Lemma 101.45.6. Let
be a cartesian diagram of algebraic stacks. If f induces an isomorphism between automorphism groups at points (Remark 101.19.5), then
is injective on isomorphism classes for any field k.
Lemma 101.45.6. Let
be a cartesian diagram of algebraic stacks. If f induces an isomorphism between automorphism groups at points (Remark 101.19.5), then
is injective on isomorphism classes for any field k.
Proof. We have to show that given (y', x) there is at most one x' mapping to it. By our construction of 2-fibre products, a morphism x' is given by a triple (x, y', \alpha ) where \alpha : g \circ y' \to f \circ x is a 2-morphism. Now, suppose we have a second such triple (x, y', \beta ). Then \alpha and \beta differ by a k-valued point \epsilon of the automorphism group algebraic space G_{f(x)}. Since f induces an isomorphism G_ x \to G_{f(x)} by assumption, this means we can lift \epsilon to a k-valued point \gamma of G_ x. Then (\gamma , \text{id}) : (x, y', \alpha ) \to (x, y', \beta ) is an isomorphism as desired. \square
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