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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 ?

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I've posted an answer on M.SE.

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