The Stacks project

Lemma 29.57.9. Let $f : X \to Y$ be a morphism of schemes. Assume that

  1. $f$ is locally quasi-finite, and

  2. $X$ is quasi-compact.

Then $f$ has universally bounded fibres.

Proof. Since $X$ is quasi-compact, there exists a finite affine open covering $X = \bigcup _{i = 1, \ldots , n} U_ i$ and affine opens $V_ i \subset Y$, $i = 1, \ldots , n$ such that $f(U_ i) \subset V_ i$. Because of the local nature of “local quasi-finiteness” (see Lemma 29.20.6 part (4)) we see that the morphisms $f|_{U_ i} : U_ i \to V_ i$ are locally quasi-finite morphisms of affines, hence quasi-finite, see Lemma 29.20.9. For $y \in Y$ it is clear that $X_ y = \bigcup _{y \in V_ i} (U_ i)_ y$ is an open covering. Hence it suffices to prove the lemma for a quasi-finite morphism of affines (namely, if $n_ i$ works for the morphism $f|_{U_ i} : U_ i \to V_ i$, then $\sum n_ i$ works for $f$).

Assume $f : X \to Y$ is a quasi-finite morphism of affines. By Lemma 29.56.3 we can find a diagram

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

with $Z$ affine, $\pi $ finite and $j$ an open immersion. Since $j$ has universally bounded fibres (Lemma 29.57.7) this reduces us to showing that $\pi $ has universally bounded fibres (Lemma 29.57.4).

This reduces us to a morphism of the form $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ where $A \to B$ is finite. Say $B$ is generated by $x_1, \ldots , x_ n$ over $A$ and say $P_ i(T) \in A[T]$ is a monic polynomial of degree $d_ i$ such that $P_ i(x_ i) = 0$ in $B$ (a finite ring extension is integral, see Algebra, Lemma 10.36.3). With these notations it is clear that

\[ \bigoplus \nolimits _{0 \leq e_ i < d_ i, i = 1, \ldots n} A \longrightarrow B, \quad (a_{(e_1, \ldots , e_ n)}) \longmapsto \sum a_{(e_1, \ldots , e_ n)} x_1^{e_1} \ldots x_ n^{e_ n} \]

is a surjective $A$-module map. Thus for any prime $\mathfrak p \subset A$ this induces a surjective map $\kappa (\mathfrak p)$-vector spaces

\[ \kappa (\mathfrak p)^{\oplus d_1 \ldots d_ n} \longrightarrow B \otimes _ A \kappa (\mathfrak p) \]

In other words, the integer $d_1 \ldots d_ n$ works in the definition of a morphism with universally bounded fibres. $\square$

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