Picard groups, weight structures, and (noncommutative) mixed motives

TL;DR

This paper introduces a general framework for computing Picard groups in symmetric monoidal triangulated categories with weight structures, applying it to various motivic and noncommutative categories.

Contribution

It develops a universal method to determine Picard groups via the associated heart, extending computations to multiple motivic and noncommutative categories.

Findings

Computed Picard groups for categories of motivic nature

Determined Picard groups of derived categories of symmetric ring spectra

Established a general theory linking weight structures to Picard groups

Abstract

We develop a general theory which enables the computation of the Picard group of a symmetric monoidal triangulated category, equipped with a weight structure, in terms of the Picard group of the associated heart. As an application, we compute the Picard group of several categories of motivic nature - mixed Artin motives, mixed Artin-Tate motives, motivic spectra, noncommutative mixed Artin motives, noncommutative mixed motives of central simple algebras, noncommutative mixed motives of separable algebras - as well as the Picard group of the derived categories of symmetric ring spectra.