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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Benjamin Steinberg (2026) studied this question.
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