Big projective modules over noetherian semilocal rings

TL;DR

This paper characterizes the monoid of countably generated projective modules over noetherian semilocal rings using solutions to systems of congruences and diophantine equations, establishing a precise algebraic correspondence.

Contribution

It provides a complete description of the monoid of projective modules over noetherian semilocal rings via solutions to algebraic systems, and proves the converse realization result.

Findings

Monoid $V^*(R)$ is isomorphic to solutions of certain algebraic systems.

Characterization applies to rings with a fixed number of simple modules.

Converse: any suitable solution set corresponds to a noetherian semilocal ring.

Abstract

We prove that for a noetherian semilocal ring $R$ with exactly $k$ isomorphism classes of simple right modules the monoid $V^*(R)$ of isomorphism classes of countably generated projective right (left) modules, viewed as a submonoid of $V^*(R/J(R))$, is isomorphic to the monoid of solutions in $(\No \cup\{\infty\})^k$ of a system consisting of congruences and diophantine linear equations. The converse also holds, that is, if $M$ is a submonoid of $(\No \cup\{\infty\})^k$ containing an order unit $(n_1,..., n_k)$ of $\No^k$ which is the set of solutions of a system of congruences and linear diophantine equations then it can be realized as $V^*(R)$ for a noetherian semilocal ring such that $R/J(R)\cong M_{n_1}(D_1)\times ... \times M_{n_k}(D_k)$ for suitable division rings $D_1,..., D_k$.