85.35 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 85.35.1. 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 85.14.1) associated to $La^*K$ and $La^*E$ (Lemma 85.14.2) where $a : U_{\acute{e}tale}\to X_{\acute{e}tale}$ is as in Section 85.32. Let us prove (2) first. By Lemma 85.34.4 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 85.14.5. The image of this arrow is the kernel of the second by Lemma 85.14.6. This finishes the proof of (2). Part (1) follows by applying part (2) with $K[i]$ and $E$ for $i > 0$.
$\square$
Lemma 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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