Section 3 — printed counterexample and nonradical correction

Work over $\mathbb C$ in the single projective space $\mathbb P^4$, with $s=1$, $N_0=4$, and $R=\mathbb C[X_0,X_1,X_2,X_3,X_4]$, ordinary homogeneous degree, and $B=(X_0,\ldots,X_4)$. For every ideal $J\subseteq R$, let $H_J\in\mathbb Q[t_0,\ldots,t_{s-1}]$ be the chosen polynomial, if one exists, whose value at every coordinatewise sufficiently large $d\in\mathbb N^s$ is the $K$-dimension of the image of the multidegree-$d$ homogeneous polynomial space in $R/J$; set $H_J=0$ if no such polynomial exists. Put $\delta(J)=\deg_{\mathrm{tot}}H_J$, with $\delta(J)=0$ when $H_J=0$, $\Delta_J(d)=\delta(J)!\,(H_J)_{\delta(J)}(d)$, and $c_J(\alpha)=[t^\alpha]H_J\prod_i\alpha_i!$ when $\sum_i\alpha_i=\delta(J)$, with $c_J(\alpha)=0$ otherwise; $(H_J)_{\delta(J)}$ is its total-homogeneous component of that degree. Here there is just one Hilbert variable $t$ and one degree argument. Define $f_1=X_1^2X_3-X_2^2X_0$, $f_2=X_1X_4-X_2X_3$, $f_3=X_3^3-X_4^2X_0$, $I_0=(f_1,f_2,f_3)$, $I=I_0+(X_3,X_1-X_4)$, $Q_1=(X_1^2,X_2^2,X_3,X_1-X_4)$, $Q_2=(X_0,X_1^2,X_3,X_1-X_4)$, and $w=X_1X_3^2-X_2X_4X_0$. For an ideal $J$, let $\Sigma(J)$ be the sum of $\Delta_{(JR_{\mathfrak p})\cap R}(1)$ over minimal primes $\mathfrak p$ of $J$ with $B\nsubseteq\mathfrak p$ that are contained in some maximal ideal; this is the component Hilbert sum for the entire maximal spectrum. The assertion says, simultaneously: $I_0$ is stable under all homogeneous-component projections and $B\nsubseteq\sqrt{I_0}$; both $X_3$ and $X_1-X_4$ are homogeneous of degree one; $w\notin I_0$, $w^2\in I_0$, $w\in\sqrt{I_0}$, $\sqrt{I_0}\ne I_0$, and $I_0$ is not prime; $\delta(I_0)=2$, $H_{I_0}=2t^2+3t+1$, and $\Delta_{I_0}(1)=4$; $I=(X_1^2,X_2^2X_0,X_3,X_1-X_4)=Q_1\cap Q_2$; $Q_1,Q_2$ are primary, have unequal radicals, and $\operatorname{Min}(I)=\{\sqrt{Q_1},\sqrt{Q_2}\}$; both radicals are relevant, meaning neither contains $B$; $\delta(Q_1)=\delta(Q_2)=0$, $H_{Q_1}=4$, $H_{Q_2}=2$, $\Delta_{Q_1}(1)=4$, and $\Delta_{Q_2}(1)=2$; $\Sigma(I)=6$, $\Sigma(I_0)=4$, and $\Sigma(I_0)<\Sigma(I)$; and $I_0$ is not locally Cohen–Macaulay on the entire maximal spectrum. The last negation means it is false that for every maximal ideal $\mathfrak m$ the local quotient $R_{\mathfrak m}/I_0R_{\mathfrak m}$ is the zero ring or has a finite regular sequence of nonunits whose length equals its extended-valued Krull dimension. There are no hypotheses or variable choices in this theorem: all rings, polynomials, ideals, dimensions, and degree arguments in the conjunction are the specified concrete ones.