Powers of componentwise linear ideals: the Herzog–Hibi–Ohsugi conjecture and related problems

In 1999, Herzog and Hibi introduced componentwise linear ideals. A homogeneous ideal I is componentwise linear if for all nonnegative integers d, the ideal generated by the homogeneous elements of degree d in I has a linear resolution. For square-free monomial ideals, componentwise linearity is related via Alexander duality to the property of being sequentially Cohen–Macaulay for the corresponding simplicial complexes. In general, the property …