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Let $R=k[x_1,\ldots,x_n]$ be a standard graded polynomial over field $k$ and $I$ an unmixed homogeneous ideal of $R$. Let $x\in R$ be an $R/I$-regular element. Can we conclude that $x+I$ is an unmixed ideal?

Height unmixed ideal and a non-zero divisor

Background:

A proper ideal $I$ in a Noetherian ring $R$ is said to be height unmixed if the heights of its prime divisors are all equal. i.e., $\operatorname{height} I=\operatorname{height}\mathfrak{p}$ for all $\mathfrak{p}\in \operatorname{Ass} I$.

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No. For instance, take $R = k[x^4,x^3y, xy^3, y^4]$. This is a $2$-dimensional domain which is not Cohen-Macaulay at the origin. In fact the depth of $R$ at the origin is $1$. Since $R$ is a domain, the ideal $0$ is unmixed. Let $f$ be a homogeneous non zero element in $R$. Then $R/(f)$ is $1$-dimensional with depth $0$. Therefore, the ideal $(0)$ in $R/(f)$ is not unmixed.

The condition is equivalent to $R/I$ satisfies Serre's condition $S_2$.

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  • $\begingroup$ @user121097: Let $S = k[a,b,c,d]$ and $\phi:S \to R$. Let $I = \ker \phi$. $\endgroup$
    – Youngsu
    Commented Mar 26, 2014 at 21:09
  • $\begingroup$ Why the condition is equivalent to $R/I$ satisfies Serre's condition $S_2$. $\endgroup$
    – Stella
    Commented Apr 8, 2014 at 14:03
  • $\begingroup$ @Stella: Sure, let me try. Claim $I$ unmixed and $R/I$ satisfies $S_2$. Then for any $x$ which is a non zerodivisor on $R/I$, then $I,x$ is unmixed: $\endgroup$
    – Youngsu
    Commented Apr 9, 2014 at 15:47
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    $\begingroup$ Since $R/I$ satisfies $S_2$, $R/(I,x)$ satisfies $S_1$. Therefore, it suffices to show that for $P \in Min((I,x))$, height $P$ = height $(I,x) = $ height $I + 1$. Consider the shortest irredundant primary decomposition of $I$, $Q_1 \cap Q_s$. Since $I$ is unmixed, height $Q_i =$ height $I$. We see that $I \subseteq P$ and $P$ prime. Therefore, $Q_i \subseteq P$ for some $i$. Let $p = \surd{Q_i}$. Since height $P/ (p,x) = 0$, height $P/p = 1$. Since $R_P$ is regular local domain, this implies height $P = $ height $p + 1$. Recall that height $p = $ height $I$. $\endgroup$
    – Youngsu
    Commented Apr 9, 2014 at 15:48
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    $\begingroup$ @Stella: A Cohen-Macaulay (CM for short) ring satisfies $S_n$ for all $n$. If $R$ is a CM and $x$ a non zerodivisor on $R$, then $R/(x)$ is CM. I don't really know "what circumstances" should mean other than normal or CM. $\endgroup$
    – Youngsu
    Commented Apr 11, 2014 at 17:06

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