The local-to-global principle via topological properties of the tensor triangular support

Abstract

Following the theory of tensor triangular support introduced by Sanders, which generalizes the Balmer-Favi support, we prove the local version of the result of Zou that the Balmer spectrum being Hochster weakly scattered implies the local-to-global principle. That is, given an object $t$ of a tensor triangulated category $\mathcal{T}$ we show that if the tensor triangular support $\text{Supp}(t)$ is a weakly scattered subset with respect to the inverse topology of the Balmer spectrum $\text{Spc}(\mathcal{T}^c)$, then the local-to-global principle holds for $t$. As immediate consequences, we have the analogue adaptations of the well-known statements that the Balmer spectrum being noetherian or Hausdorff scattered implies the local-to-global principle. 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.