Suppose \({\cal T}\) is a rigidly-compactly generated tensor triangulated category and \({\cal K}\) is a compactly generated triangulated category on which \({\cal T}\) acts, in the sense of Stevenson. We prove that if \({\rm Spc}({\cal T}^{\rm c})\) is Noetherian and \({\cal K}\) is stable, then each object in \({\cal K}\) has a unique functorial tower, filtered by Balmer-Favi cosupports. This is an analogy of Stevenson’s work on filtrations by Balmer-Favi supports.
