A Kochen-Specker theorem for integer matrices and noncommutative spectrum functors

We investigate the possibility of constructing Kochen-Specker uncolorable sets of idempotent matrices whose entries lie in various rings, including the rational numbers, the integers, and finite fields. Most notably, we show that there is no Kochen-Specker coloring of the $n \times n$ idempotent integer matrices for $n \geq 3$, thereby illustrating that Kochen-Specker contextuality is an inherent feature of pure matrix algebra. We apply this to generalize recent no-go results on noncommutative spectrum functors, showing that any contravariant functor from rings to sets (respectively, topological spaces or locales) that restricts to the Zariski prime spectrum functor for commutative rings must assign the empty set (respectively, empty space or locale) to the matrix ring $M_n(R)$ for any integer $n \geq 3$ and any ring $R$. An appendix by Alexandru Chirvasitu shows that Kochen-Specker colorings of idempotents in partial subalgebras of $M_3(F)$ for a perfect field $F$ can be extended to partial algebra morphisms into the algebraic closure of $F$.