Lemma 23.9.4. Consider a commutative diagram

$\xymatrix{ S \ar[r] & B \\ & A \ar[lu] \ar[u] }$

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

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

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

Proof. Omitted. Hint: the proof is identical to the argument given in the first paragraph of the proof of Lemma 23.9.1. $\square$

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