The Stacks project

Lemma 23.9.5. Let $A$ be a Noetherian ring. Let $A \to B$ be a finite type ring map. The following are equivalent

  1. $A \to B$ is a local complete intersection in the sense of More on Algebra, Definition 15.33.2,

  2. for every prime $\mathfrak q \subset B$ and with $\mathfrak p = A \cap \mathfrak q$ the ring map $(A_\mathfrak p)^\wedge \to (B_\mathfrak q)^\wedge $ is a complete intersection homomorphism in the sense defined above.

Proof. Choose a surjection $R = A[x_1, \ldots , x_ n] \to B$. Observe that $A \to R$ is flat with regular fibres. Let $I$ be the kernel of $R \to B$. Assume (2). Then we see that $I$ is locally generated by a regular sequence by Lemma 23.9.4 and Algebra, Lemma 10.68.6. In other words, (1) holds. Conversely, assume (1). Then after localizing on $R$ and $B$ we can assume that $I$ is generated by a Koszul regular sequence. By More on Algebra, Lemma 15.30.7 we find that $I$ is locally generated by a regular sequence. Hence (2) hold by Lemma 23.9.4. Some details omitted. $\square$

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