A Note on Cohen–Macaulayness of Stanley–Reisner Rings with Serre's Condition (S₂)

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 β_{2, d+2}(k[Δ]) vanishes, then the Castelnuovo–Mumford regularity reg k[Δ] is less than d.

Key Words:

• Alexander duality
• Cohen–Macaulay
• Initial degree
• Linear resolution
• Multiplicity
• Relation type
• Serre's condition
• Stanley–Reisner ring

2000 Mathematics Subject Classification:

• Primary 13F55
• Secondary 13H10

Notes

Communicated by R. H. Villarreal.