The Stacks project

Lemma 29.57.10. Consider a commutative diagram of morphisms of schemes

\[ \xymatrix{ X \ar[rd]_ g \ar[rr]_ f & & Y \ar[ld]^ h \\ & Z & } \]

If $g$ has universally bounded fibres, and $f$ is surjective and flat, then also $h$ has universally bounded fibres. More precisely, if $n$ bounds the degree of the fibres of $g$, then also $n$ bounds the degree of the fibres of $h$.

Proof. Assume $g$ has universally bounded fibres, and $f$ is surjective and flat. Say the degree of the fibres of $g$ is bounded by $n \in \mathbf{N}$. We claim $n$ also works for $h$. Let $z \in Z$. Consider the morphism of schemes $X_ z \to Y_ z$. It is flat and surjective. By assumption $X_ z$ is a finite scheme over $\kappa (z)$, in particular it is the spectrum of an Artinian ring (by Algebra, Lemma 10.53.2). By Lemma 29.11.13 the morphism $X_ z \to Y_ z$ is affine in particular quasi-compact. It follows from Lemma 29.25.12 that $Y_ z$ is a finite discrete as this holds for $X_ z$. Hence $Y_ z$ is an affine scheme (this is a nice exercise; it also follows for example from Properties, Lemma 28.29.1 applied to the set of all points of $Y_ z$). Write $Y_ z = \mathop{\mathrm{Spec}}(B)$ and $X_ z = \mathop{\mathrm{Spec}}(A)$. Then $A$ is faithfully flat over $B$, so $B \subset A$. Hence $\dim _ k(B) \leq \dim _ k(A) \leq n$ as desired. $\square$

Comments (0)

Post a comment

Your email address will not be published.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 toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

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 input field. As a reminder, this is tag 03JB. Beware of the difference between the letter 'O' and the digit '0'.