Lemma 75.18.2. Let $X$ be a quasi-compact and quasi-separated scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is quasi-compact. Let $K \in D(\mathcal{O}_ X)$ supported on $T$. The following are equivalent
$K$ is pseudo-coherent, and
$K = \text{hocolim} K_ n$ where $K_ n$ is perfect, supported on $T$, and $\tau _{\geq -n}K_ n \to \tau _{\geq -n}K$ is an isomorphism for all $n$.
Proof. The proof of this lemma is exactly the same as the proof of Lemma 75.18.1 except that in the choice of the approximations we use the triples $(T, K, m)$. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)