Support theory via actions of tensor triangulated categories (Q2848382)

Let \(\mathcal K\) be a triangulated category and \(\mathcal T\) be a tensor triangulated category. In the first part of the paper the author gives a definition of what is meant by the action of \(\mathcal T\) on \(\mathcal K\). In the case when \(\mathcal T\) is rigidly-compactly generated and \(\mathcal K\) is compactly generated, it is shown that this action gives rise to a notion of supports which gives the possibility to extend recent work of D. J.~Benson, S. K.~Iyengar and H.~Krause and of P.~Balmer and G.~Favi. The author states a suitable version of the local-to-global principle (originally due to Benson, Iyengar and Krause) for actions of triangulated categories which is shown to hold under very general assumptions. Next, he formulates a relative version of the telescope conjecture (roughly saying that certain smashing subcategories of \(\mathcal K\) are generated by compact objects of \(\mathcal K\)), and gives sufficient conditions for it to hold.