A homological characterization of AF groupoids
Benjamin Steinberg
TL;DR
This paper provides a complete homological characterization of AF ample groupoids, showing they have homological dimension zero, and extends this to groupoids over unital rings.
Contribution
It introduces a homological criterion for AF groupoids, linking their structure to homological dimension zero, a novel theoretical insight.
Findings
AF groupoids are characterized by homological dimension zero
Homological dimension zero applies to groupoids over unital rings
Complete homological characterization of AF groupoids provided
Abstract
An ample groupoid is said to be AF if it is a directed union of compact open principal subgroupoids. In this paper, we provide a complete homological characterization of these groupoids. Specifically, we prove that an ample groupoid is AF if and only if it has homological dimension zero. More generally, we characterize groupoids of homological dimension zero over a unital ring .
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