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Fröberg's Theorem, vertex splittability and higher independence complexes

Priyavrat Deshpande, Amit Roy, Anurag Singh, Adam Van Tuyl

TL;DR

The paper investigates higher-degree square-free monomial ideals arising from $r$-independence complexes by linking them to Stanley-Reisner theory and hypergraphs. It proves that for graphs $G$ whose complements are chordal, the ideal $I_r(G)$ has a linear resolution of length $r+1$ via two distinct approaches: vertex-splittable ideals and $r$-collapsibility of the associated complex, and provides explicit formulas for graded Betti numbers for several graph families. These results extend Fröberg's classical theorem to higher dimensions and yield practical Betti-number computations for complete, multipartite, and path-complement graphs, with an additional topological proof and new conjectural directions. The work highlights connections between combinatorial structures, algebraic properties, and topological collapsibility in studying linear resolutions of higher independence ideals.

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 $r$-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.

Fröberg's Theorem, vertex splittability and higher independence complexes

TL;DR

The paper investigates higher-degree square-free monomial ideals arising from -independence complexes by linking them to Stanley-Reisner theory and hypergraphs. It proves that for graphs whose complements are chordal, the ideal has a linear resolution of length via two distinct approaches: vertex-splittable ideals and -collapsibility of the associated complex, and provides explicit formulas for graded Betti numbers for several graph families. These results extend Fröberg's classical theorem to higher dimensions and yield practical Betti-number computations for complete, multipartite, and path-complement graphs, with an additional topological proof and new conjectural directions. The work highlights connections between combinatorial structures, algebraic properties, and topological collapsibility in studying linear resolutions of higher independence ideals.

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 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 -collapsibility.
Paper Structure (9 sections, 24 theorems, 46 equations, 4 figures)

This paper contains 9 sections, 24 theorems, 46 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Stanley-Reisner ideals of higher independence complexes
  3. Graph terminology
  4. Simplicial Complexes
  5. Stanley-Reisner ideals of higher independence complexes
  6. Fröberg's theorem via vertex splittable ideals
  7. The graded Betti numbers
  8. Fröberg's theorem via collapsibility
  9. Concluding Remarks

Key Result

Theorem 1.1

If $G$ is a graph whose complement is chordal, then $I_r(G)$ has a $(r+1)$-linear resolution.

Figures (4)

  • Figure 1: A co-chordal graph and its complement.
  • Figure 2: The graph $\widetilde{G}$ and its complement.
  • Figure 3: The graph $K_7$ and the Betti table of $R/I_3(K_7)$
  • Figure 4: Betti table of $R/I_2(P_7^c)$ and $R/I_3(P_7^c)$, respectively

Theorems & Definitions (60)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Definition 3.1
  • Remark 3.2
  • ...and 50 more