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narrowed the question to the special case B= dim A_m.

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edited Apr 5 2012 at 4:08

Let $A$ be a noetherian domain, $\mathfrak{m}$ а maximal ideal, $s$ a non-zero element of $\mathfrak{m}$. \mathfrak{m}$, $d= \dim A_\mathfrak{m}$. Is the following claim true?

Claim: For any $B>0$ and $\epsilon>0$, there exists a positive integer $n$ s.t. for any ideal $I$ satisfying

1) $ I \subset\mathfrak{m^n}$

2) $\sqrt I = \mathfrak{m}$

3) $I$ can be generated by (at most) $B$ d$ elements,

the following holds: $$ \mbox{length}(A/(I+As)) /\mbox{length}(A/I) < \epsilon$$

Note: The following example shows that the claim can be false if one drops the requirement that that the number of generators of $I$ be bounded.

Example: $A:= k[x,s]$, and let $\mathfrak{m}$ denote the ideal $(x,s)$. Let $I_{n,m}$ be an ideal of $A$ given by $$ I_{n,m}= s\mathfrak{m}^{n-1} + \mathfrak{m}^m$$.

We can calculate that for any $n$, $$\lim_{m\to \infty} \mbox{length}(A/(I_{n,m}+As)) /\mbox{length}(A/I_{n,m}) = 1$$

rephrased the question so that it makes sense for any (not just local) noetherian ring

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edited Apr 3 2012 at 21:00

Nico Bellic
243●5

Let $A$ be a noetherian local domain, $\mathfrak{m}$ the а maximal ideal, $s$ a non-zero , non-unit element of $A$. \mathfrak{m}$. Is the following claim true?

Claim: For any $B>0$ and $\epsilon>0$, there exists a positive integer $n$ s.t. for any parameter ideal $I$ contained in $\mathfrak{m^n}$, $satisfying

1) $ I \mbox{length}(A/(I+As)) /\mbox{length}(A/I) < subset\mathfrak{m^n}$

2) $\sqrt I = \epsilon$$

($I$ is a parameter ideal if it mathfrak{m}$

3) $I$ can be generated by (at most) $\dim A$ B$ elementsand ,

the following holds: $\sqrt I = $ \mathfrak{m}$.)mbox{length}(A/(I+As)) /\mbox{length}(A/I) < \epsilon$$

Note: regarding the requirement that $I$ be a parameter ideal, the The following example shows that the claim is can be false if one puts no restriction on drops the requirement that that the number of generators of $I$ other than it be $\mathfrak{m}$-primary. bounded.

Example: $A:= k[x,s]$, and let $\mathfrak{m}$ denote the ideal $(x,y)$. (x,s)$. Let $I_{n,m}$ be an ideal of $A$ given by $$ I_{n,m}= s\mathfrak{m}^{n-1} + \mathfrak{m}^m$$.

We can calculate that for any $n$, $$\lim_{m\to \infty} \mbox{length}(A/(I_{n,m}+As)) /\mbox{length}(A/I_{n,m}) = 1$$

Add an example showing that some restriction on $I$ is necessary

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edited Mar 30 2012 at 22:05

Nico Bellic
243●5

Let $A$ be a noetherian local domain, $\mathfrak{m}$ the maximal ideal, $s$ a non-zero, non-unit element of $A$. Is the following claim true?

Claim: For any $\epsilon>0$, there exists a positive integer $n$ s.t. for any parameter ideal $I$ contained in $\mathfrak{m^n}$, $$ \mbox{length}(A/(I+As)) /\mbox{length}(A/I) < \epsilon$$

($I$ is a parameter ideal if it can be generated by $\dim A$ elements and $\sqrt I = \mathfrak{m}$.)

Note: regarding the requirement that $I$ be a parameter ideal, the following example shows that the claim is false if one puts no restriction on $I$ other than it be $\mathfrak{m}$-primary.

Example: $A:= k[x,s]$, and let $\mathfrak{m}$ denote the ideal $(x,y)$. Let $I_{n,m}$ be an ideal of $A$ given by $$ I_{n,m}= s\mathfrak{m}^{n-1} + \mathfrak{m}^m$$.