The Stacks project

Lemma 59.48.3. Let $f : X \to Y$ be a closed immersion of schemes. Let $U \to X$ be a syntomic (resp. smooth, resp. étale) morphism. Then there exist syntomic (resp. smooth, resp. étale) morphisms $V_ i \to Y$ and morphisms $V_ i \times _ Y X \to U$ such that $\{ V_ i \times _ Y X \to U\} $ is a Zariski covering of $U$.

Proof. Let us prove the lemma when $\tau = syntomic$. The question is local on $U$. Thus we may assume that $U$ is an affine scheme mapping into an affine of $Y$. Hence we reduce to proving the following case: $Y = \mathop{\mathrm{Spec}}(A)$, $X = \mathop{\mathrm{Spec}}(A/I)$, and $U = \mathop{\mathrm{Spec}}(\overline{B})$, where $A/I \to \overline{B}$ be a syntomic ring map. By Algebra, Lemma 10.136.18 we can find elements $\overline{g}_ i \in \overline{B}$ such that $\overline{B}_{\overline{g}_ i} = A_ i/IA_ i$ for certain syntomic ring maps $A \to A_ i$. This proves the lemma in the syntomic case. The proof of the smooth case is the same except it uses Algebra, Lemma 10.137.20. In the étale case use Algebra, Lemma 10.143.10. $\square$

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