Proposition 41.19.4. Let A, B be Noetherian local rings. Let f : A \to B be an étale homomorphism of local rings. Then A is regular if and only if B is so.
Proof. If B is regular, then A is regular by Algebra, Lemma 10.110.9. Assume A is regular. Let \mathfrak m be the maximal ideal of A. Then \dim _{\kappa (\mathfrak m)} \mathfrak m/\mathfrak m^2 = \dim (A) = \dim (B) (see Lemma 41.19.1). On the other hand, \mathfrak mB is the maximal ideal of B and hence \mathfrak m_ B/\mathfrak m_ B = \mathfrak mB/\mathfrak m^2B is generated by at most \dim (B) elements. Thus B is regular. (You can also use the slightly more general Algebra, Lemma 10.112.8.) \square
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