Remark 36.18.4 (Variant with support). 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. The analogue of Theorem 36.18.3 holds for $D_{\mathit{QCoh}, T}(\mathcal{O}_ X)$. This follows from the exact same argument as in the proof of the theorem, using Lemmas 36.15.4 and 36.17.3 and a variant of Lemma 36.18.1 with supports. If we ever need this, we will precisely state the result here and give a detailed proof.

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