Tensor triangular geometry of non-commutative motives

In this article we initiate the study of the tensor triangular geometry of the categories Motka and Motkl of non-commutative motives (over a base ring k). Since the full computation of the spectrum of Motka and Motkl 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 Coreka and Corekl (i.e. the thick triangulated subcategories generated by the tensor unit) is closely related to the Zariski spectrum of Z. Moreover, we prove that if we slightly enlarge Coreka 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 Z.