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Tag 04DN

53.43. Property (B)

Please see Section 53.41 for the definition of property (B).

Lemma 53.43.1. Let $f : X \to Y$ be a morphism of schemes. Assume (B) holds. Then the functor $f_{small, *} : \mathop{\textit{Sh}}\nolimits(X_{\acute{e}tale}) \to \mathop{\textit{Sh}}\nolimits(Y_{\acute{e}tale})$ transforms surjections into surjections.

Proof. This follows from Sites, Lemma 7.40.2. $\square$

Lemma 53.43.2. Let $f : X \to Y$ be a morphism of schemes. Suppose

1. $V \to Y$ is an étale morphism of schemes,
2. $\{U_i \to X \times_Y V\}$ is an étale covering, and
3. $v \in V$ is a point.

Assume that for any such data there exists an étale neighbourhood $(V', v') \to (V, v)$, a disjoint union decomposition $X \times_Y V' = \coprod W'_i$, and morphisms $W'_i \to U_i$ over $X \times_Y V$. Then property (B) holds.

Proof. Omitted. $\square$

Lemma 53.43.3. Let $f : X \to Y$ be a finite morphism of schemes. Then property (B) holds.

Proof. Consider $V \to Y$ étale, $\{U_i \to X \times_Y V\}$ an étale covering, and $v \in V$. We have to find a $V' \to V$ and decomposition and maps as in Lemma 53.43.2. We may shrink $V$ and $Y$, hence we may assume that $V$ and $Y$ are affine. Since $X$ is finite over $Y$, this also implies that $X$ is affine. During the proof we may (finitely often) replace $(V, v)$ by an étale neighbourhood $(V', v')$ and correspondingly the covering $\{U_i \to X \times_Y V\}$ by $\{V' \times_V U_i \to X \times_Y V'\}$.

Since $X \times_Y V \to V$ is finite there exist finitely many (pairwise distinct) points $x_1, \ldots, x_n \in X \times_Y V$ mapping to $v$. We may apply More on Morphisms, Lemma 36.36.5 to $X \times_Y V \to V$ and the points $x_1, \ldots, x_n$ lying over $v$ and find an étale neighbourhood $(V', v') \to (V, v)$ such that $$X \times_Y V' = R \amalg \coprod T_a$$ with $T_a \to V'$ finite with exactly one point $p_a$ lying over $v'$ and moreover $\kappa(v') \subset \kappa(p_a)$ purely inseparable, and such that $R \to V'$ has empty fibre over $v'$. Because $X \to Y$ is finite, also $R \to V'$ is finite. Hence after shrinking $V'$ we may assume that $R = \emptyset$. Thus we may assume that $X \times_Y V = X_1 \amalg \ldots \amalg X_n$ with exactly one point $x_l \in X_l$ lying over $v$ with moreover $\kappa(v) \subset \kappa(x_l)$ purely inseparable. Note that this property is preserved under refinement of the étale neighbourhood $(V, v)$.

For each $l$ choose an $i_l$ and a point $u_l \in U_{i_l}$ mapping to $x_l$. Now we apply property (A) for the finite morphism $X \times_Y V \to V$ and the étale morphisms $U_{i_l} \to X \times_Y V$ and the points $u_l$. This is permissible by Lemma 53.42.3 This gives produces an étale neighbourhood $(V', v') \to (V, v)$ and decompositions $$X \times_Y V' = W_l \amalg R_l$$ and $X$-morphisms $a_l : W_l \to U_{i_l}$ whose image contains $u_{i_l}$. Here is a picture: $$\xymatrix{ & & & U_{i_l} \ar[d] & \\ W_l \ar[rrru] \ar[r] & W_l \amalg R_l \ar@{=}[r] & X \times_Y V' \ar[r] \ar[d] & X \times_Y V \ar[r] \ar[d] & X \ar[d] \\ & & V' \ar[r] & V \ar[r] & Y }$$ After replacing $(V, v)$ by $(V', v')$ we conclude that each $x_l$ is contained in an open and closed neighbourhood $W_l$ such that the inclusion morphism $W_l \to X \times_Y V$ factors through $U_i \to X \times_Y V$ for some $i$. Replacing $W_l$ by $W_l \cap X_l$ we see that these open and closed sets are disjoint and moreover that $\{x_1, \ldots, x_n\} \subset W_1 \cup \ldots \cup W_n$. Since $X \times_Y V \to V$ is finite we may shrink $V$ and assume that $X \times_Y V = W_1 \amalg \ldots \amalg W_n$ as desired. $\square$

Lemma 53.43.4. Let $f : X \to Y$ be an integral morphism of schemes. Then property (B) holds.

