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

JOURNAL OF ALGEBRA

卷:491

Article

Ben-Zvi, Michael¹  Ma, Alexander²  Reyes, Manuel³  Chirvasitu, Alexandru⁴ 

[1] Tufts Univ, Dept Math, Bromfield Pearson Hall,503 Boston Ave, Medford, MA 02155 USA

[2] Univ Minnesota, Dept Math, 206 Church St SE, Minneapolis, MN 55413 USA

[3] Bowdoin Coll, Dept Math, 8600 Coll Stn, Brunswick, ME 04011 USA

[4] SUNY Buffalo, Dept Math, 244 Math Bldg, Buffalo, NY 14260 USA

关键词: Kochen-Specker Theorem;    Contextuality;    Idempotent integer matrix;    Prime spectrum;    Noncommutative spectrum;    Prime partial ideal;    Partial Boolean algebra;   

DOI  :  10.1016/j.jalgebra.2017.08.008

来源: Elsevier

【 摘 要 】

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 x n idempotent integer matrices for n >= 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 >= 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. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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