Lemma 33.18.2. Let f : X \to S be a finite type morphism of affine schemes. Let s \in S. If \dim (X_ s) = d, then there exists a factorization
of f such that the morphism \pi _ s : X_ s \to \mathbf{A}^ d_{\kappa (s)} of fibres over s is finite.
Proof. Write S = \mathop{\mathrm{Spec}}(A) and X = \mathop{\mathrm{Spec}}(B) and let A \to B be the ring map corresponding to f. Let \mathfrak p \subset A be the prime ideal corresponding to s. We can choose a surjection A[x_1, \ldots , x_ r] \to B. By Algebra, Lemma 10.115.4 there exist elements y_1, \ldots , y_ d \in A in the \mathbf{Z}-subalgebra of A generated by x_1, \ldots , x_ r such that the A-algebra homomorphism A[t_1, \ldots , t_ d] \to B sending t_ i to y_ i induces a finite \kappa (\mathfrak p)-algebra homomorphism \kappa (\mathfrak p)[t_1, \ldots , t_ d] \to B \otimes _ A \kappa (\mathfrak p). This proves the lemma. \square
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