Bruns-Herzog define multiplicity When the ring and module are not necessarily graded as $e(M)=e(gr_m(M))$, see B-H 4.6. I have two questions:

Question1. Many concepts in commutative algebra have known behavior in exact sequences, such as $Ass, supp, reduction, ...$. I wonder if there is known fact about behavior of multiplicity in exact sequences? I mean:

Let $M$ be an $R$-module, $K$ its submodule, and $N$ a factor module of $M$. is there a relationship between $e(M)$, $e(K)$ and $e(N)$?

Question2. Is there inequalities between $e(M)$ and $\mu(M)$, (number of minimal generators of $M$) or between $e(M)$ and $\ell(M)$, (length of $M$)?

Thank you.

Bruns-Herzog define multiplicity When the ring and module are not necessarily graded as $e(M)=e(gr_m(M))$, see B-H 4.6. I have two questions:

Question1. Many concepts in commutative algebra have known behavior in exact sequences, such as $Ass, supp, reduction, ...$. I wonder if there is known fact about behavior of multiplicity in exact sequences? I mean:

Let $M$ be an $R$-module, $K$ its submodule, and $N$ a factor module of $M$. is there a relationship between $e(M)$, $e(K)$ and $e(N)$?

Question2. Is there inequalities between $e(M)$ and $\mu(M)$, (number of minimal generators of $M$) or between $e(M)$ and $\ell(M)$, (length of $M$)?

Thank you.

Bruns-Herzog define multiplicity When the ring and module are not necessarily graded as $e(M)=e(gr_m(M))$, see B-H 4.6.
Many concepts in commutative algebra have known behavior in exact sequences, such as Ass, supp, reduction and etc. I wonder if there is known fact about behavior of multiplicity in exact sequences? I mean:

Let $M$ be an $R$-module, $K$ its submodule, and $N$ a factor module of $M$. is there a relationship between $e(M)$, $e(K)$ and $e(N)$?

Thank you.

Bruns-Herzog define multiplicity When the ring and module are not necessarily graded as $e(M)=e(gr_m(M))$, see B-H 4.6. I have two questions:

Question1. Many concepts in commutative algebra have known behavior in exact sequences, such as $Ass, supp, reduction, ...$. I wonder if there is known fact about behavior of multiplicity in exact sequences? I mean:

Let $M$ be an $R$-module, $K$ its submodule, and $N$ a factor module of $M$. is there a relationship between $e(M)$, $e(K)$ and $e(N)$?

Question2. Is there inequalities between $e(M)$ and $\mu(M)$, (number of minimal generators of $M$) or between $e(M)$ and $\ell(M)$, (length of $M$)?

Thank you.