Lemma 77.12.2 (0CX5)—The Stacks project

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Part 4: Algebraic Spaces
Chapter 77: Flatness on Algebraic Spaces
Section 77.12: Grothendieck's Existence Theorem
Lemma 77.12.2 (cite)

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Lemma 77.12.2. In Situation 77.12.1 consider

$K = R\mathop{\mathrm{lim}}\nolimits _{D_\mathit{QCoh}(\mathcal{O}_ X)}(\mathcal{F}_ n) = DQ_ X(R\mathop{\mathrm{lim}}\nolimits _{D(\mathcal{O}_ X)}\mathcal{F}_ n)$

Then $K$ is in $D^ b_{\mathit{QCoh}}(\mathcal{O}_ X)$ and in fact $K$ has nonzero cohomology sheaves only in degrees $\geq 0$ .

Proof. Special case of Derived Categories of Spaces, Example 75.19.5. $\square$

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