The Stacks project

Lemma 51.16.1. Let $A$ be a Noetherian ring of dimension $d$. Let $I \subset I' \subset A$ be ideals. If $I'$ is contained in the Jacobson radical of $A$ and $\text{cd}(A, I') < d$, then $\text{cd}(A, I) < d$.

Proof. By Lemma 51.4.7 we know $\text{cd}(A, I) \leq d$. We will use Lemma 51.2.6 to show

\[ H^ d_{V(I')}(A) \to H^ d_{V(I)}(A) \]

is surjective which will finish the proof. Pick $\mathfrak p \in V(I) \setminus V(I')$. By our assumption on $I'$ we see that $\mathfrak p$ is not a maximal ideal of $A$. Hence $\dim (A_\mathfrak p) < d$. Then $H^ d_{\mathfrak pA_\mathfrak p}(A_\mathfrak p) = 0$ by Lemma 51.4.7. $\square$

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