The Stacks project

Proof. Since pullback is transitive and $f_{small} = \pi _ X \circ i_ f$ (see above) we see that $i_ f^{-1} \pi _ X^{-1}\mathcal{F} = f_{small}^{-1}\mathcal{F}$. This shows that $\pi _ X^{-1}$ has the description given in the lemma.

To prove that $\pi _ X^{-1}\mathcal{F}$ is a sheaf for the ph topology it suffices by Topologies on Spaces, Lemma 73.8.7 to show that for a surjective proper morphism $V \to U$ of algebraic spaces over $X$ we have $(\pi _ X^{-1}\mathcal{F})(U)$ is the equalizer of the two maps $(\pi _ X^{-1}\mathcal{F})(V) \to (\pi _ X^{-1}\mathcal{F})(V \times _ U V)$. This we have seen in Lemma 84.4.1.

The case of smooth, syntomic, fppf coverings follows from the case of ph coverings by Topologies on Spaces, Lemma 73.8.2.

Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be an fpqc covering of algebraic spaces over $X$. Let $s_ i \in (\pi _ X^{-1}\mathcal{F})(U_ i)$ be sections which agree over $U_ i \times _ U U_ j$. We have to prove there exists a unique $s \in (\pi _ X^{-1}\mathcal{F})(U)$ restricting to $s_ i$ over $U_ i$. Case I: $U$ and $U_ i$ are schemes. This case follows from Étale Cohomology, Lemma 59.39.2. Case II: $U$ is a scheme. Here we choose surjective étale morphisms $T_ i \to U_ i$ where $T_ i$ is a scheme. Then $\mathcal{T} = \{ T_ i \to U\} $ is an fpqc covering by schemes and by case I the result holds for $\mathcal{T}$. We omit the verification that this implies the result for $\mathcal{U}$. Case III: general case. Let $W \to U$ be a surjective étale morphism, where $W$ is a scheme. Then $\mathcal{W} = \{ U_ i \times _ U W \to W\} $ is an fpqc covering (by algebraic spaces) of the scheme $W$. By case II the result hold for $\mathcal{W}$. We omit the verification that this implies the result for $\mathcal{U}$. $\square$

Proof. Parts (1)(b) and (2)(b) follow formally from the fact that the restriction functor $\pi _{X, *} = i_ X^{-1}$ is a right adjoint of the exact functor $\pi _ X^{-1}$, see Homology, Lemma 12.29.1 and Derived Categories, Lemma 13.31.9.

Parts (1)(a) and (2)(a) can be seen in two ways. First proof: We can use that $i_ f^{-1}$ is a right adjoint of the exact functor $i_{f, !}$. This functor is constructed in Topologies, Lemma 34.4.13 for sheaves of sets and for abelian sheaves in Modules on Sites, Lemma 18.16.2. It is shown in Modules on Sites, Lemma 18.16.3 that it is exact. Second proof. We can use that $i_ f = i_ Y \circ f_{big}$ as is shown in Topologies, Lemma 34.4.17. Since $f_{big}$ is a localization, we see that pullback by it preserves injectives and K-injectives, see Cohomology on Sites, Lemmas 21.7.1 and 21.20.1. Then we apply the already proved parts (1)(b) and (2)(b) to the functor $i_ Y^{-1}$ to conclude.

To see a counter example for the case of modules we refer to Étale Cohomology, Lemma 59.99.1. $\square$

Proof. Part (1) follows from Lemma 84.5.2 and (84.5.2.1) on choosing a K-injective complex of abelian sheaves representing $K$.

Part (3) follows from Lemma 84.5.2 and Topologies, Lemma 34.4.19 on choosing a K-injective complex of abelian sheaves representing $K$.

Part (5) is Cohomology on Sites, Lemma 21.21.1.

Part (6) is Cohomology on Sites, Lemma 21.21.2.

Part (2) can be proved as follows. Above we have seen that $\pi _ X \circ f_{big} = f_{small} \circ \pi _ Y$ as morphisms of ringed sites. Hence we obtain $R\pi _{X, *} \circ Rf_{big, *} = Rf_{small, *} \circ R\pi _{Y, *}$ by Cohomology on Sites, Lemma 21.19.2. Since the restriction functors $\pi _{X, *}$ and $\pi _{Y, *}$ are exact, we conclude.

Part (4) follows from part (6) and part (2) applied to $f' : Y' \to X'$. $\square$

Proof. To prove (5) represent $M$ by a K-injective complex of abelian sheaves and apply Lemma 84.5.2 and work out the definitions. Part (3) follows from this as $i_ f^{-1}\pi _ X^{-1} = f_{small}^{-1}$. Part (1) is a special case of (3).

Part (6) follows from the very general Cohomology on Sites, Lemma 21.37.5. Then part (4) follows because $Lf_{small}^* = i_ f^* \circ L\pi _ X^*$. Part (2) is a special case of (4). $\square$

Proof. This is true because both $\pi _ X^{-1}$ and $\pi _{X, *} = i_ X^{-1}$ are exact functors and the composition $\pi _{X, *} \circ \pi _ X^{-1}$ is the identity functor. $\square$

Proof. Proof of (1). Let $\mathcal{F}$ be a sheaf on $Y_{\acute{e}tale}$. Let $g : X' \to X$ be an object of $(\textit{Spaces}/X)_{\acute{e}tale}$. Consider the fibre product

Then we have

the second equality by Lemma 84.5.1. On the other hand

again by Lemma 84.5.1. Hence by proper base change for sheaves of sets (Lemma 84.4.4) we conclude the two sets are canonically isomorphic. The isomorphism is compatible with restriction mappings and defines an isomorphism $\pi _ X^{-1}f_{small, *}\mathcal{F} = f_{big, *}\pi _ Y^{-1}\mathcal{F}$. Thus an isomorphism of functors $\pi _ X^{-1} \circ f_{small, *} = f_{big, *} \circ \pi _ Y^{-1}$.

Proof of (2). There is a canonical base change map $\pi _ X^{-1}Rf_{small, *}K \to Rf_{big, *}\pi _ Y^{-1}K$ for any $K$ in $D(Y_{\acute{e}tale})$, see Cohomology on Sites, Remark 21.19.3. To prove it is an isomorphism, it suffices to prove the pull back of the base change map by $i_ g : \mathop{\mathit{Sh}}\nolimits (X'_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/X)_{\acute{e}tale})$ is an isomorphism for any object $g : X' \to X$ of $(\mathit{Sch}/X)_{\acute{e}tale}$. Let $T', g', f'$ be as in the previous paragraph. The pullback of the base change map is

where we have used $\pi _ X \circ i_ g = g_{small}$, $\pi _ Y \circ i_{g'} = g'_{small}$, and Lemma 84.5.3. This map is an isomorphism by the proper base change theorem (Lemma 84.4.7) provided $K$ is bounded below and the cohomology sheaves of $K$ are torsion.

Proof of (3). If $f$ is finite, then the functors $f_{small, *}$ and $f_{big, *}$ are exact. This follows from Cohomology of Spaces, Lemma 69.4.1 for $f_{small}$. Since any base change $f'$ of $f$ is finite too, we conclude from Lemma 84.5.3 part (3) that $f_{big, *}$ is exact too (as the higher derived functors are zero). Thus this case follows from part (1). $\square$