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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 $\dim \Delta \le 2$: Let $\Delta$ be a $(d-1)$-dimensional simplicial complex on $[n]$. Fix an integer $\ell$ with $0 \le \ell \le n-d-1$. If $e(K[\Delta]) \le (\ell+1) d-\ell$ and $\beta_{\ell,\ell+d}(K[\Delta]) =0$, then $\mathrm{reg\,} K[\Delta] \le d-1$. \par Moreover, we discuss the relationship between the above conjecture and the lower bound theorem.