Lemma 15.75.6. A local complete intersection homomorphism is perfect.

**Proof.**
Let $A \to B$ be a local complete intersection homomorphism. By Definition 15.32.2 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 15.75.2 and 15.69.3 it suffices to show that $I$ is a perfect module over $A[x_1, \ldots , x_ n]$. By Lemma 15.69.11 this is a local question. Hence we may assume that $I$ is generated by a Koszul-regular sequence (by Definition 15.31.1). Of course this means that $I$ has a finite free resolution and we win.
$\square$

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