## 84.35 Fppf descent of complexes

In this section we pull some of the previously shown results together for fppf coverings of algebraic spaces and derived categories of quasi-coherent modules.

Lemma 84.35.1. Let $X$ be an algebraic space over a scheme $S$. Let $K, E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $a : U \to X$ be an fppf hypercovering. Assume that for all $n \geq 0$ we have

\[ \mathop{\mathrm{Ext}}\nolimits _{\mathcal{O}_{U_ n}}^ i(La_ n^*K, La_ n^*E) = 0 \text{ for } i < 0 \]

Then we have

$\mathop{\mathrm{Ext}}\nolimits _{\mathcal{O}_ X}^ i(K, E) = 0$ for $i < 0$, and

there is an exact sequence

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(K, E) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{U_0}}(La_0^*K, La_0^*E) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{U_1}}(La_1^*K, La_1^*E) \]

**Proof.**
Write $K_ n = La_ n^*K$ and $E_ n = La_ n^*E$. Then these are the simplicial systems of the derived category of modules (Definition 84.14.1) associated to $La^*K$ and $La^*E$ (Lemma 84.14.2) where $a : U_{\acute{e}tale}\to X_{\acute{e}tale}$ is as in Section 84.32. Let us prove (2) first. By Lemma 84.34.4 we have

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(K, E) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(La^*K, La^*E) \]

Thus the sequence looks like this:

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ U}(La^*K, La^*E) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{U_0}}(K_0, E_0) \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{U_1}}(K_1, E_1) \]

The first arrow is injective by Lemma 84.14.5. The image of this arrow is the kernel of the second by Lemma 84.14.6. This finishes the proof of (2). Part (1) follows by applying part (2) with $K[i]$ and $E$ for $i > 0$.
$\square$

Lemma 84.35.2. 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 84.14.3. 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 84.14.7. Finally, we apply Lemma 84.34.4 to see that $K' = La^*K$ for some $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ as desired.
$\square$

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