The Stacks project

Lemma 37.35.3. Let $S$ be a scheme, and let $s$ be a point of $S$. The category of étale neighborhoods has the following properties:

  1. Let $(U_ i, u_ i)_{i=1, 2}$ be two étale neighborhoods of $s$ in $S$. Then there exists a third étale neighborhood $(U, u)$ and morphisms $(U, u) \to (U_ i, u_ i)$, $i = 1, 2$.

  2. Let $h_1, h_2: (U, u) \to (U', u')$ be two morphisms between étale neighborhoods of $s$. Assume $h_1$, $h_2$ induce the same map $\kappa (u') \to \kappa (u)$ of residue fields. Then there exist an étale neighborhood $(U'', u'')$ and a morphism $h : (U'', u'') \to (U, u)$ which equalizes $h_1$ and $h_2$, i.e., such that $h_1 \circ h = h_2 \circ h$.

Proof. For part (1), consider the fibre product $U = U_1 \times _ S U_2$. It is étale over both $U_1$ and $U_2$ because étale morphisms are preserved under base change, see Morphisms, Lemma 29.36.4. There is a point of $U$ mapping to both $u_1$ and $u_2$ for example by the description of points of a fibre product in Schemes, Lemma 26.17.5. For part (2), define $U''$ as the fibre product

\[ \xymatrix{ U'' \ar[r] \ar[d] & U \ar[d]^{(h_1, h_2)} \\ U' \ar[r]^-\Delta & U' \times _ S U'. } \]

Since $h_1$ and $h_2$ induce the same map of residue fields $\kappa (u') \to \kappa (u)$ there exists a point $u'' \in U''$ lying over $u'$ with $\kappa (u'') = \kappa (u')$. In particular $U'' \not= \emptyset $. Moreover, since $U'$ is étale over $S$, so is the fibre product $U'\times _ S U'$ (see Morphisms, Lemmas 29.36.4 and 29.36.3). Hence the vertical arrow $(h_1, h_2)$ is étale by Morphisms, Lemma 29.36.18. Therefore $U''$ is étale over $U'$ by base change, and hence also étale over $S$ (because compositions of étale morphisms are étale). Thus $(U'', u'')$ is a solution to the problem. $\square$

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