COMPONENTWISE LINEARITY UNDER SQUARE-FREE GRÖBNER DEGENERATIONS

Abstract

Using the recent results on square-free Gröbner degenerations by Conca and Varbaro, we prove that if a homogeneous ideal $I$ of a polynomial ring is such that its initial ideal ${\mathrm{in}}_{<}\left(I\right)$ is square-free and ${\beta }_0\left(I\right)={\beta }_0\left({\mathrm{in}}_{<}\right(I\left)\right)$, then $I$ is a componentwise linear ideal if and only if ${\mathrm{in}}_{<}\left(I\right)$ is a componentwise linear ideal. In particular, if furthermore one of $I$ and ${\mathrm{in}}_{<}\left(I\right)$ is componentwise linear, then their graded Betti numbers coincide.