Let $k$ be a field and $R=k[X_1,...,X_m]$ be a polynomial ring. Let $f_1,...,f_n\in R$ and $I$ be the ideal generated by these elements. By Krull's principal ideal theorem, $\text{ht}(I)\leq n$. Suppose now $\text{ht}(I)=n$.

Let $J_l$ (for $1\leq l\leq n$) be the ideal generated by $f_1,...,f_{l-1},f_{l+1},...,f_n$, i.e. we leave out one generator. We know that $\text{ht}(J_l)\leq n-1$. However, equality need not hold for all $l$ even though $\text{ht}(I)=n$ (for example, consider $I=(Y(1-X),Z(1-X),X)$ in $\mathbb{C}[X,Y,Z]$).

My question is: Is it still true that under these conditions, we have $\text{ht}(J_l)=n-1$ for at least one $l$?

This may well be a very basic question. I would be grateful for any comments, counterexamples etc.


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