Abstract

Tensor-triangular geometry provides a broad framework to study tensor-triangulated categories arising in nature. Just as any commutative ring has its Zariski spectrum, any tensor-triangulated category has a space called its Balmer spectrum which carries the universal support theory classifying thick tensor ideals. There is also a closely related space called the homological spectrum from which one can recover the Balmer spectrum. Another space associated to any triangulated category is the Ziegler spectrum which contains all the data about the pure structure. I will explain how the homological spectrum may be constructed from the Ziegler spectrum, thus giving a bridge between tensor-triangular geometry and purity. I’ll explain an application of this to functoriality in tensor-triangular geometry. Time permitting, I’ll explain how the ideas behind this also may be applied in representation theory in the study of rank functions and t-structures. This is based on joint work with Isaac Bird.