Lemma 51.4.10. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be a proper ideal. Let $\mathfrak p \subset A$ be a prime ideal such that $V(\mathfrak p) \cap V(I) = \{ \mathfrak m\}$. Then $\dim (A/\mathfrak p) \leq \text{cd}(A, I)$.

Proof. By Lemma 51.4.5 we have $\text{cd}(A, I) \geq \text{cd}(A/\mathfrak p, I(A/\mathfrak p))$. Since $V(I) \cap V(\mathfrak p) = \{ \mathfrak m\}$ we have $\text{cd}(A/\mathfrak p, I(A/\mathfrak p)) = \text{cd}(A/\mathfrak p, \mathfrak m/\mathfrak p)$. By Lemma 51.4.9 this is equal to $\dim (A/\mathfrak p)$. $\square$

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