A Note on Cohen–Macaulayness of Stanley–Reisner Rings with Serre's Condition (S 2)

Abstract

Let Δ be a (d − 1)-dimensional simplicial complex on the vertex set V = {1, 2,…, n}. In this article, using Alexander duality, we prove that the Stanley–Reisner ring k[Δ] is Cohen–Macaulay if it satisfies Serre's condition (S 2) and the multiplicity e(k[Δ]) is “sufficiently large”, that is, . We also prove that if e(k[Δ]) ≤ 3d − 2 and the graded Betti number β2, d+2(k[Δ]) vanishes, then the Castelnuovo–Mumford regularity reg k[Δ] is less than d.