The Stacks project

Lemma 85.35.2. Let $X$ be an algebraic space over a scheme $S$. Let $a : U \to X$ be an fppf hypercovering. Suppose given $K_0 \in D_\mathit{QCoh}(U_0)$ and an isomorphism

\[ \alpha : L(f_{\delta _1^1})^*K_0 \longrightarrow L(f_{\delta _0^1})^*K_0 \]

satisfying the cocycle condition on $U_1$. Set $\tau ^ n_ i : [0] \to [n]$, $0 \mapsto i$ and set $K_ n = Lf_{\tau ^ n_ n}^*K_0$. Assume $\mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_{U_ n}}(K_ n, K_ n) = 0$ for $i < 0$. Then there exists an object $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ and an isomorphism $La_0^*K \to K$ compatible with $\alpha $.

Proof. The objects $K_ n$ form the members of a simplicial system of the derived category of modules by Lemma 85.14.3. Then we obtain an object $K' \in D_\mathit{QCoh}(\mathcal{O}_{U_{\acute{e}tale}})$ such that $(K_ n, K_\varphi )$ is the system deduced from $K'$, see Lemma 85.14.7. Finally, we apply Lemma 85.34.4 to see that $K' = La^*K$ for some $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ as desired. $\square$

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