Binomial edge ideals are F-pure in char 2 iff the graph is weakly closed, and graphs with asteroidal triples yield non-F-pure quotients in all positive characteristics.
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2 Pith papers cite this work. Polarity classification is still indexing.
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An ample groupoid is AF if and only if it has homological dimension zero.
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A homological characterization of AF groupoids
An ample groupoid is AF if and only if it has homological dimension zero.