The Stacks project

Theorem 59.32.1. Let $A\to B$ be finite type ring map and $\mathfrak p \subset A$ a prime ideal. Then there exist an étale ring map $A \to A'$ and a prime $\mathfrak p' \subset A'$ lying over $\mathfrak p$ such that

1. $\kappa (\mathfrak p) = \kappa (\mathfrak p')$,

2. $ B \otimes _ A A' = B_1\times \ldots \times B_ r \times C$,

3. $ A'\to B_ i$ is finite and there exists a unique prime $q_ i\subset B_ i$ lying over $\mathfrak p'$, and

4. all irreducible components of the fibre $\mathop{\mathrm{Spec}}(C \otimes _{A'} \kappa (\mathfrak p'))$ of $C$ over $\mathfrak p'$ have dimension at least 1.