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.

Proof. See Algebra, Lemma 10.145.3, or see [Théorème 18.12.1, EGA4]. For a slew of versions in terms of morphisms of schemes, see More on Morphisms, Section 37.41. $\square$

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