TY - JOUR
T1 - Uniformly Cohen-Macaulay simplicial complexes and almost Gorenstein* simplicial complexes
AU - Matsuoka, Naoyuki
AU - Murai, Satoshi
N1 - Funding Information:
The first author was partially supported by JSPS KAKENHI 26400054 . The second author was partially supported by JSPS KAKENHI 25400043 .
Publisher Copyright:
© 2016 Elsevier Inc.
PY - 2016/6/1
Y1 - 2016/6/1
N2 - 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.
AB - 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.
KW - Almost Gorenstein rings
KW - Doubly Cohen-Macaulay rings
KW - Stanley-Reisner rings
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U2 - 10.1016/j.jalgebra.2016.02.005
DO - 10.1016/j.jalgebra.2016.02.005
M3 - Article
AN - SCOPUS:84963836263
VL - 455
SP - 14
EP - 31
JO - Journal of Algebra
JF - Journal of Algebra
SN - 0021-8693
ER -