The Cuntz semigroup of a ring

Antoine, Ramon ; Ara, Pere ; Bosa, Joan (Universidad de Zaragoza) ; Perera, Francesc ; Vilalta, Eduard

Resumen: For any ring R, we introduce an invariant in the form of a partially ordered abelian semigroup built from an equivalence relation on the class of countably generated projective modules. We call 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 . The Cuntz semigroup of R is part of a new invariant which includes an ambient semigroup in the category of abstract Cuntz semigroups that provides additional information. We provide computations for both and 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.

Idioma: Inglés

DOI: 10.1007/s00029-024-01002-9

Año: 2025

Publicado en: Selecta Mathematica-New Series 31, 1 (2025), [46 pp.]
ISSN: 1022-1824

Tipo y forma: Article (Published version)

Área (Departamento): Área Algebra (Dpto. Matemáticas)