Lemma 41.17.1. Let $f : X \to S$ be a morphism of schemes. Let $x_1, \ldots , x_ n \in X$ be points having the same image $s$ in $S$. Assume $f$ is unramified at each $x_ i$. Then there exists an étale neighbourhood $(U, u) \to (S, s)$ and opens $V_{i, j} \subset X_ U$, $i = 1, \ldots , n$, $j = 1, \ldots , m_ i$ such that

$V_{i, j} \to U$ is a closed immersion passing through $u$,

$u$ is not in the image of $V_{i, j} \cap V_{i', j'}$ unless $i = i'$ and $j = j'$, and

any point of $(X_ U)_ u$ mapping to $x_ i$ is in some $V_{i, j}$.

## Comments (0)

There are also: