The Stacks project

Lemma 15.33.4. Let $R$ be a ring. If $R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ is a relative global complete intersection, then $f_1, \ldots , f_ c$ is a Koszul regular sequence.

Proof. Recall that the homology groups $H_ i(K_\bullet (f_\bullet ))$ are annihilated by the ideal $(f_1, \ldots , f_ c)$. Hence it suffices to show that $H_ i(K_\bullet (f_\bullet ))_\mathfrak q$ is zero for all primes $\mathfrak q \subset R[x_1, \ldots , x_ n]$ containing $(f_1, \ldots , f_ c)$. This follows from Algebra, Lemma 10.136.12 and the fact that a regular sequence is Koszul regular (Lemma 15.30.2). $\square$

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