On Green-Lazarsfeld property $N_p$ for Hibi rings

Abstract: Let $L$ be a finite distributive lattice. By Birkhoff's fundamental structure theorem, $L$ is the ideal lattice of its subposet $P$ of join-irreducible elements. Write $P=\{p_1,\ldots,p_n\}$ and let $K[t,z_1,\ldots,z_n]$ be a polynomial ring in $n+1$ variables over a field $K.$ The {\em Hibi ring} associated with $L$ is the subring of $K[t,z_1,\ldots,z_n]$ 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. We also discuss a few results about the property $N_p$ of Hibi rings for $p=2$ and 3. For example, 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 $N_2$.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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