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I am trying to understand if a certain condition on quotient rings is sufficient for a sequence to be regular. Here is the setting:

Let $\mathbb{C}[u_1,...,u_n]$ be the ring of regular functions on $\mathbb{C}^n$, and $I$ be the ideal of a linear subspace $\mathbb{C}^k \subset \mathbb{C}^n$ (thus $I$ is generated by $n-k$ polynomials of degree $1$). Let moreover $J$ be an ideal, and suppose that the following holds:

  1. $\mathbb{C}[u] / J$ has Krull dimension at least $n-k$;
  2. $\mathbb{C}[u] / (I+J)$ has Krull dimension equal to $0$.

Is it sufficient to show that the $n-k$ generators of $I$ form a regular sequence in $\mathbb{C}[u] / J$ ? If yes, what is the argument ? Is the fact that these generators are homogeneous (since $\mathbb{C}^k \subset \mathbb{C}^n$ is linear) important ?

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  • $\begingroup$ No. Take $n=2$, $k=1$, $J=(u_1^2,u_1u_2)$, $I=(u_2)$. Then 1. and 2. are satisfied, but $u_2$ is a zero divisor in $\mathbb{C}[u]/J$. $\endgroup$
    – abx
    Commented Oct 3, 2018 at 14:48
  • $\begingroup$ Thank you for your comment. What would be a sufficient additional condition then ? $\endgroup$
    – BrianT
    Commented Oct 3, 2018 at 16:07
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    $\begingroup$ I think that $\mathbb{C}[u]/J$ Cohen-Macaulay is sufficient. $\endgroup$
    – abx
    Commented Oct 3, 2018 at 16:39
  • $\begingroup$ Suppose that it is indeed Cohen-Macaulay. Don’t we need the dimension to be equal (and not at least) $n-k$ ? $\endgroup$
    – BrianT
    Commented Oct 3, 2018 at 21:26
  • $\begingroup$ Yes, sorry I didn't make that explicit. $\endgroup$
    – abx
    Commented Oct 4, 2018 at 3:47

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