The Stacks Project


Tag 02R7

41.14. Preparation for flat pullback

Recall that a morphism $f : X \to Y$ which is locally of finite type is said to have relative dimension $r$ if every nonempty fibre is equidimensional of dimension $r$. See Morphisms, Definition 28.28.1.

Lemma 41.14.1. Let $(S, \delta)$ be as in Situation 41.8.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $f$ is flat of relative dimension $r$. For any closed subset $Z \subset Y$ we have $$ \dim_\delta(f^{-1}(Z)) = \dim_\delta(Z) + r. $$ If $Z$ is irreducible and $Z' \subset f^{-1}(Z)$ is an irreducible component, then $Z'$ dominates $Z$ and $\dim_\delta(Z') = \dim_\delta(Z) + r$.

Proof. It suffices to prove the final statement. We may replace $Y$ by the integral closed subscheme $Z$ and $X$ by the scheme theoretic inverse image $f^{-1}(Z) = Z \times_Y X$. Hence we may assume $Z = Y$ is integral and $f$ is a flat morphism of relative dimension $r$. Since $Y$ is locally Noetherian the morphism $f$ which is locally of finite type, is actually locally of finite presentation. Hence Morphisms, Lemma 28.24.9 applies and we see that $f$ is open. Let $\xi \in X$ be a generic point of an irreducible component of $X$. By the openness of $f$ we see that $f(\xi)$ is the generic point $\eta$ of $Z = Y$. Note that $\dim_\xi(X_\eta) = r$ by assumption that $f$ has relative dimension $r$. On the other hand, since $\xi$ is a generic point of $X$ we see that $\mathcal{O}_{X, \xi} = \mathcal{O}_{X_\eta, \xi}$ has only one prime ideal and hence has dimension $0$. Thus by Morphisms, Lemma 28.27.1 we conclude that the transcendence degree of $\kappa(\xi)$ over $\kappa(\eta)$ is $r$. In other words, $\delta(\xi) = \delta(\eta) + r$ as desired. $\square$

Here is the lemma that we will use to prove that the flat pullback of a locally finite collection of closed subschemes is locally finite.

Lemma 41.14.2. Let $(S, \delta)$ be as in Situation 41.8.1. Let $X$, $Y$ be locally of finite type over $S$. Let $f : X \to Y$ be a morphism. Assume $\{Z_i\}_{i \in I}$ is a locally finite collection of closed subsets of $Y$. Then $\{f^{-1}(Z_i)\}_{i \in I}$ is a locally finite collection of closed subsets of $X$.

Proof. Let $U \subset X$ be a quasi-compact open subset. Since the image $f(U) \subset Y$ is a quasi-compact subset there exists a quasi-compact open $V \subset Y$ such that $f(U) \subset V$. Note that $$ \{i \in I \mid f^{-1}(Z_i) \cap U \not = \emptyset \} \subset \{i \in I \mid Z_i \cap V \not = \emptyset \}. $$ Since the right hand side is finite by assumption we win. $\square$

    The code snippet corresponding to this tag is a part of the file chow.tex and is located in lines 3664–3741 (see updates for more information).

    \section{Preparation for flat pullback}
    \label{section-preparation-flat-pullback}
    
    
    \noindent
    Recall that a morphism $f : X \to Y$ which is locally of finite type
    is said to have relative dimension $r$ if every nonempty fibre
    is equidimensional of dimension $r$. See
    Morphisms, Definition \ref{morphisms-definition-relative-dimension-d}.
    
    \begin{lemma}
    \label{lemma-flat-inverse-image-dimension}
    Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
    Let $X$, $Y$ be locally of finite type over $S$.
    Let $f : X \to Y$ be a morphism.
    Assume $f$ is flat of relative dimension $r$.
    For any closed subset $Z \subset Y$ we have
    $$
    \dim_\delta(f^{-1}(Z)) = \dim_\delta(Z) + r.
    $$
    If $Z$ is irreducible and $Z' \subset f^{-1}(Z)$ is an irreducible
    component, then $Z'$ dominates $Z$ and
    $\dim_\delta(Z') = \dim_\delta(Z) + r$.
    \end{lemma}
    
    \begin{proof}
    It suffices to prove the final statement.
    We may replace $Y$ by the integral closed subscheme $Z$ and
    $X$ by the scheme theoretic inverse image $f^{-1}(Z) = Z \times_Y X$.
    Hence we may assume $Z = Y$ is integral and $f$ is a flat morphism
    of relative dimension $r$. Since $Y$ is locally Noetherian the
    morphism $f$ which is locally of finite type,
    is actually locally of finite presentation. Hence
    Morphisms, Lemma \ref{morphisms-lemma-fppf-open}
    applies and we see that $f$ is open.
    Let $\xi \in X$ be a generic point of an irreducible component
    of $X$. By the openness of $f$ we see that $f(\xi)$ is the
    generic point $\eta$ of $Z = Y$. Note that $\dim_\xi(X_\eta) = r$
    by assumption that $f$ has relative dimension $r$. On the other
    hand, since $\xi$ is a generic point of $X$ we see that
    $\mathcal{O}_{X, \xi} = \mathcal{O}_{X_\eta, \xi}$ has only one
    prime ideal and hence has dimension $0$. Thus by
    Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-at-a-point}
    we conclude that the transcendence
    degree of $\kappa(\xi)$ over $\kappa(\eta)$ is $r$.
    In other words, $\delta(\xi) = \delta(\eta) + r$ as desired.
    \end{proof}
    
    \noindent
    Here is the lemma that we will use to prove that the flat pullback
    of a locally finite collection of closed subschemes is locally finite.
    
    \begin{lemma}
    \label{lemma-inverse-image-locally-finite}
    Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
    Let $X$, $Y$ be locally of finite type over $S$.
    Let $f : X \to Y$ be a morphism.
    Assume $\{Z_i\}_{i \in I}$ is a locally
    finite collection of closed subsets of $Y$.
    Then $\{f^{-1}(Z_i)\}_{i \in I}$ is a locally finite
    collection of closed subsets of $X$.
    \end{lemma}
    
    \begin{proof}
    Let $U \subset X$ be a quasi-compact open subset.
    Since the image $f(U) \subset Y$ is a quasi-compact subset
    there exists a quasi-compact open $V \subset Y$ such that
    $f(U) \subset V$. Note that
    $$
    \{i \in I \mid f^{-1}(Z_i) \cap U \not = \emptyset \}
    \subset
    \{i \in I \mid Z_i \cap V \not = \emptyset \}.
    $$
    Since the right hand side is finite by assumption we win.
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 02R7

    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?