Lemma 99.16.6. In Situation 99.16.3 the functor $p : \mathcal{C}\! \mathit{omplexes}_{X/B} \longrightarrow (\mathit{Sch}/S)_{fppf}$ is a stack in groupoids.
Proof. To prove that $\mathcal{C}\! \mathit{omplexes}_{X/B}$ is a stack in groupoids, we have to show that the presheaves $\mathit{Isom}$ are sheaves and that descent data are effective. The statement on $\mathit{Isom}$ follows from Lemma 99.16.5, see Algebraic Stacks, Lemma 94.10.11. Let us prove the statement on descent data.
Suppose that $\{ a_ i : T_ i \to T\} $ is an fppf covering of schemes over $S$. Let $(\xi _ i, \varphi _{ij})$ be a descent datum for $\{ T_ i \to T\} $ with values in $\mathcal{C}\! \mathit{omplexes}_{X/B}$. For each $i$ we can write $\xi _ i = (T_ i, g_ i, E_ i)$. Denote $\text{pr}_0 : T_ i \times _ T T_ j \to T_ i$ and $\text{pr}_1 : T_ i \times _ T T_ j \to T_ j$ the projections. The condition that $\xi _ i|_{T_ i \times _ T T_ j} \cong \xi _ j|_{T_ i \times _ T T_ j}$ implies in particular that $g_ i \circ \text{pr}_0 = g_ j \circ \text{pr}_1$. Thus there exists a unique morphism $g : T \to B$ such that $g_ i = g \circ a_ i$, see Descent on Spaces, Lemma 74.7.2. Denote $X_ T = T \times _{g, B} X$. Set $X_ i = X_{T_ i} = T_ i \times _{g_ i, B} X = T_ i \times _{a_ i, T} X_ T$ and
\[ X_{ij} = X_{T_ i} \times _{X_ T} X_{T_ j} = X_ i \times _{X_ T} X_ j \]
with projections $\text{pr}_ i$ and $\text{pr}_ j$ to $X_ i$ and $X_ j$. Observe that the pullback of $(T_ i, g_ i, E_ i)$ by $\text{pr}_0 : T_ i \times _ T T_ j \to T_ i$ is given by $(T_ i \times _ T T_ j, g_ i \circ \text{pr}_0, L\text{pr}_ i^*E_ i)$. Hence a descent datum for $\{ T_ i \to T\} $ in $\mathcal{C}\! \mathit{omplexes}_{X/B}$ is given by the objects $(T_ i, g \circ a_ i, E_ i)$ and for each pair $i, j$ an isomorphism in $D\mathcal{O}_{X_{ij}})$
\[ \varphi _{ij} : L\text{pr}_ i^*E_ i \longrightarrow L\text{pr}_ j^*E_ j \]
satisfying the cocycle condition over the pullback of $X$ to $T_ i \times _ T T_ j \times _ T T_ k$. Using the vanishing of negative Exts provided by (b) of Lemma 99.16.2, we may apply Simplicial Spaces, Lemma 85.35.2 to obtain descent1 for these complexes. In other words, we find there exists an object $E$ in $D_\mathit{QCoh}(\mathcal{O}_{X_ T})$ restricting to $E_ i$ on $X_{T_ i}$ compatible with $\varphi _{ij}$. Recall that being $T$-perfect signifies being pseudo-coherent and having locally finite tor dimension over $f^{-1}\mathcal{O}_ T$. Thus $E$ is $T$-perfect by an application of More on Morphisms of Spaces, Lemmas 76.54.1 and 76.54.2. Finally, we have to check condition (2) from Lemma 99.16.2 for $E$. This immediately follows from the description of the open $W$ in Lemma 99.16.1 and the fact that (2) holds for $E_ i$ on $X_{T_ i}/T_ i$. $\square$
[1] To check this, first consider the Čech fppf hypercovering $a : T_\bullet \to T$ with $T_ n = \coprod _{i_0 \ldots i_ n} T_{i_0} \times _ T \ldots \times _ T T_{i_ n}$ and then set $U_\bullet = T_\bullet \times _{a, T} X_ T$. Then $U_\bullet \to X_ T$ is an fppf hypercovering to which we may apply Simplicial Spaces, Lemma 85.35.2.
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