The Stacks project

Lemma 37.23.7. 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 a finite universal homeomorphism,

  4. $X' = (Y' \times _ Y X)_{red} = (Y' \times _ V X_ V)_{red}$,

  5. $g'$ is a finite universal homeomorphism,

  6. $Y'$ is an integral affine scheme,

  7. $f'$ is flat and of finite presentation, and

  8. the generic fibre of $f'$ is geometrically reduced.

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 $K = \kappa (\mathfrak p)$. Let $p$ be the characteristic of $K$ if positive and $1$ if the characteristic is zero. By Varieties, Lemma 33.6.11 there exists a finite purely inseparable field extension $K'/K$ such that $X_{K'}$ is geometrically reduced over $K'$. Choose elements $x_1, \ldots , x_ n \in K'$ which generate $K'$ over $K$ and such that some $p$-power of $x_ i$ is in $A/\mathfrak p$. Let $A' \subset K'$ be the finite $A$-subalgebra of $K'$ generated by $x_1, \ldots , x_ n$. Note that $A'$ is a domain with fraction field $K'$. By Algebra, Lemma 10.46.7 we see that $A \to A'$ is a universal homeomorphism. Set $Y' = \mathop{\mathrm{Spec}}(A')$. Set $X' = (Y' \times _ Y X)_{red}$. The generic fibre of $X' \to Y'$ is $(X_ K)_{red}$ by Lemma 37.23.6 which is geometrically reduced by construction. Note that $X' \to X_ V$ is a finite universal homeomorphism as the composition of the reduction morphism $X' \to Y' \times _ Y X$ (see Morphisms, Lemma 29.45.6) and the base change of $g$. At this point all of the properties of the lemma hold except for possibly (7). This can be achieved by shrinking $Y'$ and hence $V$, see Morphisms, Proposition 29.27.1. $\square$


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