The Stacks project

Proof. The statement about the étale topology is the content of Lemma 59.39.2. To finish the proof it suffices to show that $\pi _ S^{-1}\mathcal{F}$ is a sheaf for the fppf topology. This is shown in Lemma 59.39.2 as well. $\square$

Proof. The commutativity of the diagrams follows from the discussion in Topologies, Section 34.11. $\square$

Proof. You can prove this by repeating the proof of Lemma 59.99.5 part (1); we will instead deduce the result from this. As $\epsilon _{Y, *}$ is the identity functor on underlying presheaves, it reflects isomorphisms. The description in Lemma 59.100.1 shows that $\epsilon _{Y, *} \circ a_ Y^{-1} = \pi _ Y^{-1}$ and similarly for $X$. To show that the canonical map $a_ Y^{-1}f_{small, *}\mathcal{F} \to f_{big, fppf, *}a_ X^{-1}\mathcal{F}$ is an isomorphism, it suffices to show that

is an isomorphism. This is part (1) of Lemma 59.99.5. $\square$

Proof. The functor $a_ X^{-1}$ is fully faithful (as $a_{X, *}a_ X^{-1} = \text{id}$ by Lemma 59.100.1). Hence the forgetful functor $(\mathcal{G}, \mathcal{H}, \alpha ) \mapsto \mathcal{H}$ identifies the category of triples with a full subcategory of $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/Y)_{fppf})$. Moreover, the functor $a_ Y^{-1}$ is fully faithful, hence the functor in the lemma is fully faithful as well.

Suppose that we have an étale covering $\{ Y_ i \to Y\} $. Let $f_ i : X_ i \to Y_ i$ be the base change of $f$. Denote $f_{ij} = f_ i \times f_ j : X_ i \times _ X X_ j \to Y_ i \times _ Y Y_ j$. Claim: if the lemma is true for $f_ i$ and $f_{ij}$ for all $i, j$, then the lemma is true for $f$. To see this, note that the given étale covering determines an étale covering of the final object in each of the four sites $Y_{\acute{e}tale}, X_{\acute{e}tale}, (\mathit{Sch}/Y)_{fppf}, (\mathit{Sch}/X)_{fppf}$. Thus the category of sheaves is equivalent to the category of glueing data for this covering (Sites, Lemma 7.26.5) in each of the four cases. A huge commutative diagram of categories then finishes the proof of the claim. We omit the details. The claim shows that we may work étale locally on $Y$.

Note that $\{ X \to Y\} $ is an fppf covering. Working étale locally on $Y$, we may assume there exists a morphism $s : X' \to X$ such that the composition $f' = f \circ s : X' \to Y$ is surjective finite locally free, see More on Morphisms, Lemma 37.48.1. Claim: if the lemma is true for $f'$, then it is true for $f$. Namely, given a triple $(\mathcal{G}, \mathcal{H}, \alpha )$ for $f$, we can pullback by $s$ to get a triple $(s_{small}^{-1}\mathcal{G}, \mathcal{H}, s_{big, fppf}^{-1}\alpha )$ for $f'$. A solution for this triple gives a sheaf $\mathcal{F}$ on $Y_{\acute{e}tale}$ with $a_ Y^{-1}\mathcal{F} = \mathcal{H}$. By the first paragraph of the proof this means the triple is in the essential image. This reduces us to the case described in the next paragraph.

Assume $f$ is surjective finite locally free. Let $(\mathcal{G}, \mathcal{H}, \alpha )$ be a triple. In this case consider the triple

where $\alpha _1$ comes from the identifications

where the third equality is Lemma 59.100.3 and the arrow is given by $\alpha $. This triple is in the image of our functor because $\mathcal{F}_1 = f_{small, *}\mathcal{F}$ is a solution (to see this use Lemma 59.100.3 again; details omitted). There is a canonical map of triples

which uses the unit $\text{id} \to f_{big, fppf, *}f_{big, fppf}^{-1}$ on the second entry (it is enough to prescribe morphisms on the second entry by the first paragraph of the proof). Since $\{ f : X \to Y\} $ is an fppf covering the map $\mathcal{H} \to \mathcal{H}_1$ is injective (details omitted). Set

and let $\alpha _2$ be the induced isomorphism (pullback functors are exact, so this makes sense). Then $\mathcal{H}$ is the equalizer of the two maps $\mathcal{H}_1 \to \mathcal{H}_2$. Repeating the arguments above for the triple $(\mathcal{G}_2, \mathcal{H}_2, \alpha _2)$ we find an injective morphism of triples

such that this last triple is in the image of our functor. Say it corresponds to $\mathcal{F}_3$ in $\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$. By fully faithfulness we obtain two maps $\mathcal{F}_1 \to \mathcal{F}_3$ and we can let $\mathcal{F}$ be the equalizer of these two maps. By exactness of the pullback functors involved we find that $a_ Y^{-1}\mathcal{F} = \mathcal{H}$ as desired. $\square$

Proof. We first show that $\mathcal{A}'_ X \subset \textit{Ab}((\mathit{Sch}/X)_{\acute{e}tale})$ is a weak Serre subcategory by checking conditions (1), (2), (3), and (4) of Homology, Lemma 12.10.3. Parts (1), (2), (3) are immediate as $\pi _ X^{-1}$ is exact and fully faithful for example by Lemma 59.99.4. If $0 \to \pi _ X^{-1}\mathcal{F} \to \mathcal{G} \to \pi _ X^{-1}\mathcal{F}' \to 0$ is a short exact sequence in $\textit{Ab}((\mathit{Sch}/X)_{\acute{e}tale})$ then $0 \to \mathcal{F} \to \pi _{X, *}\mathcal{G} \to \mathcal{F}' \to 0$ is exact by Lemma 59.99.4. Hence $\mathcal{G} = \pi _ X^{-1}\pi _{X, *}\mathcal{G}$ is in $\mathcal{A}'_ X$ which checks the final condition.