Proof. Consider $V \to Y$ étale, $\{U_i \to X \times_Y V\}$ an étale covering, and $v \in V$. We have to find a $V' \to V$ and decomposition and maps as in Lemma 53.43.2. We may shrink $V$ and $Y$, hence we may assume that $V$ and $Y$ are affine. Since $X$ is integral over $Y$, this also implies that $X$ and $X \times_Y V$ are affine. We may refine the covering $\{U_i \to X \times_Y V\}$, and hence we may assume that $\{U_i \to X \times_Y V\}_{i = 1, \ldots, n}$ is a standard étale covering. Write $Y = \mathop{\rm Spec}(A)$, $X = \mathop{\rm Spec}(B)$, $V = \mathop{\rm Spec}(C)$, and $U_i = \mathop{\rm Spec}(B_i)$. Then $A \to B$ is an integral ring map, and $B \otimes_A C \to B_i$ are étale ring maps. By Algebra, Lemma 10.141.3 we can find a finite $A$-subalgebra $B' \subset B$ and an étale ring map $B' \otimes_A C \to B'_i$ for $i = 1, \ldots, n$ such that $B_i = B \otimes_{B'} B'_i$. Thus the question reduces to the étale covering $\{\mathop{\rm Spec}(B'_i) \to X' \times_Y V\}_{i = 1, \ldots, n}$ with $X' = \mathop{\rm Spec}(B')$ finite over $Y$. In this case the result follows from Lemma 53.43.3. $\square$

Lemma 53.43.5. Let $f : X \to Y$ be a morphism of schemes. Assume $f$ is integral (for example finite). Then

1. $f_{small, *}$ transforms surjections into surjections (on sheaves of sets and on abelian sheaves),
2. $f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}$ is surjective for any abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$,
3. $f_{small, *} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ is faithful and reflects injections and surjections, and
4. $f_{small, *} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ is exact.

Proof. Parts (2), (3) we have seen in Lemma 53.42.4. Part (1) follows from Lemmas 53.43.4 and 53.43.1. Part (4) is a consequence of part (1), see Modules on Sites, Lemma 18.15.2. $\square$

The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 5777–5960 (see updates for more information).

\section{Property (B)}
\label{section-B}

\noindent
Please see Section \ref{section-monomorphisms} for the definition of property
(B).

\begin{lemma}
\label{lemma-property-B-implies}
Let $f : X \to Y$ be a morphism of schemes. Assume (B) holds.
Then the functor
$f_{small, *} : \Sh(X_\etale) \to \Sh(Y_\etale)$
transforms surjections into surjections.
\end{lemma}

\begin{proof}
This follows from
Sites, Lemma \ref{sites-lemma-weaker}.
\end{proof}

\begin{lemma}
\label{lemma-simplify-B}
Let $f : X \to Y$ be a morphism of schemes. Suppose
\begin{enumerate}
\item $V \to Y$ is an \'etale morphism of schemes,
\item $\{U_i \to X \times_Y V\}$ is an \'etale covering, and
\item $v \in V$ is a point.
\end{enumerate}
Assume that for any such data there exists an \'etale neighbourhood
$(V', v') \to (V, v)$, a disjoint union decomposition
$X \times_Y V' = \coprod W'_i$, and morphisms $W'_i \to U_i$
over $X \times_Y V$. Then property (B) holds.
\end{lemma}

\begin{proof}
Omitted.
\end{proof}

\begin{lemma}
\label{lemma-finite-B}
Let $f : X \to Y$ be a finite morphism of schemes.
Then property (B) holds.
\end{lemma}

\begin{proof}
Consider $V \to Y$ \'etale, $\{U_i \to X \times_Y V\}$ an \'etale covering, and
$v \in V$. We have to find a $V' \to V$ and decomposition and maps as in
Lemma \ref{lemma-simplify-B}.
We may shrink $V$ and $Y$, hence we may assume that $V$ and $Y$ are affine.
Since $X$ is finite over $Y$, this also implies that $X$ is affine.
During the proof we may (finitely often) replace $(V, v)$ by an
\'etale neighbourhood $(V', v')$ and correspondingly the covering
$\{U_i \to X \times_Y V\}$ by $\{V' \times_V U_i \to X \times_Y V'\}$.

\medskip\noindent
Since $X \times_Y V \to V$ is finite there exist finitely
many (pairwise distinct) points $x_1, \ldots, x_n \in X \times_Y V$
mapping to $v$. We may apply
More on Morphisms, Lemma
\ref{more-morphisms-lemma-etale-splits-off-quasi-finite-part-technical-variant}
to $X \times_Y V \to V$ and the points $x_1, \ldots, x_n$ lying over
$v$ and find an \'etale neighbourhood $(V', v') \to (V, v)$
such that
$$X \times_Y V' = R \amalg \coprod T_a$$
with $T_a \to V'$ finite with exactly one point $p_a$ lying over $v'$
and moreover $\kappa(v') \subset \kappa(p_a)$ purely inseparable, and
such that $R \to V'$ has empty fibre over $v'$.
Because $X \to Y$ is finite, also $R \to V'$ is finite. Hence after
shrinking $V'$ we may assume that $R = \emptyset$. Thus we may
assume that $X \times_Y V = X_1 \amalg \ldots \amalg X_n$ with
exactly one point $x_l \in X_l$ lying over $v$ with moreover
$\kappa(v) \subset \kappa(x_l)$ purely inseparable. Note that this
property is preserved under refinement of the \'etale neighbourhood
$(V, v)$.

