The Stacks Project


Tag 04DN

50.43. Property (B)

Please see Section 50.41 for the definition of property (B).

Lemma 50.43.1. 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 50.43.2. 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 50.43.3. 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 50.43.2. 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 50.42.3 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 50.43.4. 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 50.43.2. 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 50.43.3. $\square$

Lemma 50.43.5. 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 50.42.4. Part (1) follows from Lemmas 50.43.4 and 50.43.1. 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}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 04DN

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?