Remark 99.7.2. In Situation 99.7.1 let $B' \to B$ be a morphism of algebraic spaces over $S$. Set $X' = X \times _ B B'$ and denote $\mathcal{F}'$ the pullback of $\mathcal{F}$ to $X'$. Thus we have the functor $Q_{\mathcal{F}'/X'/B'}$ on the category of schemes over $B'$. For a scheme $T$ over $B'$ it is clear that we have
\[ Q_{\mathcal{F}'/X'/B'}(T) = Q_{\mathcal{F}/X/B}(T) \]
where on the right hand side we think of $T$ as a scheme over $B$ via the composition $T \to B' \to B$. Similar remarks apply to $\text{Q}^{fp}_{\mathcal{F}/X/B}$. These trivial remarks will occasionally be useful to change the base algebraic space.
Comments (0)