The Stacks project

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

\[ X \xrightarrow {\pi } \mathbf{A}^ d_ S \to S \]

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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