Let $(R,m)$ be a Noetherian local ring, $M$ and $N$ finite $R$-modules, $p$ a prime ideal, and $I$ an ideal such that $IM\neq M$.

Definition: The common length of the maximal $M$-sequences in $I$ is called the grade of $I$ on $M$; denoted by $\operatorname{grade}(I,M)$.

$\operatorname{grade}(m,M)$ is denoted by $\operatorname{depth}M$. So by $\operatorname{depth}M_p$, we mean $\operatorname{grade}(pR_p,M_p)$.

Assuming $\operatorname{depth}M\ge \operatorname{depth}N$, what can one say about $\operatorname{depth}M_p$ and $\operatorname{depth}N_p$? Is there any inequality between them?

 

What if we impose additional assumptions on $M$ and $N$? For example, if $M=R/I$ and $N=R/J$?

Thank you.

Let $(R,m)$ be a Noetherian local ring, $M$ and $N$ finite $R$-modules, $p$ a prime ideal, and $I$ an ideal such that $IM\neq M$.

Definition: The common length of the maximal $M$-sequences in $I$ is called the grade of $I$ on $M$; denoted by $\operatorname{grade}(I,M)$.

$\operatorname{grade}(m,M)$ is denoted by $\operatorname{depth}M$. So by $\operatorname{depth}M_p$, we mean $\operatorname{grade}(pR_p,M_p)$.

Assuming $\operatorname{depth}M\ge \operatorname{depth}N$, what can one say about $\operatorname{depth}M_p$ and $\operatorname{depth}N_p$? Is there any inequality between them?

What if we impose additional assumptions on $M$ and $N$? For example, if $M=R/I$ and $N=R/J$?

Thank you.

Let $(R,m)$ be a Noetherian local ring􀀀, $M$ and $N$ finite $R$-modules, $p$ a prime ideal,􀀀 and $I$ an ideal such that $IM\neq M$.

Definition: The common length of the maximal $M$-sequences in $I$ is called the grade of $I$ on $M$;􀀀 denoted by $grade(I,M)$.

􀀀$grade(m,M)$ is denoted by $depth\ M$. So by $depth\ M_p$, we mean 􀀀$grade(pR_p,M_p)$.

Assuming $depth\ M\ge depth\ N$, what can one say about $depth\ M_p$ and $depth\ N_p$? Is there any inequality between them?

What if we impose additional assumptions on $M$ and $N$? For example, if $M=R/I$ and $N=R/J$?

Thank you.

Let $(R,m)$ be a Noetherian local ring, $M$ and $N$ finite $R$-modules, $p$ a prime ideal, and $I$ an ideal such that $IM\neq M$.

Definition: The common length of the maximal $M$-sequences in $I$ is called the grade of $I$ on $M$; denoted by $\operatorname{grade}(I,M)$.

$\operatorname{grade}(m,M)$ is denoted by $\operatorname{depth}M$. So by $\operatorname{depth}M_p$, we mean $\operatorname{grade}(pR_p,M_p)$.

Assuming $\operatorname{depth}M\ge \operatorname{depth}N$, what can one say about $\operatorname{depth}M_p$ and $\operatorname{depth}N_p$? Is there any inequality between them?

What if we impose additional assumptions on $M$ and $N$? For example, if $M=R/I$ and $N=R/J$?

Thank you.

Definition. Let $(R,m)$ be a Noetherian local ring􀀀, $M$ and $N$ finite R-modules, $p$ a prime ideal,􀀀 and $I$ an ideal such that $IM\neq M$. Then the common length of the maximal $M$-sequences in $I$ is called the grade of $I$ on $M$􀀀 denoted by $grade(I,M)$.
􀀀$grade(m,M)$ is denoted by $depth\ M$. So by $depth\ M_p$, we mean 􀀀$grade(pR_p,M_p)$.
Assuming $depth\ M\ge depth\ N$, what can one say about $depth\ M_p$ and $depth\ N_p$? Is there any inequality between them?
What if we impose additional assumptions on $M$ and $N$? for example if $M=R/I$ and $N=R/J$?

Thank you.

Let $(R,m)$ be a Noetherian local ring􀀀, $M$ and $N$ finite $R$-modules, $p$ a prime ideal,􀀀 and $I$ an ideal such that $IM\neq M$.

Definition: The common length of the maximal $M$-sequences in $I$ is called the grade of $I$ on $M$;􀀀 denoted by $grade(I,M)$.

􀀀$grade(m,M)$ is denoted by $depth\ M$. So by $depth\ M_p$, we mean 􀀀$grade(pR_p,M_p)$.

Assuming $depth\ M\ge depth\ N$, what can one say about $depth\ M_p$ and $depth\ N_p$? Is there any inequality between them?

What if we impose additional assumptions on $M$ and $N$? For example, if $M=R/I$ and $N=R/J$?

Thank you.