor $i_ f^* = i_ f^{-1}$ discussed in Section 84.5. The sheaf $\mathcal{H}_{\acute{e}tale}|_{U_{\acute{e}tale}}$ is equal to the homology of complex $f^*\mathcal{F}_\bullet $ in degree $1$. This is true because $i_ f \circ \pi _ X = f$ as morphisms of ringed sites $U_{\acute{e}tale}\to X_{\acute{e}tale}$. In particular we see that $\mathcal{H}_{\acute{e}tale}|_{U_{\acute{e}tale}}$ is a quasi-coherent $\mathcal{O}_ U$-module. Next, let $g : V \to U$ be a flat morphism in $(\textit{Spaces}/X)_{\acute{e}tale}$. Since

as morphisms of sites $V_{\acute{e}tale}\to X_{\acute{e}tale}$ and since $g$ is flat hence $g^*$ is exact, we obtain

With these preparations we are ready to prove the lemma.

Let $\mathcal{U} = \{ g_ i : U_ i \to U\} _{i \in I}$ be an fppf covering with $f : U \to X$ as above. The sheaf propery holds for $\mathcal{H}_{\acute{e}tale}$ and the covering $\mathcal{U}$ by (1) of Lemma 84.7.1 applied to $\mathcal{H}_{\acute{e}tale}|_{U_{\acute{e}tale}}$ and the above. Therefore we see that $\mathcal{H}_{\acute{e}tale}$ is already an fppf sheaf and this means that $\mathcal{H}_{fppf}$ is equal to $\mathcal{H}_{\acute{e}tale}$ as a presheaf. In particular $\mathcal{H}_{\acute{e}tale}= \epsilon _{X, *}\mathcal{H}_{fppf}$.

Finally, to prove the vanishing, we use Cohomology on Sites, Lemma 21.10.9. We let $\mathcal{B}$ be the affine objects of $(\textit{Spaces}/X)_{fppf}$ and we let $\text{Cov}$ be the set of finite fppf coverings $\mathcal{U} = \{ U_ i \to U\} _{i = 1, \ldots , n}$ with $U$, $U_ i$ affine. We have

because the values of $\mathcal{H}_{\acute{e}tale}$ on the affine schemes $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ flat over $U$ agree with the values of the pullback of the quasi-coherent module $\mathcal{H}_{\acute{e}tale}|_{U_{\acute{e}tale}}$ by the first paragraph. Hence we obtain vanishing by Descent, Lemma 35.9.2. This finishes the proof. $\square$

Proof. The question is étale local on $X$ hence we may assume $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$. Then we have $D_\mathit{QCoh}(\mathcal{O}_ X) = D(A)$ by Derived Categories of Spaces, Lemma 75.4.2 and Derived Categories of Schemes, Lemma 36.3.5. Hence we can choose an K-flat complex of $A$-modules $K^\bullet $ whose corresponding complex $\mathcal{K}^\bullet $ of quasi-coherent $\mathcal{O}_ X$-modules represents $K$. We claim that $\mathcal{K}^\bullet $ is a K-flat complex of $\mathcal{O}_ X$-modules.

Proof of the claim. By Derived Categories of Schemes, Lemma 36.3.6 we see that $\widetilde{K}^\bullet $ is K-flat on the scheme $(\mathop{\mathrm{Spec}}(A), \mathcal{O}_{\mathop{\mathrm{Spec}}(A)})$. Next, note that $\mathcal{K}^\bullet = \epsilon ^*\widetilde{K}^\bullet $ where $\epsilon $ is as in Derived Categories of Spaces, Lemma 75.4.2 whence $\mathcal{K}^\bullet $ is K-flat by Cohomology on Sites, Lemma 21.18.7 and the fact that the étale site of a scheme has enough points (Étale Cohomology, Remarks 59.29.11).

By the claim we see that $La_ X^*K = a_ X^*\mathcal{K}^\bullet $ and $L\pi _ X^*K = \pi _ X^*\mathcal{K}^\bullet $. Since the first part of the proof shows that the pullback $a_ X^*\mathcal{K}^ n$ of the quasi-coherent module is acyclic for $\epsilon _{X, *}$, resp. $a_{X, *}$, surely the proof is done by Leray's acyclicity lemma? Actually..., no because Leray's acyclicity lemma only applies to bounded below complexes. However, in the next paragraph we will show the result does follow from the bounded below case because our complex is the derived limit of bounded below complexes of quasi-coherent modules.

The cohomology sheaves of $\pi _ X^*\mathcal{K}^\bullet $ and $a_ X^*\mathcal{K}^\bullet $ have vanishing higher cohomology groups over affine objects of $(\textit{Spaces}/X)_{\acute{e}tale}$ by Lemma 84.7.3. Therefore we have

by Cohomology on Sites, Lemma 21.23.10.

Proof of $L\pi _ X^*K = R\epsilon _{X, *}(La_ X^*\mathcal{F})$. By the above we have

by Cohomology on Sites, Lemma 21.23.3. Note that $\tau _{\geq -n}(La_ X^*K)$ is represented by $\tau _{\geq -n}(a_ X^*\mathcal{K}^\bullet )$ which may not be the same as $a_ X^*(\tau _{\geq -n}\mathcal{K}^\bullet )$. But clearly the systems

are isomorphic as pro-systems. By Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) and the first part of the lemma we see that

Then we can use that the systems

are isomorphic as pro-systems. Finally, we put everything together as follows

Here in equalities four and six we have used that isomorphic pro-systems have the same $R\mathop{\mathrm{lim}}\nolimits $ (small detail omitted). You can avoid this step by using more about cohomology of the terms of the complex $\tau _{\geq -n}a_ X^*\mathcal{K}^\bullet $ proved in Lemma 84.7.3 as this will prove directly that $R\epsilon _{X, *}(\tau _{\geq -n}(a_ X^*\mathcal{K}^\bullet )) = \tau _{\geq -n}(\pi _ X^*\mathcal{K}^\bullet )$.

The equality $K = Ra_{X, *}(La_ X^*\mathcal{F})$ is proved in exactly the same way using in the final step that $K = R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K$ by Derived Categories of Spaces, Lemma 75.5.7. $\square$