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Multiplicity and Castelnuovo–Mumford Regularity of Stanley–Reisner Rings

Naoki Terai , $icon-email$ Ken-ichi Yoshida

Abstract

In this paper, we pose the following conjecture and give a positive answer to the case $\dim Δ \leq 2$ : Let $Δ$ be a $(d - 1)$ -dimensional simplicial complex on $[n]$ . Fix an integer $ℓ$ with $0 \leq ℓ \leq n - d - 1$ . If $e (K [Δ]) \leq (ℓ + 1) d - ℓ$ and $β_{ℓ, ℓ + d} (K [Δ]) = 0$ , then $reg K [Δ] \leq d - 1$ . \par Moreover, we discuss the relationship between the above conjecture and the lower bound theorem.

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Multiplicity and Castelnuovo–Mumford Regularity of Stanley–Reisner Rings

Abstract