The Stacks project

Proof. Let $R$, $M$, $f_1, \ldots , f_ r$ be as in the first sentence of the lemma. If $r = 1$, it is immediate that $f_1$ is $M$-Koszul-regular. Assume $r > 1$. Since $f_1$ is a nonzerodivisor on $M$, we obtain a short exact sequence of complexes:

Here $\overline{f}_ i$ is the image of $f_ i$ in $R/(f_1)$. By Lemma 15.28.8 the complex $K_\bullet (f_1, \ldots , f_ r)$ is isomorphic to the cone of multiplication by $f_1$ on $K_\bullet (f_2, \ldots , f_ r)$. Thus $K_\bullet (R, f_1, \ldots , f_ r) \otimes M$ is isomorphic to the cone on the first map. Hence $K_\bullet (\overline{f}_2, \ldots , \overline{f}_ r) \otimes M/f_1M$ is quasi-isomorphic to $K_\bullet (f_1, \ldots , f_ r) \otimes M$. As $R/(f_1)$, $M/f_1M$, $\overline{f}_2, \ldots , \overline{f}_ r$ satisfy the conditions of the lemma, by induction we conclude this complex is acyclic in postive degrees. This finishes the proof of the first statement. The second statement immediately follows from the first. $\square$