Lemma 51.4.12. 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 32.17.2. 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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)