The local-to-global principle via topological properties of the Balmer-Favi support

Following the theory of stratification of tensor triangulated categories via Balmer-Favi support inaugurated by Barthel, Heard and Sanders, we prove the local versions of the well-known statements that the Balmer spectrum being noetherian or profinite scattered implies the local-to-global principle. That is, given an object t of a tensor triangulated category T we show that if the Balmer-Favi support Supp (t) is a noetherian space, then the local-to-global principle holds for t. In the case where the Balmer spectrum Spc (Tᶜ) is profinite, if the support Supp (t) is scattered then the local-to-global principle holds for the object t. We conclude with an application of the last result to the examination of the support of injective superdecomposable modules in the derived category of an absolutely flat ring which is not semi-artinian.