The Stacks project

Lemma 68.8.2. Let $S$ be a scheme. Let $W \to X$ be a morphism of a scheme $W$ to an algebraic space $X$ which is flat, locally of finite presentation, separated, and locally quasi-finite. Then there exist open subspaces

\[ X = X_0 \supset X_1 \supset X_2 \supset \ldots \]

such that a morphism $\mathop{\mathrm{Spec}}(k) \to X$ where $k$ is a field factors through $X_ d$ if and only if $W \times _ X \mathop{\mathrm{Spec}}(k)$ has degree $\geq d$ over $k$.

Proof. Choose a scheme $U$ and a surjective ├ętale morphism $U \to X$. Apply More on Morphisms, Lemma 37.45.5 to $W \times _ X U \to U$. We obtain open subschemes

\[ U = U_0 \supset U_1 \supset U_2 \supset \ldots \]

characterized by the property stated in the lemma for the morphism $W \times _ X U \to U$. Clearly, the formation of these closed subsets commutes with base change. Setting $R = U \times _ X U$ with projection maps $s, t : R \to U$ we conclude that

\[ s^{-1}(U_ d) = t^{-1}(U_ d) \]

as open subschemes of $R$. In other words the open subschemes $U_ d \subset U$ are $R$-invariant. This means that $U_ d$ is the inverse image of an open subspace $X_ d \subset X$ (Properties of Spaces, Lemma 66.12.2). $\square$

Comments (2)

Comment #7728 by Laurent Moret-Bailly on

There seems to be an implicit convention (here and in 37.45.5) that denotes an arbitrary field. I think it should be made explicit somwhere nearby.

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