The Stacks project

Lemma 51.4.11. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $b : X' \to X = \mathop{\mathrm{Spec}}(A)$ be the blowing up of $I$. If the fibres of $b$ have dimension $\leq d - 1$, then $\text{cd}(A, I) \leq d$.

Proof. Set $U = X \setminus V(I)$. Denote $j : U \to X'$ the canonical open immersion, see Divisors, Section 31.32. Since the exceptional divisor is an effective Cartier divisor (Divisors, Lemma 31.32.4) we see that $j$ is affine, see Divisors, Lemma 31.13.3. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ U$-module. Then $R^ pj_*\mathcal{F} = 0$ for $p > 0$, see Cohomology of Schemes, Lemma 30.2.3. On the other hand, we have $R^ qb_*(j_*\mathcal{F}) = 0$ for $q \geq d$ by Limits, Lemma 32.19.2. Thus by the Leray spectral sequence (Cohomology, Lemma 20.13.8) we conclude that $R^ n(b \circ j)_*\mathcal{F} = 0$ for $n \geq d$. Thus $H^ n(U, \mathcal{F}) = 0$ for $n \geq d$ (by Cohomology, Lemma 20.13.6). This means that $\text{cd}(A, I) \leq d$ by definition. $\square$

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