Let $\nu : G \rightarrow H$ be a surjective group homomomorphism with kernel $N$, $H$ abelian, and $G$ finitely generated.

The rational abelianization of $N$, $H_1(N)$ is a $\mathbb{C}[H]$-module, and we say that it is nilpotent if some power of the augmentation ideal $I_H \subset \mathbb{C}[H] $ is in the annihilator of the module.

In some paper I read that, as an easy consequence of the Hilbert Nullstellensatz, $H_1(N)$ is nilpotent if and only if $$\operatorname{Specm}(\mathbb{C} H) \cap \operatorname{supp}_{\mathbb{C} H}\left(H_{1} N\right) \subseteq\{1\}, $$ where the support means the subset of prime ideals containing the annihilator, and "$1$" is the ideal $I_H$.

Can someone explain what version of the Nullstellensatz is applied here? I understand only the direct implication….

The result in question is Dimca, Hain, and Papadima - The abelianization of the Johnson kernel Lemma 3.3.