Abstract
This work concerns the study of properties of a group of Koszul algebras coming from the toric ideals of a chordal bipartite infinite family of graphs (alternately, these rings may be interpreted as coming from determinants of certain ladder-like structures). We determine a linear system of parameters for each ring and explicitly determine the Hilbert series for the resulting Artinian reduction. As corollaries, we obtain the multiplicity and regularity of the original rings. This work extends results easily derived from lattice theory for a subfamily coming from a two-sided ladder to a family where, as we show, lattice theory no longer applies in any obvious way and includes constructive proofs which may be useful in future study of these rings and others.
The author was partially supported by the National Science Foundation (DMS-1003384).
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Acknowledgements
Macaulay2 [11] was used for computation and hypothesis formation. We would like to thank Syracuse University for its support and hospitality and Claudia Miller for her valuable input on the original project in [1] and this condensed version. We would also like to thank the referees for noticing errors in the statement and proof of Theorem 3.16 as well as in the proof of Lemma 3.1, and for highlighting with clarity the connection between Theorem 3.4 and work done by Rafael Villarreal. We acknowledge the partial support of an NSF grant.
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Ballard, L. (2021). Properties of the Toric Rings of a Chordal Bipartite Family of Graphs. In: Miller, C., Striuli, J., Witt, E.E. (eds) Women in Commutative Algebra. Association for Women in Mathematics Series, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-030-91986-3_2
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