📝 SummarXiv

Abstract

Recent work by Miyashita and Varbaro classified the canonical traces of Stanley--Reisner rings that are Gorenstein on the punctured spectrum, under the Cohen--Macaulay assumption. We aim to generalize the result to the non--Cohen--Macaulay case. First, we establish an explicit formula for the canonical trace of graded fiber products of Noetherian rings and apply it to Stanley--Reisner rings of disconnected simplicial complexes. This allows us to reduce the problem to the case of connected simplicial complexes. In that case, we succeed in weakening the Cohen--Macaulay assumption in their result to the Serre's condition $(S_2)$, obtaining a similar classification. Finally, by combining these results, we provide a description of the canonical trace of a Stanley--Reisner ring satisfying $(S_2)$.