Tensor triangular geometry of non-commutative motives

TL;DR

This paper explores the tensor triangular geometry of non-commutative motives, revealing connections to the Zariski spectrum of integers and how the spectrum becomes richer over fields of characteristic zero.

Contribution

It initiates the study of tensor triangular geometry in non-commutative motives and relates the spectrum of certain subcategories to classical algebraic spectra.

Findings

Spectrum of monogenic cores relates to Zariski spectrum of integers

Enlarging cores with polynomial motives enriches the spectrum over characteristic zero fields

Partial spectral information for subcategories over finite fields and algebraic closures

Abstract

In this article we initiate the study of the tensor triangular geometry of the categories Mot(k)_a and Mot(k)_l of non-commutative motives (over a base ring k). Since the full computation of the spectrum of Mot(k)_a and Mot(k)_l seems completely out of reach, we provide some information about the spectrum of certain subcategories. More precisely, we show that when k is a finite field (or its algebraic closure) the spectrum of the monogenic cores Core(k)_a and Core(k)_l (i.e. the thick triangulated subcategories generated by the tensor unit) is closely related to the Zariski spectrum of the integers. Moreover, we prove that if we slightly enlarge Core(k)_a to contain the non-commutative motive associated to the ring of polynomials k[t], and assume that k is a field of characteristic zero, then the corresponding spectrum is richer than the Zariski spectrum of the integers.