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Obstructing extensions of the functor spec to noncommutative rings

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Abstract

This paper concerns contravariant functors from the category of rings to the category of sets whose restriction to the full subcategory of commutative rings is isomorphic to the prime spectrum functor Spec. The main result reveals a common characteristic of these functors: every such functor assigns the empty set to \(\mathbb{M}_n (\mathbb{C})\) for n ⩾ 3. The proof relies, in part, on the Kochen-Specker Theorem of quantum mechanics. The analogous result for noncommutative extensions of the Gel’fand spectrum functor for C*-algebras is also proved.

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Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego 9500 Gilman Drive #0112, La Jolla, CA, 92093-0112, USA

    Manuel L. Reyes

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  1. Manuel L. Reyes
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Correspondence to Manuel L. Reyes.

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The author was supported by a Ford Foundation Predoctoral Diversity Fellowship at the University of California, Berkeley, and a University of California President’s Postdoctoral Fellowship at the University of California, San Diego.

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Reyes, M.L. Obstructing extensions of the functor spec to noncommutative rings. Isr. J. Math. 192, 667–698 (2012). https://doi.org/10.1007/s11856-012-0043-y

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  • DOI: https://doi.org/10.1007/s11856-012-0043-y

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