Lemma 1

a) If $p$ is a prime ideal of $H$, $p^{gr}$ is a prime ideal.

b) Every maximal element of the set of graded ideals of $H$ distinct from $H$ is a maximal ideal of $H$ which contains $H_{\geq 1}$.

c) Every minimal prime ideal of $H$ is graded.

a) This follows from III, § 1, no. 4, prop. 4.

b) Let $m$ be a graded ideal of $H$ distinct from $H$. Then one has

If $m$ is maximal, then one has $m = m_0 + H_{\geq 1}$, where $m_0$ is a maximal ideal of $H_0$, whence $b$.

c) Let $p$ be a minimal prime ideal of $H$. Since $p^{gr}$ is prime by $a$ and contained in $p$, one has $p = p^{gr}$, whence $c$.