The Stacks project

Proof. We have $a_ X^{-1}\mathcal{F} = \epsilon _ X^{-1} \pi _ X^{-1}\mathcal{F}$. By Lemma 84.5.1 the étale sheaf $\pi _ X^{-1}\mathcal{F}$ is a sheaf for the fppf topology and therefore is equal to $a_ X^{-1}\mathcal{F}$ (as pulling back by $\epsilon _ X$ is given by fppf sheafification). Recall moreover that $\epsilon _{X, *}$ is the identity on underlying presheaves. Now part (1) is immediate from the explicit description of $\pi _ X^{-1}$ in Lemma 84.5.1.

We will prove part (2) by reducing it to the case of schemes – see part (1) of Étale Cohomology, Lemma 59.100.6. This will “clearly work” as every algebraic space is étale locally a scheme. The details are given below but we urge the reader to skip the proof.

For an abelian sheaf $\mathcal{H}$ on $(\textit{Spaces}/X)_{fppf}$ the higher direct image $R^ p\epsilon _{X, *}\mathcal{H}$ is the sheaf associated to the presheaf $U \mapsto H^ p_{fppf}(U, \mathcal{H})$ on $(\textit{Spaces}/X)_{\acute{e}tale}$. See Cohomology on Sites, Lemma 21.7.4. Since every object of $(\textit{Spaces}/X)_{\acute{e}tale}$ has a covering by schemes, it suffices to prove that given $U/X$ a scheme and $\xi \in H^ p_{fppf}(U, a_ X^{-1}\mathcal{F})$ we can find an étale covering $\{ U_ i \to U\} $ such that $\xi $ restricts to zero on $U_ i$. We have

where the second identification is Lemma 84.3.1 and the first is a general fact about restriction (Cohomology on Sites, Lemma 21.7.1). Looking at the first paragraph and the corresponding result in the case of schemes (Étale Cohomology, Lemma 59.100.1) we conclude that the sheaf $(a_ X^{-1}\mathcal{F})|_{\mathit{Sch}/U}$ matches the pullback by the “schemes version of $a_ U$”. Therefore we can find an étale covering $\{ U_ i \to U\} $ such that our class dies in $H^ p((\mathit{Sch}/U_ i)_{fppf}, (a_ X^{-1}\mathcal{F})|_{\mathit{Sch}/U_ i})$ for each $i$, see Étale Cohomology, Lemma 59.100.6 (the precise statement one should use here is that $V_ n$ holds for all $n$ which is the statement of part (2) for the case of schemes). Transporting back (using the same formulas as above but now for $U_ i$) we conclude $\xi $ restricts to zero over $U_ i$ as desired. $\square$

Proof. We only prove the second statement; the first is easier and proved in exactly the same manner. There is an immediate reduction 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 84.6.1 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 84.6.1 we have $R^ i\epsilon _{X, *}(a_ X^{-1}\mathcal{F}) = 0$ for $i > 0$. We have $\epsilon _{X, *}a_ X^{-1}\mathcal{F} = \pi _ X^{-1}\mathcal{F}$ and by Lemma 84.5.5 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 84.6.2 by Cohomology on Sites, Remark 21.14.4. $\square$

Proof. This follows immediately from working out the definitions of the morphisms involved, see Topologies on Spaces, Section 73.7 and Section 84.5. $\square$

Proof. Proof of (1). You can prove this by repeating the proof of Lemma 84.5.6 part (1); we will instead deduce the result from this. As $\epsilon _{Y, *}$ is the identity functor on underlying presheaves, it reflects isomorphisms. Lemma 84.6.1 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 84.5.6.

To see (2) we use that

The first equality by the commutative diagram in Lemma 84.6.4 and Cohomology on Sites, Lemma 21.19.2. Then second equality is Lemma 84.6.2. The third is Lemma 84.5.6 part (2). The fourth is Lemma 84.6.2 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 $D^+((\textit{Spaces}/Y)_{fppf})$ 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 Lemma 84.6.1 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. Let $V \to Y$ be a surjective étale morphism where $V$ is a scheme. It suffices to prove the base change map is an isomorphism after restricting to $V$. Hence we may assume that $Y$ is a scheme. As the morphism is finite, hence representable, we conclude that we may assume both $X$ and $Y$ are schemes. In this case the result follows from the case of schemes (Étale Cohomology, Lemma 59.100.6 part (2)) using the comparison of topoi discussed in Section 84.3 and in particular given in Lemma 84.3.1. Some details omitted. $\square$

Proof. The functor $a_ X^{-1}$ is fully faithful (as $a_{X, *}a_ X^{-1} = \text{id}$ by Lemma 84.6.1). 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 $Y$. In particular, we may assume $Y$ is a scheme.

Assume $Y$ is a scheme. Choose a scheme $X'$ and a surjective étale morphism $s : X' \to X$. Set $f' = f \circ s : X' \to Y$ and observe that $f'$ is surjective, locally of finite presentation, and flat. 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 where both $X$ and $Y$ are schemes. This case follows from Étale Cohomology, Lemma 59.100.4 via the discussion in Section 84.3 and in particular Lemma 84.3.1. $\square$