Koszul multi-Rees algebras of principal $L$-Borel Ideals

TL;DR

This paper proves that the multi-Rees algebra of certain principal L-Borel ideals, associated with chordal bipartite graphs, is Koszul, Cohen-Macaulay, and normal, by establishing a quadratic Gröbner basis under lex order.

Contribution

It generalizes previous results by showing that multi-Rees algebras of principal L-Borel ideals have quadratic Gröbner bases when associated with chordal bipartite graphs, implying Koszulness.

Findings

Multi-Rees algebra has a quadratic Gröbner basis with squarefree lead terms.

The algebra is Koszul, Cohen-Macaulay, and normal under the given conditions.

Generalizes a theorem of Ohsugi and Hibi on Koszul bipartite graphs.

Abstract

Given a monomial $m$ in a polynomial ring and a subset $L$ of the variables of the polynomial ring, the principal $L$-Borel ideal generated by $m$ is the ideal generated by all monomials which can be obtained from $m$ by successively replacing variables of $m$ by those which are in $L$ and have smaller index. Given a collection $\mathcal{I}=\{I_1,\ldots,I_r\}$ where $I_i$ is $L_i$-Borel for $i=1,\ldots,r$ (where the subsets $L_1,\ldots,L_r$ may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets $L_1,\ldots,L_r$ is chordal bipartite, then the defining equations of the multi-Rees algebra of $\mathcal{I}$ has a Gr\"obner basis of quadrics with squarefree lead terms under lexicographic order. Thus the multi-Rees algebra of such a collection of ideals is Koszul, Cohen-Macaulay, and normal. This significantly generalizes a theorem of Ohsugi and Hibi on Koszul bipartite graphs. As a corollary we obtain that the multi-Rees algebra of a collection of principal Borel ideals is Koszul. To prove our main result we use a fiber-wise Gr\"obner basis criterion for the kernel of a toric map and we introduce a modification of Sturmfels' sorting algorithm.