Lemma 51.4.6. Let $I \subset A$ be a finitely generated ideal of a ring $A$. Then $\text{cd}(A, I) = \max \text{cd}(A_\mathfrak p, I_\mathfrak p)$.

**Proof.**
Let $Y = V(I)$ and $Y' = V(I_\mathfrak p) \subset \mathop{\mathrm{Spec}}(A_\mathfrak p)$. Recall that $R\Gamma _ Y(A) \otimes _ A A_\mathfrak p = R\Gamma _{Y'}(A_\mathfrak p)$ by Dualizing Complexes, Lemma 47.9.3. Thus we conclude by Algebra, Lemma 10.22.1.
$\square$

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