Inequalities of invariants on Stanley-Reisner rings of Cohen-Macaulay simplicial complexes

2020/12/26 by Akihiro Higashitani, Higashitani, Akihiro, Hiroju Kanno +3
Mathematics · Medicine · #05C70 #05E40 #13D40 #Advanced Combinatorial Mathematics #Cholinesterase and Neurodegenerative Diseases #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Primary 13F55 #Secondary 13D02

paper · pdf · doi:10.48550/arxiv.2012.13725

openalex publication_date 2020/12/26 · openalex created_date 2021/01/05 · openalex updated_date 2026/07/28

Abstract

The goal of the present paper is the study of some algebraic invariants of Stanley-Reisner rings of Cohen-Macaulay simplicial complexes of dimension d - 1. We prove that the inequality d ≤ reg(Δ) ⋅ type(Δ) holds for any (d-1)-dimensional Cohen-Macaulay simplicial complex Δ satisfying Δ=core(Δ), where reg(Δ) (resp. type(Δ)) denotes the Castelnuovo-Mumford regularity (resp. Cohen-Macaulay type) of the Stanley-Reisner ring \Bbbk[Δ]. Moreover, for any given integers d,r,t satisfying r,t ≥ 2 and r ≤ d ≤ rt, we construct a Cohen-Macaulay simplicial complex Δ(G) as an independent complex of a graph G such that dim(Δ(G))=d-1, reg(Δ(G))=r and type(Δ(G))=t.

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