The Stacks project

Lemma 59.48.4. Let $f : X \to Y$ be a closed immersion of schemes. Let $\{ U_ i \to X\} $ be a syntomic (resp. smooth, resp. étale) covering. There exists a syntomic (resp. smooth, resp. étale) covering $\{ V_ j \to Y\} $ such that for each $j$, either $V_ j \times _ Y X = \emptyset $, or the morphism $V_ j \times _ Y X \to X$ factors through $U_ i$ for some $i$.

Proof. For each $i$ we can choose syntomic (resp. smooth, resp. étale) morphisms $g_{ij} : V_{ij} \to Y$ and morphisms $V_{ij} \times _ Y X \to U_ i$ over $X$, such that $\{ V_{ij} \times _ Y X \to U_ i\} $ are Zariski coverings, see Lemma 59.48.3. This in particular implies that $\bigcup _{ij} g_{ij}(V_{ij})$ contains the closed subset $f(X)$. Hence the family of syntomic (resp. smooth, resp. étale) maps $g_{ij}$ together with the open immersion $Y \setminus f(X) \to Y$ forms the desired syntomic (resp. smooth, resp. étale) covering of $Y$. $\square$

Comments (2)

Comment #3276 by Kevin Carlson on

Suggested slogan: Coverings extend trivially along closed immersions.

Comment #3368 by on

Nope. Not a good slogan for this result.

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