Remark 4

One says that the sequence $x = (x_1, ..., x_n)$ is $M$-coregular if (denoting by $(x_i)_M$ the homothety of ratio $x_i$ in $M$) the homothety of ratio $x_i$ in the module

is surjective for $i = 1, ..., n$. One then has $H_i(x, M) = 0$ for $i < n$: this is proved in the same way as prop. 5.

Take for example $A = k[D_1, ..., D_n]$, where $k$ is a $\mathbf{Q}$-algebra, and $M = k[T_1, ..., T_n]$, endowed with the $A$-module structure such that $D_i P = \partial P / \partial T_i$ for every $P \in M$ ($1 \leqslant i \leqslant n$). One verifies immediately that the sequence $(D_1, ..., D_n)$ is $M$-coregular; taking into account the example p. 155, one deduces that the de Rham complex of $k[T_1, ..., T_n]$ over $k$ is acyclic in degrees $> 0$.