The Stacks project

Example 36.21.5. Let $X$ be a quasi-compact and quasi-separated scheme. Let $(\mathcal{F}_ n)$ be an inverse system of quasi-coherent sheaves. Since $DQ_ X$ is a right adjoint it commutes with products and therefore with derived limits. Hence we see that

\[ DQ_ X(R\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n) = (R\mathop{\mathrm{lim}}\nolimits \text{ in }D_\mathit{QCoh}(\mathcal{O}_ X))(\mathcal{F}_ n) \]

where the first $R\mathop{\mathrm{lim}}\nolimits $ is taken in $D(\mathcal{O}_ X)$. In fact, let's write $K = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$ for this. For any affine open $U \subset X$ we have

\[ H^ i(U, K) = H^ i(R\Gamma (U, R\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n)) = H^ i(R\mathop{\mathrm{lim}}\nolimits R\Gamma (U, \mathcal{F}_ n)) = H^ i(R\mathop{\mathrm{lim}}\nolimits \Gamma (U, \mathcal{F}_ n)) \]

since cohomology commutes with derived limits and since the quasi-coherent sheaves $\mathcal{F}_ n$ have no higher cohomology on affines. By the computation of $R\mathop{\mathrm{lim}}\nolimits $ in the category of abelian groups, we see that $H^ i(U, K) = 0$ unless $i \in [0, 1]$. Then finally we conclude that the $R\mathop{\mathrm{lim}}\nolimits $ in $D_\mathit{QCoh}(\mathcal{O}_ X)$, which is $DQ_ X(K)$ by the above, is in $D^ b_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 36.21.4.

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