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

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 k [ Δ ] . Moreover, for any given integers d , r , t satisfying r , t ≥ 2 and r ≤ d ≤ r t , 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 .