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 15.88.6 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. $\square$

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