The Stacks project

Lemma 38.25.11. Let $A$ be a local ring. Let $I, J \subset A$ be ideals. If $J$ is finitely generated and $I \subset J^ n$ for all $n \geq 1$, then $V(I)$ contains the closed points of $\mathop{\mathrm{Spec}}(A) \setminus V(J)$.

Proof. Let $\mathfrak p \subset A$ be a closed point of $\mathop{\mathrm{Spec}}(A) \setminus V(J)$. We want to show that $I \subset \mathfrak p$. If not, then some $f \in I$ maps to a nonzero element of $A/\mathfrak p$. Note that $V(J) \cap \mathop{\mathrm{Spec}}(A/\mathfrak p)$ is the set of non-generic points. Hence by Lemma 38.25.10 applied to the collection of ideals $J^ nA/\mathfrak p$ we conclude that the image of $f$ is zero in $A/\mathfrak p$. $\square$

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