Torsion and complete dualizable objects in tensor-triangulated categories over a Noetherian ring
Jun Maillard, Jan Šťovíček
TL;DR
The paper develops a general framework for adic completion in tensor-triangular geometry for R-linear tt-categories, enabling a systematic reconstruction of dualizable torsion and complete objects from compact data. It proves that, under a Noetherian graded-ring action with finite-generation on Hom-sets, the categories of dualizable torsion and dualizable complete objects can be reconstructed as R^∧-linear tensor-triangulated categories from the compact torsion data via restricted Yoneda, and that strong generation of the compacts yields strong generators in the dualizable subcategories. It also establishes thick-generation results, cohomological/homological representability theorems, and a suite of finiteness and completion properties for Hom-sets, culminating in a comprehensive reconstruction of triangulated structure and monoidal structure. The framework is illustrated through a case study in finite-group algebras, connecting to known examples in modular representation theory and derived categories, and highlighting the broader impact for non-local Noetherian settings. Overall, the work provides a robust, non-local analogue of adic completion in tt-geometry with explicit reconstruction and representability results, expanding the toolkit for understanding dualizable objects in tensor-triangulated categories under Noetherian actions.
Abstract
We study categories of dualizable torsion and complete objects for compactly-rigidly generated tensor-triangulated categories T with a Noetherian central action of a graded commutative Noetherian ring R. We show that they always admit a natural Noetherian action of the completed graded ring R^ and that the categories of dualizable torsion and complete objects can be abstractly reconstructed as tensor-triangulated R^-linear categories from the category of compact torsions objects with the corresponding structure. If the category of compact objects of T in addition admits a strong generator g, we show that the torsion coreflection (resp. complete reflection) of g is a strong generator for the category of dualizable torsion (resp. dualizable complete) objects. In that case, we also show that the categories of dualizable torsion and compact torsion objects determine each other in terms of Brown-type representability theorems.