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

TL;DR

This paper generalizes the classification of canonical traces of Stanley--Reisner rings to non-Cohen--Macaulay cases by deriving explicit formulas for graded fiber products and applying them to disconnected complexes.

Contribution

It provides a new explicit formula for canonical traces of graded fiber products and extends classification results to disconnected simplicial complexes without Cohen--Macaulay assumptions.

Findings

Explicit formula for canonical trace of graded fiber products

Complete classification for connected simplicial complexes

Extended classification to disconnected simplicial complexes

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. 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.