The Stacks project

Lemma 57.18.3. 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 57.18.2 it suffices to prove that $\textit{Coh}(\mathcal{O}_ X)$ is countable. This we omit. $\square$


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