AI 中文总结
该研究给出Stanley-Reisner环Artinian截断的典范迹的组合公式，分类了对应近Gorenstein代数，还刻画了旗复形的迹生成及相关等式，分离了平方零一维情形的额外贡献。
AI 中文摘要
对于单纯复形Δ和整数nᵢ≥2，令A_{Δ,𝐧}=𝕜[x₁,…,xₘ]/(I_Δ+(x₁^{n₁},…,xₘ^{nₘ}))。我们针对任意截断指数及删除无关虚顶点后的任意单纯复形，给出典范迹的精确组合公式。该公式推广了Gasanova–Herzog–Hibi–Moradi关于平方零旗面代数的自由面公式，且在单纯形边界情形下，可得到其关于单项式几乎完全交的公式的一个特例。作为首个推论，我们对该族中的近Gorenstein代数进行分类：在Δ⁽¹⁾的每个连通分支C上，诱导复形要么是任意指数的单纯形2^C，要么是所有指数均为2的边界∂2^C。我们还计算了该近Gorenstein轨迹上的Teter数。对于旗复形，迹由任意指数的自由面单项式生成，且我们刻画了等式tr_A(ω_A)=𝔪_A^q。在平方零一维情形下，我们分离出三角形分支带来的额外贡献。
英文摘要
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.
Comments11 pages, comments are welcome
