Canonical traces of Artinian truncations of Stanley-Reisner rings

Abstract

For a simplicial complex $Δ$ and integers $n_i\ge 2$, set $A_{Δ,\mathbf n}=\mathbb{k}[x_1,\ldots,x_m]/(I_Δ+(x_1^{n_1},\ldots,x_m^{n_m}))$. We give an exact combinatorial formula for the canonical trace for arbitrary truncation exponents and for an arbitrary simplicial complex after deleting irrelevant ghost vertices. The formula extends the free-face formula of Gasanova--Herzog--Hibi--Moradi for square-zero flag face algebras and recovers, in the simplex-boundary case, a special case of their formula for monomial almost complete intersections. As a first consequence, we classify the nearly Gorenstein algebras in this family: on each connected component $C$ of $Δ^{(1)}$, the induced complex is either the simplex $2^C$, with arbitrary exponents, or the boundary $\partial 2^C$, with every exponent equal to two. We also compute the Teter number on this nearly Gorenstein locus. For flag complexes the trace is generated by the free-face monomials for arbitrary exponents, and we characterize the equalities $\operatorname{tr}_A(ω_A)=\mathfrak m_A^q$. In the square-zero one-dimensional case we isolate the additional contribution coming from triangle components.

Disclosure

“ONICAL TRACES OF ARTINIAN TRUNCATIONS 11 ACKNOWLEDGMENTS The author was supported by JSPS KAKENHI Grant Number 25KJ1744. The author used OpenAI’s ChatGPT as an interactive aid during the exploratory and editorial stages of this work, including pre- liminary mathematical brainstorming, literature discovery, language editing, and TEX preparation. All mathematical statements, proofs, and cited”

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Classification

Brainstorming or outlining

Multiplier

2

Verified

2026-08-23T05:18:34.514320Z

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