The Stacks project

Lemma 51.4.3. Let $I \subset A$ be a finitely generated ideal of a ring $A$. Then

  1. $\text{cd}(A, I)$ is at most equal to the number of generators of $I$,

  2. $\text{cd}(A, I) \leq r$ if there exist $f_1, \ldots , f_ r \in A$ such that $V(f_1, \ldots , f_ r) = V(I)$,

  3. $\text{cd}(A, I) \leq c$ if $\mathop{\mathrm{Spec}}(A) \setminus V(I)$ can be covered by $c$ affine opens.

Proof. The explicit description for $R\Gamma _ Y(-)$ given in Dualizing Complexes, Lemma 47.9.1 shows that (1) is true. We can deduce (2) from (1) using the fact that $R\Gamma _ Z$ depends only on the closed subset $Z$ and not on the choice of the finitely generated ideal $I \subset A$ with $V(I) = Z$. This follows either from the construction of local cohomology in Dualizing Complexes, Section 47.9 combined with More on Algebra, Lemma 15.82.6 or it follows from Lemma 51.2.1. To see (3) we use Lemma 51.4.1 and the vanishing result of Cohomology of Schemes, Lemma 30.4.2. $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DX9. Beware of the difference between the letter 'O' and the digit '0'.