A note on cohen-macaulayness of stanley-reisner rings with serre's condition (S2)

Let 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 (S2) and the multiplicity e(k[Δ]) is "sufficiently large", that is, [image omitted]. We also prove that if e(k[Δ])3d-2 and the graded Betti number 2, d+2(k[Δ]) vanishes, then the Castelnuovo-Mumford regularity regk[Δ] is less than d.