Averaging t-structures and extension closure of aisles

TL;DR

This paper investigates conditions under which multiple t-structures in a triangulated category can be combined into a single t-structure through averaging, providing criteria, algorithms, and implications for equivariant categories.

Contribution

It establishes when averaging t-structures is possible, offers an algorithm for piecewise tame hereditary categories, and explores the impact of group actions on t-structures.

Findings

Positive answer for compactly generated t-structures in big categories.

Provides a criterion and algorithm for averaging in piecewise tame hereditary categories.

Shows how group actions induce t-structures on equivariant categories.

Abstract

We ask when a finite set of t-structures in a triangulated category can be `averaged' into one t-structure or, equivalently, when the extension closure of a finite set of aisles is again an aisle. There is a straightforward, positive answer for a finite set of compactly generated t-structures in a big triangulated category. For piecewise tame hereditary categories, we give a criterion for when averaging is possible, and an algorithm that computes truncation triangles in this case. A finite group action on a triangulated category gives a natural way of producing a finite set of t-structures out of a given one. If averaging is possible, there is an induced t-structure on the equivariant triangulated category.