The Stacks project

Definition 59.95.1. Let $X$ be a quasi-compact and quasi-separated scheme. The cohomological dimension of $X$ is the smallest element

\[ \text{cd}(X) \in \{ 0, 1, 2, \ldots \} \cup \{ \infty \} \]

such that for any abelian torsion sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ we have $H^ i_{\acute{e}tale}(X, \mathcal{F}) = 0$ for $i > \text{cd}(X)$. If $X = \mathop{\mathrm{Spec}}(A)$ we sometimes call this the cohomological dimension of $A$.

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