Please see Section 59.41 for the definition of property (A).
Lemma 59.42.1. Let $f : X \to Y$ be a morphism of schemes. Assume (A).
$f_{small, *} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ reflects injections and surjections,
$f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}$ is surjective for any abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$,
$f_{small, *} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ is faithful.
Proof. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Let $U$ be an object of $X_{\acute{e}tale}$. By assumption we can find a covering $\{ W_ i \to U\} $ in $X_{\acute{e}tale}$ such that each $W_ i$ is an open and closed subscheme of $X \times _ Y V_ i$ for some object $V_ i$ of $Y_{\acute{e}tale}$. The sheaf condition shows that
and that $\mathcal{F}(W_ i)$ is a direct summand of $\mathcal{F}(X \times _ Y V_ i) = f_{small, *}\mathcal{F}(V_ i)$. Hence it is clear that $f_{small, *}$ reflects injections.
Next, suppose that $a : \mathcal{G} \to \mathcal{F}$ is a map of abelian sheaves such that $f_{small, *}a$ is surjective. Let $s \in \mathcal{F}(U)$ with $U$ as above. With $W_ i$, $V_ i$ as above we see that it suffices to show that $s|_{W_ i}$ is étale locally the image of a section of $\mathcal{G}$ under $a$. Since $\mathcal{F}(W_ i)$ is a direct summand of $\mathcal{F}(X \times _ Y V_ i)$ it suffices to show that for any $V \in \mathop{\mathrm{Ob}}\nolimits (Y_{\acute{e}tale})$ any element $s \in \mathcal{F}(X \times _ Y V)$ is étale locally on $X \times _ Y V$ the image of a section of $\mathcal{G}$ under $a$. Since $\mathcal{F}(X \times _ Y V) = f_{small, *}\mathcal{F}(V)$ we see by assumption that there exists a covering $\{ V_ j \to V\} $ such that $s$ is the image of $s_ j \in f_{small, *}\mathcal{G}(V_ j) = \mathcal{G}(X \times _ Y V_ j)$. This proves $f_{small, *}$ reflects surjections.
Parts (2), (3) follow formally from part (1), see Modules on Sites, Lemma 18.15.1. $\square$
Lemma 59.42.2. Let $f : X \to Y$ be a separated locally quasi-finite morphism of schemes. Then property (A) above holds.
Proof. Let $U \to X$ be an étale morphism and $u \in U$. The geometric statement (A) reduces directly to the case where $U$ and $Y$ are affine schemes. Denote $x \in X$ and $y \in Y$ the images of $u$. Since $X \to Y$ is locally quasi-finite, and $U \to X$ is locally quasi-finite (see Morphisms, Lemma 29.36.6) we see that $U \to Y$ is locally quasi-finite (see Morphisms, Lemma 29.20.12). Moreover both $X \to Y$ and $U \to Y$ are separated. Thus More on Morphisms, Lemma 37.41.5 applies to both morphisms. This means we may pick an étale neighbourhood $(V, v) \to (Y, y)$ such that
and points $w \in W$, $w' \in W'$ such that
$W$, $R$ are open and closed in $X \times _ Y V$,
$W'$, $R'$ are open and closed in $U \times _ Y V$,
$W \to V$ and $W' \to V$ are finite,
$w$, $w'$ map to $v$,
$\kappa (v) \subset \kappa (w)$ and $\kappa (v) \subset \kappa (w')$ are purely inseparable, and
no other point of $W$ or $W'$ maps to $v$.
Here is a commutative diagram
After shrinking $V$ we may assume that $W'$ maps into $W$: just remove the image the inverse image of $R$ in $W'$; this is a closed set (as $W' \to V$ is finite) not containing $v$. Then $W' \to W$ is finite because both $W \to V$ and $W' \to V$ are finite. Hence $W' \to W$ is finite étale, and there is exactly one point in the fibre over $w$ with $\kappa (w) = \kappa (w')$. Hence $W' \to W$ is an isomorphism in an open neighbourhood $W^\circ $ of $w$, see Étale Morphisms, Lemma 41.14.2. Since $W \to V$ is finite the image of $W \setminus W^\circ $ is a closed subset $T$ of $V$ not containing $v$. Thus after replacing $V$ by $V \setminus T$ we may assume that $W' \to W$ is an isomorphism. Now the decomposition $X \times _ Y V = W \amalg R$ and the morphism $W \to U$ are as desired and we win. $\square$
Lemma 59.42.3. Let $f : X \to Y$ be an integral morphism of schemes. Then property (A) holds.
Proof. Let $U \to X$ be étale, and let $u \in U$ be a point. We have to find $V \to Y$ étale, a disjoint union decomposition $X \times _ Y V = W \amalg W'$ and an $X$-morphism $W \to U$ with $u$ in the image. We may shrink $U$ and $Y$ and assume $U$ and $Y$ are affine. In this case also $X$ is affine, since an integral morphism is affine by definition. Write $Y = \mathop{\mathrm{Spec}}(A)$, $X = \mathop{\mathrm{Spec}}(B)$ and $U = \mathop{\mathrm{Spec}}(C)$. Then $A \to B$ is an integral ring map, and $B \to C$ is an étale ring map. By Algebra, Lemma 10.143.3 we can find a finite $A$-subalgebra $B' \subset B$ and an étale ring map $B' \to C'$ such that $C = B \otimes _{B'} C'$. Thus the question reduces to the étale morphism $U' = \mathop{\mathrm{Spec}}(C') \to X' = \mathop{\mathrm{Spec}}(B')$ over the finite morphism $X' \to Y$. In this case the result follows from Lemma 59.42.2. $\square$
Lemma 59.42.4. Let $f : X \to Y$ be a morphism of schemes. Denote $f_{small} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ the associated morphism of small étale topoi. Assume at least one of the following
$f$ is integral, or
$f$ is separated and locally quasi-finite.
Then the functor $f_{small, *} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ has the following properties
the map $f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}$ is always surjective,
$f_{small, *}$ is faithful, and
$f_{small, *}$ reflects injections and surjections.
Proof. Combine Lemmas 59.42.2, 59.42.3, and 59.42.1. $\square$
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