The Stacks Project


Tag 0BB6

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

Lemma 59.13.1. Any locally Noetherian decent algebraic space is quasi-separated.

Proof. Namely, let $X$ be an algebraic space (over some base scheme, for example over $\mathbf{Z}$) which is decent and locally Noetherian. Let $U \to X$ and $V \to X$ be étale morphisms with $U$ and $V$ affine schemes. We have to show that $W = U \times_X V$ is quasi-compact (Properties of Spaces, Lemma 57.3.3). Since $X$ is locally Noetherian, the schemes $U$, $V$ are Noetherian and $W$ is locally Noetherian. Since $X$ is decent, the fibres of the morphism $W \to U$ are finite. Namely, we can represent any $x \in |X|$ by a quasi-compact monomorphism $\mathop{\mathrm{Spec}}(k) \to X$. Then $U_k$ and $V_k$ are finite disjoint unions of spectra of finite separable extensions of $k$ (Remark 59.4.1) and we see that $W_k = U_k \times_{\mathop{\mathrm{Spec}}(k)} V_k$ is finite. Let $n$ be the maximum degree of a fibre of $W \to U$ at a generic point of an irreducible component of $U$. Consider the stratification $$ U = U_0 \supset U_1 \supset U_2 \supset \ldots $$ associated to $W \to U$ in More on Morphisms, Lemma 36.38.10. By our choice of $n$ above we conclude that $U_{n + 1}$ is empty. Hence we see that the fibres of $W \to U$ are universally bounded. Then we can apply More on Morphisms, Lemma 36.38.8 to find a stratification $$ \emptyset = Z_{-1} \subset Z_0 \subset Z_1 \subset Z_2 \subset \ldots \subset Z_n = U $$ by closed subsets such that with $S_r = Z_r \setminus Z_{r - 1}$ the morphism $W \times_U S_r \to S_r$ is finite locally free. Since $U$ is Noetherian, the schemes $S_r$ are Noetherian, whence the schemes $W \times_U S_r$ are Noetherian, whence $W = \coprod W \times_U S_r$ is quasi-compact as desired. $\square$

    The code snippet corresponding to this tag is a part of the file decent-spaces.tex and is located in lines 2729–2732 (see updates for more information).

    \begin{lemma}
    \label{lemma-locally-Noetherian-decent-quasi-separated}
    Any locally Noetherian decent algebraic space is quasi-separated.
    \end{lemma}
    
    \begin{proof}
    Namely, let $X$ be an algebraic space (over some base scheme, for
    example over $\mathbf{Z}$) which is decent and locally Noetherian.
    Let $U \to X$ and $V \to X$ be \'etale morphisms with $U$ and $V$
    affine schemes. We have to show that $W = U \times_X V$ is quasi-compact
    (Properties of Spaces, Lemma
    \ref{spaces-properties-lemma-characterize-quasi-separated}).
    Since $X$ is locally Noetherian, the schemes $U$, $V$ are Noetherian
    and $W$ is locally Noetherian. Since $X$ is decent, the fibres
    of the morphism $W \to U$ are finite. Namely, we can represent
    any $x \in |X|$ by a quasi-compact monomorphism $\Spec(k) \to X$.
    Then $U_k$ and $V_k$ are finite disjoint unions of spectra of
    finite separable extensions of $k$ (Remark \ref{remark-recall})
    and we see that $W_k = U_k \times_{\Spec(k)} V_k$ is finite.
    Let $n$ be the maximum degree of a fibre of $W \to U$ at a generic
    point of an irreducible component of $U$. Consider the stratification
    $$
    U = U_0 \supset U_1 \supset U_2 \supset \ldots
    $$
    associated to $W \to U$ in
    More on Morphisms, Lemma \ref{more-morphisms-lemma-stratify-flat-fp-lqf}.
    By our choice of $n$ above we conclude that $U_{n + 1}$ is empty.
    Hence we see that the fibres of $W \to U$ are universally bounded.
    Then we can apply More on Morphisms, Lemma
    \ref{more-morphisms-lemma-stratify-flat-fp-lqf-universally-bounded}
    to find a stratification
    $$
    \emptyset = Z_{-1} \subset Z_0 \subset Z_1 \subset Z_2 \subset
    \ldots \subset Z_n = U
    $$
    by closed subsets such that with $S_r = Z_r \setminus Z_{r - 1}$
    the morphism $W \times_U S_r \to S_r$ is finite locally free.
    Since $U$ is Noetherian, the schemes $S_r$ are Noetherian,
    whence the schemes $W \times_U S_r$ are Noetherian, whence
    $W = \coprod W \times_U S_r$ is quasi-compact as desired.
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 0BB6

    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?