The Stacks project

Proof. Let $R$ be a ring and let $M$ be an $R$-module. Let $f_1, \ldots , f_ r$ be an $M$-$H_1$-regular sequence. Denote $J = (f_1, \ldots , f_ r)$. The assumption means that we have an exact sequence

\[ \wedge ^2(R^ r) \otimes M \to R^{\oplus r} \otimes M \to JM \to 0 \]

where the first arrow is given by $e_ i \wedge e_ j \otimes m \mapsto (f_ ie_ j - f_ je_ i) \otimes m$. In particular this implies that

\[ JM/J^2M = JM \otimes _ R R/J = (M/JM)^{\oplus r} \]

is a finite free module. To finish the proof we have to prove for every $n \geq 2$ the following: if

\[ \xi = \sum \nolimits _{|I| = n, I = (i_1, \ldots , i_ r)} m_ I f_1^{i_1} \ldots f_ r^{i_ r} \in J^{n + 1}M \]

then $m_ I \in JM$ for all $I$. Note that $f_1, \ldots , f_{r - 1}, f_ r^ n$ is an $M$-$H_1$-regular sequence by Lemma 15.29.4. Hence we see that the required result holds for the multi-index $I = (0, \ldots , 0, n)$. It turns out that we can reduce the general case to this case as follows.

Let $S = R[x_1, x_2, \ldots , x_ r, 1/x_ r]$. The ring map $R \to S$ is faithfully flat, hence $f_1, \ldots , f_ r$ is an $M$-$H_1$-regular sequence in $S$, see Lemma 15.29.5. By Lemma 15.28.4 we see that

\[ g_1 = f_1 - x_1/x_ r f_ r, \ldots g_{r - 1} = f_{r - 1} - x_{r - 1}/x_ r f_ r, g_ r = (1/x_ r)f_ r \]

is an $M$-$H_1$-regular sequence in $S$. Finally, note that our element $\xi $ can be rewritten

\[ \xi = \sum \nolimits _{|I| = n, I = (i_1, \ldots , i_ r)} m_ I (g_1 + x_ r g_ r)^{i_1} \ldots (g_{r - 1} + x_ r g_ r)^{i_{r - 1}} (x_ rg_ r)^{i_ r} \]

and the coefficient of $g_ r^ n$ in this expression is

\[ \sum m_ I x_1^{i_1} \ldots x_ r^{i_ r} \in J(M \otimes _ R S). \]

Since the monomials $x_1^{i_1} \ldots x_ r^{i_ r}$ form part of an $R$-basis of $S$ over $R$ we conclude that $m_ I \in J$ for all $I$ as desired. $\square$


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