Regular sequence on a module

Definition

Let $R$ be a commutative unital ring, $M$ a $R$-module, and $x_{1},x_{2},\ldots ,x_{n}$ 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:

• $(x_{1},x_{2},\ldots ,x_{n})M\neq M$
• For $1\leq i\leq n$, $x_{i}$ is a nonzerodivisor on $M/(x_{1},x_{2},\ldots ,x_{i-1})M$

When no module is specified, we assume the module to be $R$ itself. Further information: regular sequence in a ring

Facts

• If $R$ is a Noetherian local ring and $x_{1},x_{2},\ldots ,x_{n}$ 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 $x_{1},x_{2},\ldots ,x_{n}$ 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 $x_{1},x_{2},\ldots ,x_{d}$ 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 $M/(x_{1},x_{2},\ldots ,x_{d})$ is at least $d$.