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

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

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

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

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