Torsion and complete dualizable objects in tensor-triangulated categories over a Noetherian ring

TL;DR

This paper investigates dualizable torsion and complete objects in tensor-triangulated categories with Noetherian central actions, establishing their structure, reconstruction, and relationships under certain conditions.

Contribution

It introduces a framework for understanding dualizable torsion and complete objects in tensor-triangulated categories over Noetherian rings, including their reconstruction and generator properties.

Findings

Categories admit a natural Noetherian action of the completed ring R^

Dualizable torsion and complete objects can be reconstructed from compact torsion objects

Strong generators induce generators for dualizable torsion and complete categories

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.