The Stacks project

Lemma 15.82.6. A local complete intersection homomorphism is perfect.

Proof. Let $A \to B$ be a local complete intersection homomorphism. By Definition 15.33.2 this means that $B = A[x_1, \ldots , x_ n]/I$ where $I$ is a Koszul ideal in $A[x_1, \ldots , x_ n]$. By Lemmas 15.82.2 and 15.74.3 it suffices to show that $I$ is a perfect module over $A[x_1, \ldots , x_ n]$. By Lemma 15.74.12 this is a local question. Hence we may assume that $I$ is generated by a Koszul-regular sequence (by Definition 15.32.1). Of course this means that $I$ has a finite free resolution and we win. $\square$

Comments (2)

Comment #2954 by Ko Aoki on

Typo in the proof: "Let he a ..." should be replaced by "Let $A \to B" be a ...".

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