Lemma 23.9.4. Consider a commutative diagram

of Noetherian local rings with $S \to B$ surjective, $A \to S$ flat, and $S/\mathfrak m_ A S$ a regular local ring. The following are equivalent

$\mathop{\mathrm{Ker}}(S \to B)$ is generated by a regular sequence, and

$A^\wedge \to B^\wedge $ is a complete intersection homomorphism as defined above.

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