Lemma 97.12.4. In the situation of Lemma 97.12.1. Assume that G, H are representable by algebraic spaces and étale. Then \mathcal{H}_ d(\mathcal{X}'/\mathcal{Y}') \to \mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) is representable by algebraic spaces and étale. If also H is surjective and the induced functor \mathcal{X}' \to \mathcal{Y}' \times _\mathcal {Y} \mathcal{X} is surjective, then \mathcal{H}_ d(\mathcal{X}'/\mathcal{Y}') \to \mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) is surjective.
Proof. Set \mathcal{X}'' = \mathcal{Y}' \times _\mathcal {Y} \mathcal{X}. By Lemma 97.4.1 the 1-morphism \mathcal{X}' \to \mathcal{X}'' is representable by algebraic spaces and étale (in particular the condition in the second statement of the lemma that \mathcal{X}' \to \mathcal{X}'' be surjective makes sense). We obtain a 2-commutative diagram
\xymatrix{ \mathcal{X}' \ar[r] \ar[d] & \mathcal{X}'' \ar[r] \ar[d] & \mathcal{X} \ar[d] \\ \mathcal{Y}' \ar[r] & \mathcal{Y}' \ar[r] & \mathcal{Y} }
It follows from Lemma 97.12.2 that \mathcal{H}_ d(\mathcal{X}''/\mathcal{Y}') is the base change of \mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) by \mathcal{Y}' \to \mathcal{Y}. In particular we see that \mathcal{H}_ d(\mathcal{X}''/\mathcal{Y}') \to \mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) is representable by algebraic spaces and étale, see Algebraic Stacks, Lemma 94.10.6. Moreover, it is also surjective if H is. Hence if we can show that the result holds for the left square in the diagram, then we're done. In this way we reduce to the case where \mathcal{Y}' = \mathcal{Y} which is the content of Lemma 97.12.3. \square
Comments (0)