Betti numbers of strongly color-stable ideals and squarefree strongly color-stable ideals

摘要

In this paper, we will show that the color-squarefree operation does not change the graded Betti numbers of strongly color-stable ideals. In addition, we will give an example of a nonpure balanced complex which shows that colored algebraic shifting, which was introduced by Babson and Novik, does not always preserve the dimension of reduced homology groups of balanced simplicial complexes.