Singular Ideals over arbitrary fields for the two- and three-headed snakes

TL;DR

This paper investigates the structure of singular ideals in Steinberg algebras over arbitrary fields for specific non-Hausdorff groupoids called two- and three-headed snakes, revealing field-dependent properties of these ideals.

Contribution

It characterizes when the singular ideal in Steinberg algebras over two- and three-headed snake groupoids contains non-zero ideals, depending on the base field's properties.

Findings

In the two-headed snake case, the singular ideal contains no non-zero ideals.

In the three-headed snake case, the singular ideal contains non-zero ideals iff the field is a splitting field of x^2 + x + 1.

Non-zero proper subset ideals exist when the field's characteristic is a prime not congruent to -1 mod 3.

Abstract

We study the Steinberg algebra with coefficients in an arbitrary field K for the two- and three-headed snake groupoids, which are basic examples of non-Hausdorff groupoids. We are particularly interested in elements of this algebra that are no longer continuous, known as singular functions. We prove that the ideal of singular functions in the Steinberg algebra of the two-headed snake does not properly contain any non-zero ideal, regardless of the choice of the base field K. In the case of the three-headed snake, we prove that the ideal of singular functions of the Steinberg algebra properly contains non-zero ideals if, and only if, the base field K is a splitting field of x^2 + x + 1. In particular, there are always non-zero proper subset ideals when K has characteristic a prime not congruent to -1 mod 3.