Cohomology on Sites, Property (1) holds by the existence of fibre products of schemes and the fact that the base change of a finite morphism of schemes is a finite morphism of schemes, see Morphisms, Lemma 29.44.6.

Cohomology on Sites, Property (2) follows from the commutative diagram (3) in Topologies, Lemma 34.4.17.

Cohomology on Sites, Property (3) is Lemma 59.100.1.

Cohomology on Sites, Property (4) holds by Lemma 59.99.5 part (4).

Cohomology on Sites, Property (5) is implied by More on Morphisms, Lemma 37.48.1. $\square$

Proof. By Lemma 59.100.5 the lemmas in Cohomology on Sites, Section 21.30 all apply to our current setting. To translate the results observe that the category $\mathcal{A}_ X$ of Cohomology on Sites, Lemma 21.30.2 is the essential image of $a_ X^{-1} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}((\mathit{Sch}/X)_{fppf})$.

Part (1) is equivalent to $(V_ n)$ for all $n$ which holds by Cohomology on Sites, Lemma 21.30.8.

Part (2) follows by applying $\epsilon _ Y^{-1}$ to the conclusion of Cohomology on Sites, Lemma 21.30.3.

Part (3) follows from Cohomology on Sites, Lemma 21.30.8 part (1) because $\pi _ X^{-1}K$ is in $D^+_{\mathcal{A}'_ X}((\mathit{Sch}/X)_{\acute{e}tale})$ and $a_ X^{-1} = \epsilon _ X^{-1} \circ a_ X^{-1}$.

Part (4) follows from Cohomology on Sites, Lemma 21.30.8 part (2) for the same reason.

Part (5). We use that

The first equality by the commutative diagram in Lemma 59.100.2 and Cohomology on Sites, Lemma 21.19.2. The second equality is (3). The third is Lemma 59.99.5 part (2). The fourth is (3) again. Thus the base change map $a_ Y^{-1}(Rf_{small, *}K) \to Rf_{big, fppf, *}(a_ X^{-1}K)$ induces an isomorphism

The proof is finished by the following remark: a map $\alpha : a_ Y^{-1}L \to M$ with $L$ in $D^+(Y_{\acute{e}tale})$ and $M$ in $D^+((\mathit{Sch}/Y)_{fppf})$ such that $R\epsilon _{Y, *}\alpha $ is an isomorphism, is an isomorphism. Namely, we show by induction on $i$ that $H^ i(\alpha )$ is an isomorphism. This is true for all sufficiently small $i$. If it holds for $i \leq i_0$, then we see that $R^ j\epsilon _{Y, *}H^ i(M) = 0$ for $j > 0$ and $i \leq i_0$ by (1) because $H^ i(M) = a_ Y^{-1}H^ i(L)$ in this range. Hence $\epsilon _{Y, *}H^{i_0 + 1}(M) = H^{i_0 + 1}(R\epsilon _{Y, *}M)$ by a spectral sequence argument. Thus $\epsilon _{Y, *}H^{i_0 + 1}(M) = \pi _ Y^{-1}H^{i_0 + 1}(L) = \epsilon _{Y, *}a_ Y^{-1}H^{i_0 + 1}(L)$. This implies $H^{i_0 + 1}(\alpha )$ is an isomorphism (because $\epsilon _{Y, *}$ reflects isomorphisms as it is the identity on underlying presheaves) as desired. $\square$

Proof. We first reduce the statement to the case where $K$ is given by a single abelian sheaf. Namely, represent $K$ by a bounded below complex $\mathcal{F}^\bullet $. By the case of a sheaf we see that $\mathcal{F}^ n = a_{X, *} a_ X^{-1} \mathcal{F}^ n$ and that the sheaves $R^ qa_{X, *}a_ X^{-1}\mathcal{F}^ n$ are zero for $q > 0$. By Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) applied to $a_ X^{-1}\mathcal{F}^\bullet $ and the functor $a_{X, *}$ we conclude. From now on assume $K = \mathcal{F}$.

By Lemma 59.100.1 we have $a_{X, *}a_ X^{-1}\mathcal{F} = \mathcal{F}$. Thus it suffices to show that $R^ qa_{X, *}a_ X^{-1}\mathcal{F} = 0$ for $q > 0$. For this we can use $a_ X = \epsilon _ X \circ \pi _ X$ and the Leray spectral sequence (Cohomology on Sites, Lemma 21.14.7). By Lemma 59.100.6 we have $R^ i\epsilon _{X, *}(a_ X^{-1}\mathcal{F}) = 0$ for $i > 0$ and $\epsilon _{X, *}a_ X^{-1}\mathcal{F} = \pi _ X^{-1}\mathcal{F}$. By Lemma 59.99.4 we have $R^ j\pi _{X, *}(\pi _ X^{-1}\mathcal{F}) = 0$ for $j > 0$. This concludes the proof. $\square$

Proof. This follows from Lemma 59.100.7 by Cohomology on Sites, Remark 21.14.4. $\square$