Fröberg's Theorem, vertex splittability and higher independence complexes
arxiv(2023)
摘要
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.
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