The Stacks project

Example 36.35.2. Let $k$ be a field. Let $X$ be a scheme of finite presentation over $k$ (in particular $X$ is quasi-compact). Then an object $E$ of $D(\mathcal{O}_ X)$ is $k$-perfect if and only if it is bounded and pseudo-coherent (by definition), i.e., if and only if it is in $D^ b_{\textit{Coh}}(X)$ (by Lemma 36.10.3). Thus being relatively perfect does not mean “perfect on the fibres”.

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View Example 36.35.2 as pdf