Lemma 10.145.4. Let $R \to S$ be a ring map. Let $\mathfrak p \subset R$ be a prime. Assume $R \to S$ finite type. Then there exists

1. an étale ring map $R \to R'$,

2. a prime $\mathfrak p' \subset R'$ lying over $\mathfrak p$,

3. a product decomposition

$R' \otimes _ R S = A_1 \times \ldots \times A_ n \times B$

with the following properties

1. each $A_ i$ is finite over $R'$,

2. each $A_ i$ has exactly one prime $\mathfrak r_ i$ lying over $\mathfrak p'$,

3. the finite field extensions $\kappa (\mathfrak r_ i)/\kappa (\mathfrak p')$ are purely inseparable, and

4. $R' \to B$ not quasi-finite at any prime lying over $\mathfrak p'$.

Proof. The strategy of the proof is to make two étale ring extensions: first we control the residue fields, then we apply Lemma 10.145.3.

Denote $F = S \otimes _ R \kappa (\mathfrak p)$ the fibre ring of $S/R$ at the prime $\mathfrak p$. As in the proof of Lemma 10.145.3 there are finitely may primes, say $\mathfrak q_1, \ldots , \mathfrak q_ n$ of $S$ lying over $R$ at which the ring map $R \to S$ is quasi-finite. Let $\kappa (\mathfrak p) \subset L_ i \subset \kappa (\mathfrak q_ i)$ be the subfield such that $\kappa (\mathfrak p) \subset L_ i$ is separable, and the field extension $\kappa (\mathfrak q_ i)/L_ i$ is purely inseparable. Let $L/\kappa (\mathfrak p)$ be a finite Galois extension into which $L_ i$ embeds for $i = 1, \ldots , n$. By Lemma 10.144.3 we can find an étale ring extension $R \to R'$ together with a prime $\mathfrak p'$ lying over $\mathfrak p$ such that the field extension $\kappa (\mathfrak p')/\kappa (\mathfrak p)$ is isomorphic to $\kappa (\mathfrak p) \subset L$. Thus the fibre ring of $R' \otimes _ R S$ at $\mathfrak p'$ is isomorphic to $F \otimes _{\kappa (\mathfrak p)} L$. The primes lying over $\mathfrak q_ i$ correspond to primes of $\kappa (\mathfrak q_ i) \otimes _{\kappa (\mathfrak p)} L$ which is a product of fields purely inseparable over $L$ by our choice of $L$ and elementary field theory. These are also the only primes over $\mathfrak p'$ at which $R' \to R' \otimes _ R S$ is quasi-finite, by Lemma 10.122.8. Hence after replacing $R$ by $R'$, $\mathfrak p$ by $\mathfrak p'$, and $S$ by $R' \otimes _ R S$ we may assume that for all primes $\mathfrak q$ lying over $\mathfrak p$ for which $S/R$ is quasi-finite the field extensions $\kappa (\mathfrak q)/\kappa (\mathfrak p)$ are purely inseparable.

Next apply Lemma 10.145.3. The result is what we want since the field extensions do not change under this étale ring extension. $\square$

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