Uniformly Cohen-Macaulay simplicial complexes and almost Gorenstein* simplicial complexes

TL;DR

This paper introduces uniformly Cohen-Macaulay simplicial complexes and demonstrates that almost Gorenstein* complexes are necessarily uniformly Cohen-Macaulay, providing new insights into their algebraic and combinatorial structure.

Contribution

The paper defines uniformly Cohen-Macaulay complexes and proves that all almost Gorenstein* complexes are uniformly Cohen-Macaulay, offering a new framework for their analysis.

Findings

Almost Gorenstein* complexes are uniformly Cohen-Macaulay.

Decomposition of almost Gorenstein* complexes based on top homology.

A combinatorial criterion for dimension ≤ 2 complexes.

Abstract

In this paper, we study simplicial complexes whose Stanley-Reisner rings are almost Gorenstein and have $a$-invariant zero. We call such a simplicial complex an almost Gorenstein* simplicial complex. To study the almost Gorenstein* property, we introduce a new class of simplicial complexes which we call uniformly Cohen-Macaulay simplicial complexes. A $d$-dimensional simplicial complex $\Delta$ is said to be uniformly Cohen-Macaulay if it is Cohen-Macaulay and, for any facet $F$ of $\Delta$, the simplicial complex $\Delta \setminus\{F\}$ is Cohen-Macaulay of dimension $d$. We investigate fundamental algebraic, combinatorial and topological properties of these simplicial complexes, and show that almost Gorenstein* simplicial complexes must be uniformly Cohen-Macaulay. By using this fact, we show that every almost Gorenstein* simplicial complex can be decomposed into those of having one dimensional top homology. Also, we give a combinatorial criterion of the almost Gorenstein* property for simplicial complexes of dimension $\leq 2$.