Let $A$ be a Noetherian ring and $R$ is a standard graded ring over $A.$ Let $M$ be a finitely generated graded $R$-module and $I$ be a graded ideal of $R.$ Then $x_1,\ldots,x_r\in I$ is called $M$-filter regular sequence with respect to $I$ if $x_i\notin P$ for all $P\in Ass(M/(x_1,\ldots,x_{i-1})M)\setminus V(I)$ for all $i=1,\ldots,r.$

What can we say about maximal length of $M$-filter regular sequence with respect to $I.$

Let $A$ be a Noetherian ring and $R$ is a standard graded ring over $A.$ Let $M$ be a finitely generated graded $R$-module and $I$ be a graded ideal of $R.$ Then $x_1,\ldots,x_r\in I$ is called $M$-filter regular sequence with respect to $I$ if $x_i\notin P$ for all $P\in Ass(M/(x_1,\ldots,x_{i-1})M)\setminus V(I)$ for all $i=1,\ldots,r.$

Q: What can we say about maximal length of $M$-filter regular sequence with respect to $I.$

Let $A$ be a Noetherian ring and $R$ is a standard graded ring over $A.$ Let $M$ be a finitely generated graded $R$-module and $I$ be a graded ideal of $R.$ Then $x_1,\ldots,x_r\in I$ is called $M$-filter regular sequence with respect to $I$ if $x_i\notin P$ for all $P\in Ass(M/(x_1,\ldots,x_{i-1})M)\setminus V(I)$ for all $i=1,\ldots,s.$

What can we say about maximal length of $M$-filter regular sequence with respect to $I.$

Let $A$ be a Noetherian ring and $R$ is a standard graded ring over $A.$ Let $M$ be a finitely generated graded $R$-module and $I$ be a graded ideal of $R.$ Then $x_1,\ldots,x_r\in I$ is called $M$-filter regular sequence with respect to $I$ if $x_i\notin P$ for all $P\in Ass(M/(x_1,\ldots,x_{i-1})M)\setminus V(I)$ for all $i=1,\ldots,r.$

What can we say about maximal length of $M$-filter regular sequence with respect to $I.$