Year: 2024
Type: article
Abstract: For any ring R, we introduce an invariant in the form of a partially ordered abelian semigroup $${\mathrm S}(R)$$ S ( R ) built from an equivalence relation on the class of countably generated projective modules. We call $${\mathrm S}(R)$$ 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 $${\mathrm S}(R)$$ S ( R ) . The Cuntz semigroup of R is part of a new invariant $$\textrm{SCu}(R)$$ SCu ( R ) which includes an ambient semigroup in the category of abstract Cuntz semigroups that provides additional information. We provide computations for both $${\mathrm S}(R)$$ S ( R ) and $$\textrm{SCu}(R)$$ 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.
Source: Selecta Mathematica
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4405319188