Deriving some properties of Stanley-Reisner rings from their squarefree zero-divisor graphs

Authors : Ashkan NIKSERESHT

Pages : 121-133

Doi:10.24330/ieja.1058421

View : 17 | Download : 6

Publication Date : 2022-01-17

Article Type : Research Paper

Abstract :Let Δ Δ be a simplicial complex, I Δ IΔ its Stanley-Reisner ideal and R = K [ Δ ] R=K[Δ] its Stanley-Reisner ring over a field K K . In 2018, the author introduced the squarefree zero-divisor graph of R R , denoted by Γ s f insert ignore into journalissuearticles values( R ); Γsfinsert ignore into journalissuearticles values(R); , and proved that if Δ Δ and Δ ′ Δ′ are two simplicial complexes, then the graphs Γ s f insert ignore into journalissuearticles values( K [ Δ ] ); Γsfinsert ignore into journalissuearticles values(K[Δ]); and  Γ s f insert ignore into journalissuearticles values( K [ Δ ′ ] ); Γsfinsert ignore into journalissuearticles values(K[Δ′]); are isomorphic if and only if the rings K [ Δ ] K[Δ] and K [ Δ ′ ] K[Δ′] are isomorphic. Here we derive some algebraic properties of R R using combinatorial properties of Γ s f insert ignore into journalissuearticles values( R ); Γsfinsert ignore into journalissuearticles values(R); . In particular, we state combinatorial conditions on Γ s f insert ignore into journalissuearticles values( R ); Γsfinsert ignore into journalissuearticles values(R); which are necessary or sufficient for R R to be Cohen-Macaulay. Moreover, we investigate when Γ s f insert ignore into journalissuearticles values( R ); Γsfinsert ignore into journalissuearticles values(R); is in some well-known classes of graphs and show that in these cases, I Δ IΔ has a linear resolution or is componentwise linear. Also we study the diameter and girth of Γ s f insert ignore into journalissuearticles values( R ); Γsfinsert ignore into journalissuearticles values(R); and their algebraic interpretations.
Keywords : Squarefree monomial ideal, simplicial complex, squarefree zero divisor graph, Cohen Macaulay ring, linear resolution