Canonical traces of graded fiber products: applications to disconnected Stanley--Reisner rings
Shinya Kumashiro, Sora Miyashita
TL;DR
The paper extends the classification of canonical traces from Cohen–Macaulay Stanley–Reisner rings to the non-Cohen–Macaulay regime by deriving an explicit formula for the canonical trace of graded fiber products $R=A\times_\mathbb{k}B$, and applying it to disconnected Stanley–Reisner rings to reduce questions to connected components. In the connected case, it weakens Cohen–Macaulay to Serre's $(S_2)$ and establishes a parallel classification for when $R$ is Gorenstein on the punctured spectrum, describing $\operatorname{tr}_R(\omega_R)$ in terms of the maximal ideal powers $\mathfrak{m}_R^i$. The work also generalizes the notion of Noetherian rings of Teter type to non-Cohen–Macaulay rings and applies these ideas to Stanley–Reisner rings, yielding a detailed combinatorial-Topological characterization: for normal connected $\Delta$, $\operatorname{tr}_R(\omega_R)$ reflects whether $\Delta$ is a path or a non-orientable $\mathbb{k}$-homology manifold, and for disconnected $\Delta$ the trace decomposes additively across components. Overall, the results provide a cohesive framework linking canonical traces, fiber products, and the topology of simplicial complexes in the non-CM setting.
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 $(S_2)$.