The Stacks project

Proof. To prove that $\mathcal{C}\! \mathit{oh}_{X/B}$ 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 99.5.3, 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 $\mathcal{C}\! \mathit{oh}_{X/B}$. For each $i$ we can write $\xi _ i = (T_ i, g_ i, \mathcal{F}_ i)$. 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 $\xi _ i|_{T_ i \times _ T T_ j} = \xi _ j|_{T_ i \times _ T T_ j}$ 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

with projections $\text{pr}_ i$ and $\text{pr}_ j$ to $X_ i$ and $X_ j$. Observe that the pullback of $(T_ i, g_ i, \mathcal{F}_ i)$ by $\text{pr}_0 : T_ i \times _ T T_ j \to T_ i$ is given by $(T_ i \times _ T T_ j, g_ i \circ \text{pr}_0, \text{pr}_ i^*\mathcal{F}_ i)$. Hence a descent datum for $\{ T_ i \to T\} $ in $\mathcal{C}\! \mathit{oh}_{X/B}$ is given by the objects $(T_ i, g \circ a_ i, \mathcal{F}_ i)$ and for each pair $i, j$ an isomorphism of $\mathcal{O}_{X_{ij}}$-modules

satisfying the cocycle condition over (the pullback of $X$ to) $T_ i \times _ T T_ j \times _ T T_ k$. Ok, and now we simply use that $\{ X_ i \to X_ T\} $ is an fppf covering so that we can view $(\mathcal{F}_ i, \varphi _{ij})$ as a descent datum for this covering. By Descent on Spaces, Proposition 74.4.1 this descent datum is effective and we obtain a quasi-coherent sheaf $\mathcal{F}$ over $X_ T$ restricting to $\mathcal{F}_ i$ on $X_ i$. By Morphisms of Spaces, Lemma 67.31.5 we see that $\mathcal{F}$ is flat over $T$ and Descent on Spaces, Lemma 74.6.2 guarantees that $\mathcal{Q}$ is of finite presentation as an $\mathcal{O}_{X_ T}$-module. Finally, by Descent on Spaces, Lemma 74.11.19 we see that the scheme theoretic support of $\mathcal{F}$ is proper over $T$ as we've assumed the scheme theoretic support of $\mathcal{F}_ i$ is proper over $T_ i$ (note that taking scheme theoretic support commutes with flat base change by Morphisms of Spaces, Lemma 67.30.10). In this way we obtain our desired object over $T$. $\square$