Regular sequence on a module

Definition

Let $R$ be a commutative unital ring, $M$ a $R$-module, and $x_1,x_2,\dots ,x_n$ be a sequence of elements in $R$. We say that the $x_i$s form a regular sequence on $M$ if the following two conditions hold:

• $(x_1,x_2,\dots ,x_n)M\ne M$
• For $1\le i\le n$, $x_i$ is a nonzerodivisor on $M/(x_1,x_2,\dots ,x_{i-1})$

Facts

• If $R$ is a Noetherian local ring and $x_1,x_2,\dots ,x_n$ form a regular sequence in its unique maximal ideal, then any permutation of the $x_i$s also forms a regular sequence in the maximal ideal. In general, a permutation of a regular sequence need not be regular. For full proof, refer: Permutation of regular sequence is not necessarily regular
• If $R$ is a graded ring, and $x_1,x_2,\dots ,x_n$ form a regular sequence and all the $x_i$s are homogeneous elements, then any permutation of the $x_i$s is also a regular sequence.
• If $x_1,x_2,\dots ,x_d$ are a regular sequence on a module $M$ over a Noetherian local ring, then the difference of degrees of the Hilbert-Samuel polynomial for $M$ and for $M/(x_1,x_2,\dots ,x_d)$ is at least $d$.