The Stacks project

Lemma 37.22.8. Let $f : X \to Y$ be a morphism of schemes. Assume that $Y$ is irreducible and $f$ is of finite type. There exists a diagram

\[ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X_ V \ar[r] \ar[d] & X \ar[d]^ f \\ Y' \ar[r]^ g & V \ar[r] & Y } \]

where

  1. $V$ is a nonempty open of $Y$,

  2. $X_ V = V \times _ Y X$,

  3. $g : Y' \to V$ is surjective finite étale,

  4. $X' = Y' \times _ Y X = Y' \times _ V X_ V$,

  5. $g'$ is surjective finite étale,

  6. $Y'$ is an irreducible affine scheme, and

  7. all irreducible components of the generic fibre of $f'$ are geometrically irreducible.

Proof. Let $V = \mathop{\mathrm{Spec}}(A)$ be a nonempty affine open of $Y$. By assumption the Jacobson radical of $A$ is a prime ideal $\mathfrak p$. Let $K = \kappa (\mathfrak p)$. By Varieties, Lemma 33.8.15 there exists a finite separable field extension $K \subset K'$ such that all irreducible components of $X_{K'}$ are geometrically irreducible over $K'$. Choose an element $\alpha \in K'$ which generates $K'$ over $K$, see Fields, Lemma 9.19.1. Let $P(T) \in K[T]$ be the minimal polynomial for $\alpha $ over $K$. After replacing $\alpha $ by $f \alpha $ for some $f \in A$, $f \not\in \mathfrak p$ we may assume that there exists a monic polynomial $T^ d + a_1T^{d - 1} + \ldots + a_ d \in A[T]$ which maps to $P(T) \in K[T]$ under the map $A[T] \to K[T]$. Set $A' = A[T]/(P)$. Then $A \to A'$ is a finite free ring map such that there exists a unique prime $\mathfrak q$ lying over $\mathfrak p$, such that $K = \kappa (\mathfrak p) \subset \kappa (\mathfrak q) = K'$ is finite separable, and such that $\mathfrak pA'_{\mathfrak q}$ is the maximal ideal of $A'_{\mathfrak q}$. Hence $g : Y' = \mathop{\mathrm{Spec}}(A') \to V = \mathop{\mathrm{Spec}}(A)$ is étale at $\mathfrak q$, see Algebra, Lemma 10.143.7. This means that there exists an open $W \subset \mathop{\mathrm{Spec}}(A')$ such that $g|_ W : W \to \mathop{\mathrm{Spec}}(A)$ is étale. Since $g$ is finite and since $\mathfrak q$ is the only point lying over $\mathfrak p$ we see that $Z = g(Y' \setminus W)$ is a closed subset of $V$ not containing $\mathfrak p$. Hence after replacing $V$ by a principal affine open of $V$ which does not meet $Z$ we obtain that $g$ is finite étale. $\square$


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