On Stanley–Reisner Rings with Linear Resolutions

Abstract For a graph G , Bayer–Denker–Milutinović–Rowlands–Sundaram–Xue study in [1] a new graph complex $$\Delta _k^t(G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>Δ</mml:mi> <mml:mi>k</mml:mi> <mml:mi>t</mml:mi> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , namely the simplicial complex with facets that are complements to independent sets of size k in G . They are interested in topological properties such as shellability, vertex decomposability, homotopy type, and homology of these complexes. In this paper we study more algebraic properties, such as Cohen-Macaulayness, Betti numbers, and linear resolutions of the Stanley-Reisner ring of these complexes and their Alexander duals.