The Stacks project

Lemma 10.153.6. Let $(R, \mathfrak m, \kappa )$ be a strictly 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 $\kappa \subset \kappa (\mathfrak m_{A_ i})$ finite purely inseparable and $R \to B$ not quasi-finite at any prime of $B$ lying over $\mathfrak m$.

Proof. First write $S = A_1 \times \ldots \times A_ n \times B$ as in Lemma 10.153.5. The field extension $\kappa (\mathfrak m_{A_ i})/\kappa $ is finite and $\kappa $ is separably algebraically closed, hence it is finite purely inseparable. $\square$

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