Lemma 51.4.5. Let $A \to B$ be a ring map. Let $I \subset A$ be a finitely generated ideal. Then $\text{cd}(B, IB) \leq \text{cd}(A, I)$. If $A \to B$ is faithfully flat, then equality holds.

**Proof.**
Use the definition and Dualizing Complexes, Lemma 47.9.3.
$\square$

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