In this paper, we pose the following conjecture and give a positive answer to the case dimΔ≤2: Let Δ be a (d−1)-dimensional simplicial complex on [n]. Fix an integer ℓ with 0≤ℓ≤n−d−1. If e(K[Δ])≤(ℓ+1)d−ℓ and βℓ,ℓ+d(K[Δ])=0, then reg K[Δ]≤d−1. Moreover, we discuss the relationship between the above conjecture and the lower bound theorem.
- Arithmetic degree
- Castelnuovo–Mumford regularity
- Lower bound theorem
- Stanley–Reisner ring
ASJC Scopus subject areas