The Stacks Project


Tag 08AL

Chapter 59: Decent Algebraic Spaces > Section 59.13: Decent spaces

Lemma 59.13.5. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Consider a commutative diagram $$ \xymatrix{ \mathop{\rm Spec}(k) \ar[rr] \ar[rd] & & X \ar[ld] \\ & S } $$ Assume that the image point $s \in S$ of $\mathop{\rm Spec}(k) \to S$ is a closed point and that $\kappa(s) \subset k$ is finite. Then $\mathop{\rm Spec}(k) \to X$ is finite morphism. If $\kappa(s) = k$ then $\mathop{\rm Spec}(k) \to X$ is closed immersion.

Proof. By Lemma 59.13.4 the image point $x \in |X|$ is closed. Let $Z \subset X$ be the reduced closed subspace with $|Z| = \{x\}$ (Properties of Spaces, Lemma 57.11.4). Note that $Z$ is a decent algebraic space by Lemma 59.6.5. By Lemma 59.13.2 we see that $Z = \mathop{\rm Spec}(k')$ for some field $k'$. Of course $k \supset k' \supset \kappa(s)$. Then $\mathop{\rm Spec}(k) \to Z$ is a finite morphism of schemes and $Z \to X$ is a finite morphism as it is a closed immersion. Hence $\mathop{\rm Spec}(k) \to X$ is finite (Morphisms of Spaces, Lemma 58.44.4). If $k = \kappa(s)$, then $\mathop{\rm Spec}(k) = Z$ and $\mathop{\rm Spec}(k) \to X$ is a closed immersion. $\square$

    The code snippet corresponding to this tag is a part of the file decent-spaces.tex and is located in lines 2766–2780 (see updates for more information).

    \begin{lemma}
    \label{lemma-finite-residue-field-extension-finite}
    Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$.
    Consider a commutative diagram
    $$
    \xymatrix{
    \Spec(k) \ar[rr] \ar[rd] & & X \ar[ld] \\
    & S
    }
    $$
    Assume that the image point $s \in S$ of $\Spec(k) \to S$ is
    a closed point and that $\kappa(s) \subset k$ is finite.
    Then $\Spec(k) \to X$ is finite morphism. If $\kappa(s) = k$
    then $\Spec(k) \to X$ is closed immersion.
    \end{lemma}
    
    \begin{proof}
    By Lemma \ref{lemma-algebraic-residue-field-extension-closed-point}
    the image point $x \in |X|$ is closed. Let $Z \subset X$ be the
    reduced closed subspace with $|Z| = \{x\}$ (Properties of Spaces,
    Lemma \ref{spaces-properties-lemma-reduced-closed-subspace}).
    Note that $Z$ is a decent algebraic space by
    Lemma \ref{lemma-representable-named-properties}.
    By Lemma \ref{lemma-when-field} we see that $Z = \Spec(k')$
    for some field $k'$. Of course $k \supset k' \supset \kappa(s)$.
    Then $\Spec(k) \to Z$ is a finite morphism of schemes
    and $Z \to X$ is a finite morphism as it is a closed immersion.
    Hence $\Spec(k) \to X$ is finite (Morphisms of Spaces, Lemma
    \ref{spaces-morphisms-lemma-composition-integral}).
    If $k = \kappa(s)$, then $\Spec(k) = Z$ and $\Spec(k) \to X$
    is a closed immersion.
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 08AL

    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?