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The following proposition was given in Liu's Algebraic Geometry and Arithmetic Curves:

Let $A$ be a Noetherian ring, $I$ an ideal of $A$, and $M$ a finitely generated $A$-module endowed with stable $I$-filtration $(M_n)_n$. Then for any submodule $N$ of $M$, the $I$-filtration $(M_n\cap N)_n$ of $N$ is also stable.

But exercise 1.3.6. asks why this is not true if $N=A$ is an integral domain and $M$ the field of fractions of $N$. How can I prove that this gives a counterexample for the proposition?

user26857
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Jaakko Seppälä
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1 Answers1

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I think the exercise is not clearly formulated. In my opinion the author considers the filtration on $M=K$ given by $M_n=M$ for all $n\ge 0$. We thus have $IM_n=M_{n+1}$ for all $n\ge 0$, so it is $I$-stable, but $(M_n\cap A)_{n\ge 0}$ it's not: we have $M_n\cap A=A$ for all $n\ge 0$ and it's impossible to have $I(M_n\cap A)=M_{n+1}\cap A$, that is, $I=A$. (I suppose $I$ has been also chosen different from $A$.)

user26857
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