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.