Nearly Gorenstein Stanley-Reisner rings of dimension at least three are Gorenstein, and canonical traces of punctured-Gorenstein Stanley-Reisner rings are exactly the ring, the maximal ideal, or its square.
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The canonical trace of Stanley-Reisner rings that are Gorenstein on the punctured spectrum
Nearly Gorenstein Stanley-Reisner rings of dimension at least three are Gorenstein, and canonical traces of punctured-Gorenstein Stanley-Reisner rings are exactly the ring, the maximal ideal, or its square.