Koszul multi-Rees algebras of principal $L$-Borel Ideals
Michael DiPasquale, Babak Jabbar Nezhad

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 in a polynomial ring and a subset of the variables of the polynomial ring, the principal -Borel ideal generated by is the ideal generated by all monomials which can be obtained from by successively replacing variables of by those which are in and have smaller index. Given a collection where is -Borel for (where the subsets may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets is chordal bipartite, then the defining equations of the multi-Rees algebra of 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…
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