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The Stacks project

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.


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