Corollary 18.6]).

The ideal $C$ is the canonical module of $R$ (see [@BH Chapter 3]). This fact can be proved in the same way as in the proof of [@BH Theorem 5.7.1], and gives an alternative proof of Lemma \[2.3.1\].

Since $C$ is a submodule of $R$ with $\dim C=\dim R$ and since $C$ is Cohen-Macaulay, $\Theta$ is an l.s.o.p. of $C$ and we have $$\dim_K \big(C/(\Theta C)\big)_i=h_{d-i}$$ for $i=0,1,\dots,d$ by Lemma \[2.3.1\]. This fact shows

\[E2.5\] If $h_j=0$ for some $1 \leq j \leq d-1$ then $C_{d-j}=\Theta C_{d-j-1}.$

\[2.3.2\] Suppose $h_j=0$ for some $1 \leq j \leq d-1$. If $\theta_d f  \in (C+\overline \Theta R)_{d-j}$ then, for any linear form $l \in R_1$, one has $l f \in C+\overline \Theta R$.

By Lemma \[E2.5\], $\theta_d f \in \overline \Theta R + \theta_d C$. Thus there is $g \in C$ such that $\theta_d (f-g) \in \overline \Theta R$. Since $\theta_d$ is a non-zero divisor of $R/(\overline \Theta R)$, we have $f-g \in \overline \Theta R$ and $f \in C + \overline \Theta R$. Hence $l f \in C+\overline \Theta R$.

The next theorem completes the proof of the necessity of Theorem \[1.2\].

\[necessity6\] Suppose $\partial h_i=0$ and $h_j=0$ for some 