2014/05/29 by Matsuoka, Naoyuki, Murai, Satoshi
#13F55 #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1405.7438
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 Δ is said to be uniformly Cohen-Macaulay if it is Cohen-Macaulay and, for any facet F of Δ, the simplicial complex Δ∖\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 ≤ 2.