Toric rings attached to simplicial complexes

2023/02/07 by Jürgen Herzog, Herzog, Jürgen, Somayeh Moradi +3
Mathematics · #05E40 #13A02 #13P10 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2302.03653

openalex publication_date 2023/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider standard graded toric rings R_Δ whose generators correspond to the faces of a simplicial complex Δ. When R_Δ is normal, it is shown that its divisor class group is free. For a flag complex Δ which is the clique complex of a perfect graph, a nice description for the class group and the canonical module of R_Δ in terms of the minimal vertex covers of the graph is given. Moreover, for a quasi-forest simplicial complex a quadratic Gröbner basis for the defining ideal of R_Δ is presented. Using this fact we give combinatorial descriptions for the a-invariant and the Gorenstein property of R_Δ.

Related

• The toric ring of one dimensional simplicial complexes
• Sortable simplicial complexes and their associated toric rings
• Subdivisions of toric complexes
• Algebraic properties of toric rings of graphs
• Cellular free resolutions for normalizations of toric ideals