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Tensor triangular geometry of non-commutative motives

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.