The Cuntz semigroup of a ring

• Ramon Antoine ^[1] ; Pere Ara ^[1] ; Joan Bosa ^[2] ; Francesc Perera ^[1] ; Eduard Vilalta ^[3]
  1. [1] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     
  2. [2] Universidad de Zaragoza

     Universidad de Zaragoza

     Zaragoza, España

     
  3. [3] Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 1, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-024-01002-9
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• Resumen

  • For any ring R, we introduce an invariant in the form of a partially ordered abelian semigroup S(R) built from an equivalence relation on the class of countably generated projective modules. We call S(R)the Cuntz semigroup of the ring R. This construction is akin to the manufacture of the Cuntz semigroup of a C*-algebra using countably generated Hilbert modules. To circumvent the lack of a topology in a general ring R, we deepen our understanding of countably projective modules over R, thus uncovering new features in their direct limit decompositions, which in turn yields two equivalent descriptions of S(R). The Cuntz semigroup of R is part of a new invariant SCu(R) which includes an ambient semigroup in the category of abstract Cuntz semigroups that provides additional information. We provide computations for both S(R) and SCu(R) in a number of interesting situations, such as unit-regular rings, semilocal rings, and in the context of nearly simple domains. We also relate our construcion to the Cuntz semigroup of a C*-algebra.

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