The Stacks project

Proof. Choose an affine open neighbourhood $U = \mathop{\mathrm{Spec}}(A) \subset X$ of $x$. Write $V = f^{-1}(U) = \mathop{\mathrm{Spec}}(B)$. Then $B$ is a finite locally free $A$-module and the inclusion $A \subset B$ is a locally direct summand. Thus after shrinking $U$ we can choose a basis $1 = e_1, e_2, \ldots , e_ d$ of $B$ as an $A$-module. Write $e_ i e_ j = \sum \alpha _{ij}^ l e_ l$ for unique elements $\alpha _{ij}^ l \in A$ which satisfy the relations $\sum _ l \alpha _{ij}^ l \alpha _{lk}^ m = \sum _ l \alpha _{il}^ m \alpha _{jk}^ l$ and $\alpha _{ij}^ k = \alpha _{ji}^ k$ and $\alpha _{i1}^ j - \delta _{ij}$ in $A$. This determines a morphism $\mathop{\mathrm{Spec}}(A) \to X_ d$ by sending $a_{ij}^ l \in A_ d$ to $\alpha _{ij}^ l \in A$. By construction $V \cong \mathop{\mathrm{Spec}}(A) \times _{X_ d} Y_ d$. By the definition of $U_ d$ we see that $\mathop{\mathrm{Spec}}(A) \to X_ d$ factors through $U_ d$. This finishes the proof. $\square$