Higher comparison maps for the spectrum of a tensor triangulated category

TL;DR

This paper introduces higher comparison maps for tensor triangulated categories, enabling an iterative analysis of their spectra through a hierarchy of maps and filtrations, exemplified in stable homotopy theory.

Contribution

It constructs a new family of comparison maps for the spectrum of tensor triangulated categories, extending Balmer's work and providing a method for iterative spectral analysis.

Findings

Constructed natural continuous maps from objects' support to Zariski spectra.

Developed an iterative approach to analyze spectra via higher comparison maps.

Applied the method to the stable homotopy category of finite spectra.

Abstract

For each object in a tensor triangulated category, we construct a natural continuous map from the object's support---a closed subset of the category's triangular spectrum---to the Zariski spectrum of a certain commutative ring of endomorphisms. When applied to the unit object this recovers a construction of P. Balmer. These maps provide an iterative approach for understanding the spectrum of a tensor triangulated category by starting with the comparison map for the unit object and iteratively analyzing the fibers of this map via "higher" comparison maps. We illustrate this approach for the stable homotopy category of finite spectra. In fact, the same underlying construction produces a whole collection of new comparison maps, including maps associated to (and defined on) each closed subset of the triangular spectrum. These latter maps provide an alternative strategy for analyzing the spectrum by iteratively building a filtration of closed subsets by pulling back filtrations of affine schemes.