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I am trying to calculate the length of two modules: $k[x,y]_{(x,y)}/(y-x^2,y)$ and $k[x,y]_{(x,y)}/(y-x^2,x)$. The claim is that the former has length 2 and the latter has length 1. But I am not sure why this is true: the chain of ideals of $k[x,y]_{(x,y)}$ containing $(y-x^2,y)$ can be $(y-x^2,y) \subset (x,y) \subset k[x,y]_{(x,y)}$, so it should have length 3. Similarly I have $(y-x^2,x) \subset (x,y) \subset k[x,y]_{(x,y)}$ so the length of $k[x,y]_{(x,y)}/(y-x^2,x)$ should also be 3. Can somebody please point out what's wrong with my method?

MathChopper
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1 Answers1

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The first filtration is true, but the length is 2 because $(y-x^2,y)=0$ in the ring.

The second filtration is not true, since $(y-x^2,x)=(x,y)$, hence the length is 1 by the same reason.

Jian
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  • Thanks, then shouldn't the answer of this question https://math.stackexchange.com/questions/2459063/calculate-length-of-module be 2 as well? – MathChopper Jan 13 '20 at 00:30
  • @MathChopper Yes, the length should be 2 in that answer. – Jian Jan 13 '20 at 02:47