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Paper IPM / M / 16065  


Abstract:  
Using the concept of dcollapsibility from combinatorial topology, we define chordal simplicial complexes and show that their StanleyReisner ideals are componentwise linear. Our construction is inspired by and an extension of ï¿½??chordal cluttersï¿½?ï¿½ which was defined by Bigdeli, Yazdan Pour and ZaareNahandi in 2017, and characterizes Betti tables of all ideals with a linear resolution in a polynomial ring.
We show dcollapsible and drepresentable complexes produce componentwise linear ideals for appropriate d. Along the way, we prove that there are generators that when added to the ideal, do not change Betti numbers in certain degrees.
We then show that large classes of componentwise linear ideals, such as Gotzmann ideals and squarefree stable ideals have chordal StanleyReisner complexes, that Alexander duals of vertex decomposable complexes are chordal, and conclude that the Betti table of every componentwise linear ideal is identical to that of the StanleyReisner ideal of a chordal complex.
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