Lemma 57.17.3. Let X be a scheme of finite type over a countable Noetherian ring. Then the categories D_{perf}(\mathcal{O}_ X) and D^ b_{\textit{Coh}}(\mathcal{O}_ X) are countable.
Proof. Observe that X is Noetherian by Morphisms, Lemma 29.15.6. Hence D_{perf}(\mathcal{O}_ X) is a full subcategory of D^ b_{\textit{Coh}}(\mathcal{O}_ X) by Derived Categories of Schemes, Lemma 36.11.6. Thus it suffices to prove the result for D^ b_{\textit{Coh}}(\mathcal{O}_ X). Recall that D^ b_{\textit{Coh}}(\mathcal{O}_ X) = D^ b(\textit{Coh}(\mathcal{O}_ X)) by Derived Categories of Schemes, Proposition 36.11.2. Hence by Lemma 57.17.2 it suffices to prove that \textit{Coh}(\mathcal{O}_ X) is countable. This we omit. \square
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