Difference between two constructions of the core tt-category for full Y-completion

Determine whether, for an arbitrary rigidly-compactly generated tensor-triangulated category T and a Thomason subset Y, the subcategory of dualizable objects in the Y-complete category T^Y (equivalently, (TY)^⊥⊥) coincides with the smallest thick tt-subcategory of the compact objects in T^Y that contains the image of the compact subcategory T^c under the Y-completion functor. In other words, ascertain whether there is any difference in general between taking the core tt-category to be the dualizable objects in T^Y (as in Naumann–Pol–Ramzi) versus taking it to be the tt-closure of the image of T^c inside the compact subcategory of T^Y. Theorem 4.17 shows these choices agree when T = D(R), but the general case remains unsettled.

Background

The paper studies how to recover perfect complexes over the I-adic completion of a ring using tensor-triangular (tt) completion, and proves precise equivalences under a Koszul-completeness criterion (always satisfied for noetherian rings). Beyond the derived category of a ring, the authors discuss how to build a ‘full’ completion T^ in general tt-geometry by Ind-completing an appropriate rigid tt-subcategory that plays the role of dualizable objects in the completed category.

They present two natural candidates for this core tt-subcategory: (1) the subcategory of dualizable objects in the Y-complete category T^Y (the choice adopted by Naumann–Pol–Ramzi), and (2) the smallest tt-subcategory of the compact objects in T^Y that contains the image of the compact subcategory T^c of T. Theorem 4.17 establishes that, for T = D(R), these candidates coincide. However, the authors explicitly note that it is unclear whether a difference exists in general, motivating further investigation in the broader framework of tt-geometry.