Fröberg's Theorem, vertex splittability and higher independence complexes
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909821723213824 |
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| author | Deshpande, Priyavrat Roy, Amit Singh, Anurag Van Tuyl, Adam |
| author_facet | Deshpande, Priyavrat Roy, Amit Singh, Anurag Van Tuyl, Adam |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_02430 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fröberg's Theorem, vertex splittability and higher independence complexes Deshpande, Priyavrat Roy, Amit Singh, Anurag Van Tuyl, Adam Commutative Algebra Combinatorics 13F55, 05E45 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. |
| title | Fröberg's Theorem, vertex splittability and higher independence complexes |
| topic | Commutative Algebra Combinatorics 13F55, 05E45 |
| url | https://arxiv.org/abs/2311.02430 |