Lemma 10.153.5. Let $(R, \mathfrak m, \kappa )$ be a henselian local ring. Any finite type $R$-algebra $S$ can be written as $S = A_1 \times \ldots \times A_ n \times B$ with $A_ i$ local and finite over $R$ and $R \to B$ not quasi-finite at any prime of $B$ lying over $\mathfrak m$.

Proof. This is a combination of parts (11) and (10) of Lemma 10.153.3. $\square$

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