COMPONENTWISE LINEARITY UNDER SQUARE-FREE GRÖBNER DEGENERATIONS

Using the recent results on square-free Gr & ouml;bner degenerations by Conca and Varbaro, we prove that if a homogeneous ideal I of a polynomial ring is such that its initial ideal in<(I) is square-free and beta(0)(I)= beta (0 )( in < ( I )) , then I is a componentwise linear ideal if and only if in<(I) is a componentwise linear ideal. In particular, if furthermore one of I and in<(I) is componentwise linear, then their graded Betti numbers coincide.