Lemma 99.7.6 (08IV)—The Stacks project

Table of contents
Part 7: Algebraic Stacks
Chapter 99: Quot and Hilbert Spaces
Section 99.7: The functor of quotients
Lemma 99.7.6 (cite)

Lemma 99.7.6. In Situation 99.7.1 let $\{ X_ i \to X\} _{i \in I}$ be an fpqc covering and for each $i, j \in I$ let $\{ X_{ijk} \to X_ i \times _ X X_ j\} $ be an fpqc covering. Denote $\mathcal{F}_ i$, resp. $\mathcal{F}_{ijk}$ the pullback of $\mathcal{F}$ to $X_ i$, resp. $X_{ijk}$. For every scheme $T$ over $B$ the diagram

\[ \xymatrix{ Q_{\mathcal{F}/X/B}(T) \ar[r] & \prod \nolimits _ i Q_{\mathcal{F}_ i/X_ i/B}(T) \ar@<1ex>[r]^-{\text{pr}_0^*} \ar@<-1ex>[r]_-{\text{pr}_1^*} & \prod \nolimits _{i, j, k} Q_{\mathcal{F}_{ijk}/X_{ijk}/B}(T) } \]

presents the first arrow as the equalizer of the other two. The same is true for the functor $\text{Q}^{fp}_{\mathcal{F}/X/B}$.

Proof. Let $\mathcal{F}_{i, T} \to \mathcal{Q}_ i$ be an element in the equalizer of $\text{pr}_0^*$ and $\text{pr}_1^*$. By Remark 99.7.3 we obtain a surjection $\mathcal{F}_ T \to \mathcal{Q}$ of quasi-coherent $\mathcal{O}_{X_ T}$-modules whose restriction to $X_{i, T}$ recovers $\mathcal{F}_ i \to \mathcal{Q}_ i$. By Morphisms of Spaces, Lemma 67.31.5 we see that $\mathcal{Q}$ is flat over $T$ as desired. In the case of the functor $\text{Q}^{fp}_{\mathcal{F}/X/B}$, i.e., if $\mathcal{Q}_ i$ is of finite presentation, then $\mathcal{Q}$ is of finite presentation too by Descent on Spaces, Lemma 74.6.2. $\square$

The Stacks project

Comments (0)

Post a comment