The Stacks project

Proof. We will prove this by induction on $r$. If $r = 1$ this follows from the following two easy facts: (a) a power of a nonzerodivisor on $M$ is a nonzerodivisor on $M$ and (b) a divisor of a nonzerodivisor on $M$ is a nonzerodivisor on $M$. If $r > 1$, then by induction applied to $M/f_1M$ we have that $f_1, f_2, \ldots , f_ r$ is an $M$-regular sequence if and only if $f_1, f_2^{e_2}, \ldots , f_ r^{e_ r}$ is an $M$-regular sequence. Thus it suffices to show, given $e > 0$, that $f_1^ e, f_2, \ldots , f_ r$ is an $M$-regular sequence if and only if $f_1, \ldots , f_ r$ is an $M$-regular sequence. We will prove this by induction on $e$. The case $e = 1$ is trivial. Since $f_1$ is a nonzerodivisor under both assumptions (by the case $r = 1$) we have a short exact sequence

Suppose that $f_1, f_2, \ldots , f_ r$ is an $M$-regular sequence. Then by induction the elements $f_2, \ldots , f_ r$ are $M/f_1M$ and $M/f_1^{e - 1}M$-regular sequences. By Lemma 10.68.8 $f_2, \ldots , f_ r$ is $M/f_1^ eM$-regular. Hence $f_1^ e, f_2, \ldots , f_ r$ is $M$-regular. Conversely, suppose that $f_1^ e, f_2, \ldots , f_ r$ is an $M$-regular sequence. Then $f_2 : M/f_1^ eM \to M/f_1^ eM$ is injective, hence $f_2 : M/f_1M \to M/f_1M$ is injective, hence by induction(!) $f_2 : M/f_1^{e - 1}M \to M/f_1^{e - 1}M$ is injective, hence

is a short exact sequence by Lemma 10.4.1. This proves the converse for $r = 2$. If $r > 2$, then we have $f_3 : M/(f_1^ e, f_2)M \to M/(f_1^ e, f_2)M$ is injective, hence $f_3 : M/(f_1, f_2)M \to M/(f_1, f_2)M$ is injective, and so on. Some details omitted. $\square$