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Mathematics, Algebraic Geometry, Category Theory, K-Theory and Homology, Representation Theory
Abstract:
We analyze the spectrum of the tensor-triangulated category of Artin-Tate
motives over the base field R of real numbers, with integral coefficients. Away
from 2, we obtain the same spectrum as for complex Tate motives, previously
studied by the second-named author. So the novelty is concentrated at the prime
2, where modular representation theory enters the picture via work of
Positselski, based on Voevodsky's resolution of the Milnor Conjecture. With
coefficients in k=Z/2, our spectrum becomes homeomorphic to the spectrum of the
derived category of filtered kC_2-modules with a peculiar exact structure, for
the cyclic group C_2=Gal(C/R). This spectrum consists of six points organized
in an interesting way. As an application, we find exactly fourteen classes of
mod-2 real Artin-Tate motives, up to the tensor-triangular structure. Among
those, three special motives stand out, from which we can construct all others.
We also discuss the spectrum of Artin motives and of Tate motives.
