Navid Nemati, Mohammad Reza Pournaki, Siamak Yassemi: Componentwise linearity and the gcd condition are preserved by the polarization, 391-399

Abstract:

A graded ideal $I$ of a polynomial ring over a field is componentwise linear if for every nonnegative integer $j$, the ideal generated by all homogeneous polynomials of degree $j$ belonging to $I$ admits a linear resolution. In this paper, we show that the componentwise linearity of monomial ideals is preserved by the polarization. As an application, we give a condition to guarantee that none of the powers of a monomial ideal is componentwise linear.

Key Words: Monomial ideal, linear resolution, componentwise linear, the gcd condition, polarization.

2010 Mathematics Subject Classification: Primary 13F55, 13D02; Secondary 13F20.