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This is a problem I quote from P107 of Matsumura's book Commutative ring theory.

Let $(A,m)$ be a Noetherian local ring. $q$ is a $m$-primary ideal. Then we have $l(q^n/q^nm)l(A/q)\geq l(q^n/q^{n+1})$, where $l$ means the length of module.

The book mentioned that $l(q^n/q^nm)$ is the number of minimal basis since $q^n/q^nm$ is a $A/m$-vector space. So I tried to get a composition series of $q^n/q^{n+1}$. But I can't get further.

Isosceles
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Intoks Liobein
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1 Answers1

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Use Nakayama to show that $q^n/q^{n+1}$ is generated by $e=l(q^n/q^nm)$ elements. Thus we have a surjection $A^e\to q^n/q^{n+1}$. Since $qA^e$ maps to zero under this map, we get a surjection $(A/q)^e\to q^n/q^{n+1}$, proving the inequality you want.

Mohan
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