A tour of support theory for triangulated categories through tensor triangular geometry

TL;DR

This paper surveys support theory and the structure of triangulated subcategories using tensor triangular geometry, focusing on the local-to-global principle, spectrum construction, and support for compactly generated categories.

Contribution

It introduces the machinery of tensor triangular geometry to study support theory and lattices of triangulated subcategories, including the spectrum and local-to-global principles.

Findings

Construction of the spectrum for tensor triangulated categories

Development of support theory for compactly generated triangulated categories

Examples illustrating the application of tensor triangular geometry

Abstract

These notes attempt to give a short survey of the approach to support theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the material necessary to state and prove the local-to-global principle. In particular, we discuss Balmer's construction of the spectrum, generalised Rickard idempotents and support for compactly generated triangulated categories, and actions of tensor triangulated categories. Several examples are also given along the way. These notes are based on a series of lectures given during the Spring 2015 program on 'Interactions between Representation Theory, Algebraic Topology and Commutative Algebra' (IRTATCA) at the CRM in Barcelona.