Let $A$ be a commutative Noetherian ring, $R$ be a standard graded algebra over $A$, $M$ be finitely generated graded $R$-module. Let $R\_{+}$ be the irrelevant ideal. The Castelnuovo-Mumford regularity of $M$ or regularity for short is defined to be : $\text{reg}(M):=\text{max}\lbrace a(H\_{R\_+}^{i}(M))+i|i\ge 0\rbrace$.
Let $x\_1,...,x\_s$ be linear form in $R$. This set of elements is called a $M$-filter regular sequence if $x\_{i}\notin \mathfrak{p}$ for any associated prime $\mathfrak{p}\nsupseteq R\_{+}$ of $(x\_1,...,x\_{i-1})M$ for $i=1,...,s$


Then, people claim that : $\text{reg}(M)=\text{max}\lbrace a((x\_1,...,x\_i)M:R\_{+}/(x\_1,...,x\_{i})M)|i=1,...,s\rbrace$


Could you explain for me why is it ? What is the motivation of filter regular sequence ?



Filter-regular sequence and regularity
I've posted an answer on [M.SE](https://math.stackexchange.com/questions/219265/equivalent-definiton-of-castelnuovo-mumford-regularity/219302#219302).


