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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