Remarks on Brauer–Manin Obstruction for Weil Restrictions

Given a finite extension [Formula: see text] of number fields and a smooth quasi-projective variety [Formula: see text] over [Formula: see text], if the abelianization of [Formula: see text] is trivial, we prove that there is a natural identification between Brauer–Manin sets of [Formula: see text] and its Weil restriction [Formula: see text]. If [Formula: see text] is projective and [Formula: see text] is a torsion-free abelian group, we prove that there is a natural identification between algebraic Brauer–Manin sets of [Formula: see text] and [Formula: see text].