Lemma 29.32.20. Let $f : X \to S$ be a morphism of schemes. Let $\sigma : S \to X$ be a section of $f$. Let $s \in S$ be a point such that $f$ is smooth at $x = \sigma (s)$. Then there exist affine open neighbourhoods $\mathop{\mathrm{Spec}}(A) = U \subset S$ of $s$ and $\mathop{\mathrm{Spec}}(B) = V \subset X$ of $x$ such that

$f(V) \subset U$ and $\sigma (U) \subset V$,

with $I = \mathop{\mathrm{Ker}}(\sigma ^\# : B \to A)$ the module $I/I^2$ is a free $A$-module, and

$B^\wedge \cong A[[x_1, \ldots , x_ d]]$ as $A$-algebras where $B^\wedge $ denotes the completion of $B$ with respect to $I$.

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