LAUSR

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(∆) · type(∆) holds for any (d − 1)-dimensional Cohen–Macaulay simplicial complex ∆ satisfying ∆ = core(∆), where reg(∆) (resp. type(∆)) denotes the Castelnuovo–Mumford regularity (resp. Cohen–Macaulay type) of the Stanley–Reisner ring k[∆]. Moreover, for any given integers d, r, t satisfying r, t ≥ 2 and r ≤ d ≤ rt, we construct a Cohen–Macaulay simplicial complex ∆(G) as an independent complex of a graph G such that dim(∆(G)) = d − 1, reg(∆(G)) = r and type(∆(G)) = t.