We investigate the Tate–Shafarevich group of a multinorm-one torus T over a global field k. We explore a few functorial maps among cohomology groups and their relations. Using these properties and the approach in Bayer-Fluckiger–Lee–Parimala [Adv. in Math., 2019], we obtain a similar basic structural result for and extends a few results in loc. cit. for some more general multinorm-one tori. We also give an alternative proof of previous results on the vanishing of due to Demarche–Wei and the case of a product of two abelian extensions due to Pollio. Moreover, we improve the explicit description of of Lee by removing an intersection condition.
