The Stacks Project


Tag 0CM6

Chapter 61: Limits of Algebraic Spaces > Section 61.3: Morphisms of finite presentation

Lemma 61.3.11. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If for every directed limit $T = \mathop{\rm lim}\nolimits_{i \in I} T_i$ of affine schemes over $S$ the map $$ \mathop{\rm colim}\nolimits X(T_i) \longrightarrow X(T) \times_{Y(T)} \mathop{\rm colim}\nolimits Y(T_i) $$ is surjective, then $f$ is locally of finite presentation. In other words, in Proposition 61.3.9 part (2) it suffices to check surjectivity in the criterion of Lemma 61.3.2.

Proof. Choose a scheme $V$ and a surjective étale morphism $g : V \to Y$. Next, choose a scheme $U$ and a surjective étale morphism $h : U \to V \times_Y X$. It suffices to show for $T = \mathop{\rm lim}\nolimits T_i$ as in the lemma that the map $$ \mathop{\rm colim}\nolimits U(T_i) \longrightarrow U(T) \times_{V(T)} \mathop{\rm colim}\nolimits V(T_i) $$ is surjective, because then $U \to V$ will be locally of finite presentation by Limits, Lemma 31.6.3 (modulo a set theoretic remark exactly as in the proof of Proposition 61.3.9). Thus we take $a : T \to U$ and $b_i : T_i \to V$ which determine the same morphism $T \to V$. Picture $$ \xymatrix{ T \ar[d]_a \ar[rr]_{p_i} & & T_i \ar[d]^{b_i} \ar@{..>}[ld] \\ U \ar[r]^-h & X \times_Y V \ar[d] \ar[r] & V \ar[d]^g \\ & X \ar[r]^f & Y } $$ By the assumption of the lemma after increasing $i$ we can find a morphism $c_i : T_i \to X$ such that $h \circ a = (b_i, c_i) \circ p_i : T_i \to V \times_Y X$ and such that $f \circ c_i = g \circ b_i$. Since $h$ is an étale morphism of algebraic spaces (and hence locally of finite presentation), we have the surjectivity of $$ \mathop{\rm colim}\nolimits U(T_i) \longrightarrow U(T) \times_{(X \times_Y V)(T)} \mathop{\rm colim}\nolimits (X \times_Y V)(T_i) $$ by Proposition 61.3.9. Hence after increasing $i$ again we can find the desired morphism $a_i : T_i \to U$ with $a = a_i \circ p_i$ and $b_i = (U \to V) \circ a_i$. $\square$

    The code snippet corresponding to this tag is a part of the file spaces-limits.tex and is located in lines 554–568 (see updates for more information).

    \begin{lemma}
    \label{lemma-surjection-is-enough}
    Let $S$ be a scheme.
    Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.
    If for every directed limit $T = \lim_{i \in I} T_i$
    of affine schemes over $S$ the map
    $$
    \colim X(T_i) \longrightarrow X(T) \times_{Y(T)} \colim Y(T_i)
    $$
    is surjective, then $f$ is locally of finite presentation.
    In other words, in
    Proposition \ref{proposition-characterize-locally-finite-presentation}
    part (2) it suffices to check surjectivity in the criterion of
    Lemma \ref{lemma-characterize-relative-limit-preserving}.
    \end{lemma}
    
    \begin{proof}
    Choose a scheme $V$ and a surjective \'etale morphism $g : V \to Y$.
    Next, choose a scheme $U$ and a surjective \'etale morphism
    $h : U \to V \times_Y X$. It suffices to show for $T = \lim T_i$
    as in the lemma that the map
    $$
    \colim U(T_i) \longrightarrow U(T) \times_{V(T)} \colim V(T_i)
    $$
    is surjective, because then $U \to V$ will be locally of finite
    presentation by Limits, Lemma \ref{limits-lemma-surjection-is-enough}
    (modulo a set theoretic remark exactly as in the proof of
    Proposition \ref{proposition-characterize-locally-finite-presentation}).
    Thus we take $a : T \to U$ and $b_i : T_i \to V$ which determine
    the same morphism $T \to V$. Picture
    $$
    \xymatrix{
    T \ar[d]_a \ar[rr]_{p_i} & & T_i \ar[d]^{b_i} \ar@{..>}[ld] \\
    U \ar[r]^-h & X \times_Y V \ar[d] \ar[r] & V \ar[d]^g \\
    & X \ar[r]^f & Y
    }
    $$
    By the assumption of the lemma after increasing $i$
    we can find a morphism $c_i : T_i \to X$ such that
    $h \circ a = (b_i, c_i) \circ p_i : T_i \to V \times_Y X$
    and such that $f \circ c_i = g \circ b_i$.
    Since $h$ is an \'etale morphism of algebraic spaces
    (and hence locally of finite presentation), we have the surjectivity of
    $$
    \colim U(T_i) \longrightarrow U(T) \times_{(X \times_Y V)(T)}
    \colim (X \times_Y V)(T_i)
    $$
    by Proposition \ref{proposition-characterize-locally-finite-presentation}.
    Hence after increasing $i$ again we can find the desired
    morphism $a_i : T_i \to U$ with $a = a_i \circ p_i$ and
    $b_i = (U \to V) \circ a_i$.
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 0CM6

    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?