Fr\"oberg's Theorem, vertex splittability and higher independence complexes

TL;DR

This paper explores the relationship between combinatorial structures and algebraic properties of monomial ideals, extending Fr"oberg's theorem to higher degrees and using topological and hypergraph methods to identify linear resolutions.

Contribution

It introduces the concept of $r$-independence to construct hypergraphs, proves vertex splittability for co-chordal graphs, and provides new proofs for linear resolutions using topological methods.

Findings

Vertex splittability of edge ideals for co-chordal graphs

Explicit computation of graded Betti numbers for various graph classes

Topological proof of linear resolution existence via $r$-collapsibility

Abstract

A celebrated theorem of Fr\"oberg 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 $r$-independence to construct an $r$-uniform hypergraph from the given graph. We then show that when the underlying graph is co-chordal, 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 $r$-collapsibility.