A classification of ideals in Steinberg and Leavitt path algebras over arbitrary rings

TL;DR

This paper establishes a comprehensive correspondence between ideals in Steinberg algebras and group algebra ideals, enabling a complete graph-theoretic classification of Leavitt path algebra ideals over any commutative ring.

Contribution

It generalizes ideal correspondence theorems to broader classes of groupoids and rings, extending the classification of Leavitt path algebra ideals beyond fields.

Findings

Established a one-to-one ideal correspondence for Hausdorff ample groupoids.

Provided a complete graph-theoretic description of Leavitt path algebra ideals.

Generalized ideal classification from fields to arbitrary commutative rings.

Abstract

We give a one-to-one correspondence between ideals in the Steinberg algebra of a Hausdorff ample groupoid $G$, and certain families of ideals in the group algebras of isotropy groups in $G$. This generalises a known ideal correspondence theorem for Steinberg algebras of strongly effective groupoids. We use this to give a complete graph-theoretic description of the ideal lattice of Leavitt path algebras over arbitrary commutative rings, generalising the classification of ideals in Leavitt path algebras over fields.