The Stacks project

Lemma 37.44.4. Let $f : X \to S$ be a morphism of schemes which is flat, locally of finite presentation, separated, and quasi-finite. Then there exist closed subsets

\[ \emptyset = Z_{-1} \subset Z_0 \subset Z_1 \subset Z_2 \subset \ldots \subset S \]

such that with $S_ r = Z_ r \setminus Z_{r - 1}$ the stratification $S = \coprod S_ r$ is characterized by the following universal property: Given a morphism $g : T \to S$ the projection $X \times _ S T \to T$ is finite locally free of degree $r$ if and only if $g(T) \subset S_ r$ (set theoretically). Moreover, the inclusion maps $S_ r \to S$ are quasi-compact.

Proof. The question is local on $S$, hence we may assume that $S$ is affine. By Morphisms, Lemma 29.56.9 the fibres of $f$ are universally bounded in this case. Hence the existence of the stratification follows from Lemma 37.44.3.

We will show that $U_ r = S \setminus Z_ r \to S$ is quasi-compact for each $r \geq 0$. This will prove the final statement by elementary topology. Since a composition of quasi-compact maps is quasi-compact it suffices to prove that $U_ r \to U_{r - 1}$ is quasi-compact. Choose an affine open $W \subset U_{r - 1}$. Write $W = \mathop{\mathrm{Spec}}(A)$. Then $Z_ r \cap W = V(I)$ for some ideal $I \subset A$ and $X \times _ S \mathop{\mathrm{Spec}}(A/I) \to \mathop{\mathrm{Spec}}(A/I)$ is finite locally free of degree $r$. Note that $A/I = \mathop{\mathrm{colim}}\nolimits A/I_ i$ where $I_ i \subset I$ runs through the finitely generated ideals. By Limits, Lemma 32.8.8 we see that $X \times _ S \mathop{\mathrm{Spec}}(A/I_ i) \to \mathop{\mathrm{Spec}}(A/I_ i)$ is finite locally free of degree $r$ for some $i$. (This uses that $X \to S$ is of finite presentation, as it is locally of finite presentation, separated, and quasi-compact.) Hence $\mathop{\mathrm{Spec}}(A/I_ i) \to \mathop{\mathrm{Spec}}(A) = W$ factors (set theoretically) through $Z_ r \cap W$. It follows that $Z_ r \cap W = V(I_ i)$ is the zero set of a finite subset of elements of $A$. This means that $W \setminus Z_ r$ is a finite union of standard opens, hence quasi-compact, as desired. $\square$


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