The Stacks Project


Tag 02LA

Chapter 34: Descent > Section 34.20: Properties of morphisms local in the fpqc topology on the target

Lemma 34.20.23. The property $\mathcal{P}(f) =$''$f$ is finite'' is fpqc local on the base.

Proof. An finite morphism is the same thing as an integral morphism which is locally of finite type. See Morphisms, Lemma 28.42.4. Hence the lemma follows on combining Lemmas 34.20.10 and 34.20.22. $\square$

    The code snippet corresponding to this tag is a part of the file descent.tex and is located in lines 5168–5172 (see updates for more information).

    \begin{lemma}
    \label{lemma-descending-property-finite}
    The property $\mathcal{P}(f) =$``$f$ is finite''
    is fpqc local on the base.
    \end{lemma}
    
    \begin{proof}
    An finite morphism is the same thing as an integral
    morphism which is locally of finite type. See
    Morphisms, Lemma \ref{morphisms-lemma-finite-integral}.
    Hence the lemma follows on combining
    Lemmas \ref{lemma-descending-property-locally-finite-type}
    and \ref{lemma-descending-property-integral}.
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 02LA

    Your email address will not be published.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?