\medskip\noindent
For each $l$ choose an $i_l$ and a point $u_l \in U_{i_l}$ mapping to $x_l$.
Now we apply property (A) for the finite morphism
$X \times_Y V \to V$ and the \'etale
morphisms $U_{i_l} \to X \times_Y V$ and the points $u_l$.
This is permissible by
Lemma \ref{lemma-integral-A}
This gives produces an \'etale neighbourhood $(V', v') \to (V, v)$
and decompositions
$$X \times_Y V' = W_l \amalg R_l$$
and $X$-morphisms $a_l : W_l \to U_{i_l}$ whose image contains $u_{i_l}$.
Here is a picture:
$$\xymatrix{ & & & U_{i_l} \ar[d] & \\ W_l \ar[rrru] \ar[r] & W_l \amalg R_l \ar@{=}[r] & X \times_Y V' \ar[r] \ar[d] & X \times_Y V \ar[r] \ar[d] & X \ar[d] \\ & & V' \ar[r] & V \ar[r] & Y }$$
After replacing $(V, v)$ by $(V', v')$ we conclude that each
$x_l$ is contained in an open and closed neighbourhood $W_l$ such that
the inclusion morphism $W_l \to X \times_Y V$ factors through
$U_i \to X \times_Y V$ for some $i$. Replacing $W_l$ by $W_l \cap X_l$
we see that these open and closed sets are disjoint and moreover
that $\{x_1, \ldots, x_n\} \subset W_1 \cup \ldots \cup W_n$.
Since $X \times_Y V \to V$ is finite we may shrink $V$ and assume that
$X \times_Y V = W_1 \amalg \ldots \amalg W_n$ as desired.
\end{proof}

\begin{lemma}
\label{lemma-integral-B}
Let $f : X \to Y$ be an integral morphism of schemes.
Then property (B) holds.
\end{lemma}

\begin{proof}
Consider $V \to Y$ \'etale, $\{U_i \to X \times_Y V\}$ an \'etale covering, and
$v \in V$. We have to find a $V' \to V$ and decomposition and maps as in
Lemma \ref{lemma-simplify-B}.
We may shrink $V$ and $Y$, hence we may assume that $V$ and $Y$ are affine.
Since $X$ is integral over $Y$, this also implies that $X$ and
$X \times_Y V$ are affine. We may refine the covering
$\{U_i \to X \times_Y V\}$, and hence we may assume that
$\{U_i \to X \times_Y V\}_{i = 1, \ldots, n}$ is a standard \'etale covering.
Write $Y = \Spec(A)$, $X = \Spec(B)$,
$V = \Spec(C)$, and $U_i = \Spec(B_i)$.
Then $A \to B$ is an integral ring map, and $B \otimes_A C \to B_i$ are
\'etale ring maps. By
Algebra, Lemma \ref{algebra-lemma-etale}
we can find a finite $A$-subalgebra $B' \subset B$ and an \'etale ring
map $B' \otimes_A C \to B'_i$ for $i = 1, \ldots, n$
such that $B_i = B \otimes_{B'} B'_i$. Thus the question
reduces to the \'etale covering
$\{\Spec(B'_i) \to X' \times_Y V\}_{i = 1, \ldots, n}$
with $X' = \Spec(B')$ finite over $Y$.
In this case the result follows from
Lemma \ref{lemma-finite-B}.
\end{proof}

\begin{lemma}
\label{lemma-what-integral}
Let $f : X \to Y$ be a morphism of schemes.
Assume $f$ is integral (for example finite).
Then
\begin{enumerate}
\item $f_{small, *}$ transforms surjections into surjections (on sheaves
of sets and on abelian sheaves),
\item $f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}$
is surjective for any abelian sheaf $\mathcal{F}$ on $X_\etale$,
\item
$f_{small, *} : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$
is faithful and reflects injections and surjections, and
\item
$f_{small, *} : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$
is exact.
\end{enumerate}
\end{lemma}

\begin{proof}
Parts (2), (3) we have seen in
Lemma \ref{lemma-when-push-pull-surjective}.
Part (1) follows from
Lemmas \ref{lemma-integral-B} and \ref{lemma-property-B-implies}.
Part (4) is a consequence of part (1), see
Modules on Sites, Lemma \ref{sites-modules-lemma-exactness}.
\end{proof}

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