For any connected linear algebraic group G over a number field, strong approximation with the Brauer-Manin obstruction holds for the classifying stack BG off any nonempty finite set of places.
On the Motivic Homotopy Type of Algebraic Stacks
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We construct smooth presentations of algebraic stacks that are local epimorphisms in the Morel-Voevodsky $\mathbb{A}^1$-homotopy category. As a consequence we show that the motive of a smooth stack (in Voevodsky's triangulated category of motives) has many of the same properties as the motive of a smooth scheme.
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Approximation theorems for classifying stacks over number fields
For any connected linear algebraic group G over a number field, strong approximation with the Brauer-Manin obstruction holds for the classifying stack BG off any nonempty finite set of places.