The Stacks project

Lemma 57.89.2. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a directed limit of a system of quasi-compact and quasi-separated schemes with affine transition morphisms. Then $\text{cd}(X) \leq \max \text{cd}(X_ i)$.

Proof. Denote $f_ i : X \to X_ i$ the projections. Let $\mathcal{F}$ be an abelian torsion sheaf on $X_{\acute{e}tale}$. Then we have $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits f_ i^{-1}f_{i, *}\mathcal{F}$ by Lemma 57.51.9. Thus $H^ q_{\acute{e}tale}(X, \mathcal{F}) = \mathop{\mathrm{colim}}\nolimits H^ q_{\acute{e}tale}(X_ i, f_{i, *}\mathcal{F})$ by Theorem 57.51.3. The lemma follows. $\square$

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