Filter-regular sequence and regularity - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T20:58:08Z http://mathoverflow.net/feeds/question/110381 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110381/filter-regular-sequence-and-regularity Filter-regular sequence and regularity Knot 2012-10-23T02:05:56Z 2012-10-23T13:32:21Z <p>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 $i=1,...,s$</p> <p>Then, people claim that : $\text{reg}(M)=\text{max}\lbrace a((x_1,...,x_i)M:R_{+}/(x_1,...,x_{i})M)|i=1,...,s\rbrace$</p> <p>Could you explain for me why is it ? What is the motivation of filter regular sequence ?</p> http://mathoverflow.net/questions/110381/filter-regular-sequence-and-regularity/110411#110411 Answer by YACP for Filter-regular sequence and regularity YACP 2012-10-23T10:35:07Z 2012-10-23T10:35:07Z <p>I've posted an answer here: <a href="http://math.stackexchange.com/questions/219265/equivalent-definiton-of-castelnuovo-mumford-regularity/219302#219302" rel="nofollow">http://math.stackexchange.com/questions/219265/equivalent-definiton-of-castelnuovo-mumford-regularity/219302#219302</a></p> <p>(Sorry, but I still can't post comments.)</p>