Chennai Mathematical Institute


Research Seminar 8
Date/Time: 16.09.2021, 12:00 pm.
Green-Lazarsfeld property $N_p$ for Hibi rings

Dharm Veer
Chennai Mathematical Institute.


Let $L$ be a finite distributive lattice. By Birkhoff's fundamental structure theorem, $L$ is the ideal lattice $\MI(P)$ of its subposet $P$ of join-irreducible elements. Write $P=\{p_1,\ldots,p_n\}$ and let $R=K[t,z_1,\ldots,z_n]$ be a polynomial ring in $n+1$ variables over a field $K.$ The {\em Hibi ring} associated with $L$, denoted by $R[L]$, is the subring of $R$ generated by the monomials $u_{\alpha}=t\prod_{p_i\in \alpha}z_i$ where $\alpha\in L$. In this talk, we show that a Hibi ring satisfies property $N_4$ if and only if it is a polynomial ring or it has a linear resolution. Therefore, it satisfies property $N_p$ for all $p\geq 4$ as well. Moreover, we show that if a Hibi ring satisfies property $N_2$, then its Segre product with a polynomial ring in finitely many variables also satisfies property $N_2$. *Zoom details.*