The Stacks project

Lemma 15.41.5. Let $R$ be a ring. Let $(A_ i, \varphi _{ii'})$ be a directed system of smooth $R$-algebras. Set $\Lambda = \mathop{\mathrm{colim}}\nolimits A_ i$. If the fibre rings $\Lambda \otimes _ R \kappa (\mathfrak p)$ are Noetherian for all $\mathfrak p \subset R$, then $R \to \Lambda $ is regular.

Proof. Note that $\Lambda $ is flat over $R$ by Algebra, Lemmas 10.39.3 and 10.137.10. Let $\kappa (\mathfrak p) \subset k$ be a finite purely inseparable extension. Note that

\[ \Lambda \otimes _ R \kappa (\mathfrak p) \otimes _{\kappa (\mathfrak p)} k = \Lambda \otimes _ R k = \mathop{\mathrm{colim}}\nolimits A_ i \otimes _ R k \]

is a colimit of smooth $k$-algebras, see Algebra, Lemma 10.137.4. Since each local ring of a smooth $k$-algebra is regular by Algebra, Lemma 10.140.3 we conclude that all local rings of $\Lambda \otimes _ R k$ are regular by Algebra, Lemma 10.106.8. This proves the lemma. $\square$

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