Ring Theory Seminar
Speaker: Guillem Quingles (UAB)
Title: The Cuntz semigroup of rings with stable rank one
Date: 4/5/2026
Time: 11:30
Web: http://mat.uab.cat/web/ligat/
Abstract: The Cuntz semigroup of a C*-algebra A with stable rank one enjoys a key structural property due to Coward, Elliott and Ivanescu: the order relation in Cu(A) among countably generated Hilbert A-modules is simply the order of isometric embeddings: $[X] \leq [Y]$ if and only if $X$ is isomorphic to a submodule of $Y$, and $[X] = [Y]$ iff $X$ is isomorphic to $Y$. In the talk I will present the algebraic analogue for the Cuntz semigroup of rings with stable rank one: for countably generated projective modules $P$ and $Q$, $[P] \leq [Q]$ if and only if $P$ is isomorphic to a pure submodule of $Q$, and $[P] = [Q]$ iff $P$ is isomorphic to $Q$. The proof follows the same broad strategy as the C*-algebraic one, but now set in a purely algebraic framework. A key tool in the proof is the study of K(P), a two-sided ideal of the ring of endomorphisms of a module P, which plays the role of “compact operators". We show that if R has stable rank one, then so does K(P). In the talk we will sketch the proof of the main result and see some applications. This is part of an ongoing work of my Phd under the supervision of Pere Ara and Francesc Perera.