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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).