The tensor triangular geometry of fully faithful functors

TL;DR

This paper investigates the geometric properties of fully faithful functors between tensor triangulated categories, establishing connectedness results for induced spectra and applying these to equivariant and motivic contexts.

Contribution

It introduces the concept of concentration at a set of generators, computes spectra for specific categories, and analyzes the comparison map's properties in tensor triangulated geometry.

Findings

The map on Balmer spectra induced by a fully faithful functor is a quotient map with connected fibers.

The spectrum of the unitation of the equivariant stable homotopy category is computed.

The comparison map of a connective category is a quotient map with connected fibers.

Abstract

We prove that the map on Balmer spectra induced by a fully faithful geometric functor is a quotient map whose fibers are connected. This is an analogue of the Zariski Connectedness Theorem in algebraic geometry and it can be applied to a plethora of examples in equivariant and motivic mathematics. We isolate a significant source of examples by introducing the "concentration" of a tt-category at a well-behaved chosen set of compact generators. Various categories of cellular objects arise in this way. In particular, the "unitation" of a tt-category is the concentration at the unit object. We compute the Balmer spectrum of the unitation of the equivariant stable homotopy category as well as related categories arising in equivariant higher algebra. We also apply our results to the study of the comparison map of a tt-category. Among other results, we prove that the comparison map of a connective category is a quotient map with connected fibers. This involves studying the tt-geometry of weight complex functors, which may be of independent interest. We also study the relationship between the comparison map and the affinization of the Balmer spectrum viewed as a locally ringed space. These results provide a layered approach to understanding the spectrum of a given tt-category, by starting with the Zariski spectrum of the endomorphism ring of the unit, and then passing backwards to larger and larger concentrations (through quotient maps with connected fibers). Significant stages along the way include the passage to the unitation and the passage to the concentration at the Picard group.