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 $\mathcal{U}$. $\square$