Canonical traces of graded fiber products: applications to disconnected Stanley--Reisner rings

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. The purpose of this paper is 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 succeeded in giving a complete classification without assuming the Cohen--Macaulay property. Finally, we combine these results to obtain a classification for disconnected simplicial complexes, complementing the work of Miyashita and Varbaro.