Lemma 59.44.1. Let $f : X \to Y$ be a morphism of schemes. Assume (C) holds. Then the functor $f_{small, *} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ reflects injections and surjections.
59.44 Property (C)
Please see Section 59.41 for the definition of property (C).
Proof. Follows from Sites, Lemma 7.41.4. We omit the verification that property (C) implies that the functor $Y_{\acute{e}tale}\to X_{\acute{e}tale}$, $V \mapsto X \times _ Y V$ satisfies the assumption of Sites, Lemma 7.41.4. $\square$
Remark 59.44.2. Property (C) holds if $f : X \to Y$ is an open immersion. Namely, if $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$, then we can view $U$ also as an object of $Y_{\acute{e}tale}$ and $U \times _ Y X = U$. Hence property (C) does not imply that $f_{small, *}$ is exact as this is not the case for open immersions (in general).
Lemma 59.44.3. Let $f : X \to Y$ be a morphism of schemes. Assume that for any $V \to Y$ étale we have that
$X \times _ Y V \to V$ has property (C), and
$X \times _ Y V \to V$ is closed.
Then the functor $Y_{\acute{e}tale}\to X_{\acute{e}tale}$, $V \mapsto X \times _ Y V$ is almost cocontinuous, see Sites, Definition 7.42.3.
Proof. Let $V \to Y$ be an object of $Y_{\acute{e}tale}$ and let $\{ U_ i \to X \times _ Y V\} _{i \in I}$ be a covering of $X_{\acute{e}tale}$. By assumption (1) for each $i$ we can find an étale morphism $h_ i : V_ i \to V$ and a surjective morphism $X \times _ Y V_ i \to U_ i$ over $X \times _ Y V$. Note that $\bigcup h_ i(V_ i) \subset V$ is an open set containing the closed set $Z = \mathop{\mathrm{Im}}(X \times _ Y V \to V)$. Let $h_0 : V_0 = V \setminus Z \to V$ be the open immersion. It is clear that $\{ V_ i \to V\} _{i \in I \cup \{ 0\} }$ is an étale covering such that for each $i \in I \cup \{ 0\} $ we have either $V_ i \times _ Y X = \emptyset $ (namely if $i = 0$), or $V_ i \times _ Y X \to V \times _ Y X$ factors through $U_ i \to X \times _ Y V$ (if $i \not= 0$). Hence the functor $Y_{\acute{e}tale}\to X_{\acute{e}tale}$ is almost cocontinuous. $\square$
Lemma 59.44.4. Let $f : X \to Y$ be an integral morphism of schemes which defines a homeomorphism of $X$ with a closed subset of $Y$. Then property (C) holds.
Proof. Let $g : U \to X$ be an étale morphism. We need to find an object $V \to Y$ of $Y_{\acute{e}tale}$ and a surjective morphism $X \times _ Y V \to U$ over $X$. Suppose that for every $u \in U$ we can find an object $V_ u \to Y$ of $Y_{\acute{e}tale}$ and a morphism $h_ u : X \times _ Y V_ u \to U$ over $X$ with $u \in \mathop{\mathrm{Im}}(h_ u)$. Then we can take $V = \coprod V_ u$ and $h = \coprod h_ u$ and we win. Hence given a point $u \in U$ we find a pair $(V_ u, h_ u)$ as above. To do this we may shrink $U$ and assume that $U$ is affine. In this case $g : U \to X$ is locally quasi-finite. Let $g^{-1}(g(\{ u\} )) = \{ u, u_2, \ldots , u_ n\} $. Since there are no specializations $u_ i \leadsto u$ we may replace $U$ by an affine neighbourhood so that $g^{-1}(g(\{ u\} )) = \{ u\} $.
The image $g(U) \subset X$ is open, hence $f(g(U))$ is locally closed in $Y$. Choose an open $V \subset Y$ such that $f(g(U)) = f(X) \cap V$. It follows that $g$ factors through $X \times _ Y V$ and that the resulting $\{ U \to X \times _ Y V\} $ is an étale covering. Since $f$ has property (B) , see Lemma 59.43.4, we see that there exists an étale covering $\{ V_ j \to V\} $ such that $X \times _ Y V_ j \to X \times _ Y V$ factor through $U$. This implies that $V' = \coprod V_ j$ is étale over $Y$ and that there is a morphism $h : X \times _ Y V' \to U$ whose image surjects onto $g(U)$. Since $u$ is the only point in its fibre it must be in the image of $h$ and we win. $\square$
We urge the reader to think of the following lemma as a way station1 on the journey towards the ultimate truth regarding $f_{small, *}$ for integral universally injective morphisms.
Lemma 59.44.5. Let $f : X \to Y$ be a morphism of schemes. Assume that $f$ is universally injective and integral (for example a closed immersion). Then
$f_{small, *} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ reflects injections and surjections,
$f_{small, *} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ commutes with pushouts and coequalizers (and more generally finite connected colimits),
$f_{small, *}$ transforms surjections into surjections (on sheaves of sets and on abelian sheaves),
the map $f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}$ is surjective for any sheaf (of sets or of abelian groups) $\mathcal{F}$ on $X_{\acute{e}tale}$,
the functor $f_{small, *}$ is faithful (on sheaves of sets and on abelian sheaves),
$f_{small, *} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ is exact, and
the functor $Y_{\acute{e}tale}\to X_{\acute{e}tale}$, $V \mapsto X \times _ Y V$ is almost cocontinuous.
Proof. By Lemmas 59.42.3, 59.43.4 and 59.44.4 we know that the morphism $f$ has properties (A), (B), and (C). Moreover, by Lemma 59.44.3 we know that the functor $Y_{\acute{e}tale}\to X_{\acute{e}tale}$ is almost cocontinuous. Now we have
property (C) implies (1) by Lemma 59.44.1,
almost continuous implies (2) by Sites, Lemma 7.42.6,
property (B) implies (3) by Lemma 59.43.1.
Properties (4), (5), and (6) follow formally from the first three, see Sites, Lemma 7.41.1 and Modules on Sites, Lemma 18.15.2. Property (7) we saw above. $\square$
[1] A way station is a place where people stop to eat and rest when they are on a long journey.
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