Winter 2024 FRÖBERG’S THEOREM, VERTEX SPLITTABILITY AND HIGHER INDEPENDENCE COMPLEXES
Priyavrat Deshpande, Amit Roy, Anurag Singh, Adam Van Tuyl
J. Commut. Algebra 16(4): 391-410 (Winter 2024). DOI: 10.1216/jca.2024.16.391

Abstract

A celebrated theorem of Fröberg gives a complete combinatorial classification of quadratic square-free monomial ideals with a linear resolution. A generalization of this theorem to higher degree square-free monomial ideals is an active area of research. The existence of a linear resolution of such ideals often depends on the field over which the polynomial ring is defined. Hence, it is too much to expect that in the higher degree case a linear resolution can be identified purely using a combinatorial feature of an associated combinatorial structure. However, some classes of ideals having linear resolutions have been identified using combinatorial structures. In the present paper, we use the notion of -independence to construct an -uniform hypergraph from the given graph. We then show that when the underlying graph is cochordal, the corresponding edge ideal is vertex splittable, a condition stronger than having a linear resolution. We use this result to explicitly compute graded Betti numbers for various graph classes. Finally, we give a different proof for the existence of a linear resolution using the topological notion of -collapsibility.

Citation

Download Citation

Priyavrat Deshpande. Amit Roy. Anurag Singh. Adam Van Tuyl. "FRÖBERG’S THEOREM, VERTEX SPLITTABILITY AND HIGHER INDEPENDENCE COMPLEXES." J. Commut. Algebra 16 (4) 391 - 410, Winter 2024. https://doi.org/10.1216/jca.2024.16.391

Information

Received: 16 February 2024; Revised: 7 March 2024; Accepted: 8 March 2024; Published: Winter 2024
First available in Project Euclid: 6 January 2025

MathSciNet: MR4767371
Digital Object Identifier: 10.1216/jca.2024.16.391

Subjects:
Primary: 05E45 , 13F55

Keywords: collapsible complex , Edge ideal , independence complex , linear resolution , Stanley–Reisner ideal , vertex splittable

Rights: Copyright © 2024 Rocky Mountain Mathematics Consortium

Vol.16 • No. 4 • Winter 2024
Back to Top