The Stacks project

Lemma 59.72.1. Let $U \to X$ be an étale morphism of quasi-compact and quasi-separated schemes (for example an étale morphism of Noetherian schemes). Then there exists a partition $X = \coprod _ i X_ i$ by constructible locally closed subschemes such that $X_ i \times _ X U \to X_ i$ is finite étale for all $i$.

Proof. If $U \to X$ is separated, then this is More on Morphisms, Lemma 37.45.4. In general, we may assume $X$ is affine. Choose a finite affine open covering $U = \bigcup U_ j$. Apply the previous case to all the morphisms $U_ j \to X$ and $U_ j \cap U_{j'} \to X$ and choose a common refinement $X = \coprod X_ i$ of the resulting partitions. After refining the partition further we may assume $X_ i$ affine as well. Fix $i$ and set $V = U \times _ X X_ i$. The morphisms $V_ j = U_ j \times _ X X_ i \to X_ i$ and $V_{jj'} = (U_ j \cap U_{j'}) \times _ X X_ i \to X_ i$ are finite étale. Hence $V_ j$ and $V_{jj'}$ are affine schemes and $V_{jj'} \subset V_ j$ is closed as well as open (since $V_{jj'} \to X_ i$ is proper, so Morphisms, Lemma 29.41.7 applies). Then $V = \bigcup V_ j$ is separated because $\mathcal{O}(V_ j) \to \mathcal{O}(V_{jj'})$ is surjective, see Schemes, Lemma 26.21.7. Thus the previous case applies to $V \to X_ i$ and we can further refine the partition if needed (it actually isn't but we don't need this). $\square$

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