The main object of interest in this talk is a rigidly compactly generated tensor triangulated category with an action of a graded commutative Noetherian ring R making (graded) homomorphism groups between compacts finitely generated R-modules. There are various examples of this setup coming from commutative algebra, modular representation theory of groups or homotopy theory. In joint work with Jun Maillard, we studied the categories of dualizable objects inside the mutually equivalent categories of a-torsion and a-complete objects with respect to a homogeneous ideal a of R. This is a surprisingly well behaved setup. For example, the categories of a-torsion/complete dualizable objects are always Hom-finite over the a-adic completion of R and we can recover them using a form of Cauchy completions from the categories of a-torsion/comlete compact objects. The best results are achieved when we in addition assume that the category of compact objects is strongly generated (or equivalently, has finite Rouquier dimension). Then so is the category of a-torsion/complete dualizable objects and the so-called local regularity condition introduced by Benson, Iyengar, Krause and Pevtsova is automatically satisfied. Moreover, there are Brown representability results providing a "perfect pairing" between the categories of a-torsion/complete compact objects and a-torsion/complete dualizable objects which are formally very similar to Neeman's arXiv:1804.02240.