The Stacks project

Proposition 36.11.2. Let $X$ be a Noetherian scheme. Then the functors

\[ D^-(\textit{Coh}(\mathcal{O}_ X)) \longrightarrow D^-_{\textit{Coh}}(\mathcal{O}_ X) \quad \text{and}\quad D^ b(\textit{Coh}(\mathcal{O}_ X)) \longrightarrow D^ b_{\textit{Coh}}(\mathcal{O}_ X) \]

are equivalences.

Proof. Consider the commutative diagram

\[ \xymatrix{ D^-(\textit{Coh}(\mathcal{O}_ X)) \ar[r] \ar[d] & D^-_{\textit{Coh}}(\mathcal{O}_ X) \ar[d] \\ D^-(\mathit{QCoh}(\mathcal{O}_ X)) \ar[r] & D^-_\mathit{QCoh}(\mathcal{O}_ X) } \]

By Lemma 36.11.1 the left vertical arrow is fully faithful. By Proposition 36.8.3 the bottom arrow is an equivalence. By construction the right vertical arrow is fully faithful. We conclude that the top horizontal arrow is fully faithful. If $K$ is an object of $D^-_{\textit{Coh}}(\mathcal{O}_ X)$ then the object $K'$ of $D^-(\mathit{QCoh}(\mathcal{O}_ X))$ which corresponds to it by Proposition 36.8.3 will have coherent cohomology sheaves. Hence $K'$ is in the essential image of the left vertical arrow by Lemma 36.11.1 and we find that the top horizontal arrow is essentially surjective. This finishes the proof for the bounded above case. The bounded case follows immediately from the bounded above case. $\square$

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