** Multiplicity and Castelnuovo–Mumford Regularity of Stanley–Reisner Rings **

* Naoki Terai, Ken-ichi Yoshida *

**Abstract**

In this paper, we pose the following conjecture and give a positive answer to the case dim*Δ*≤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 *β* _{ ℓ,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.