Gr\"obner bases of contraction ideals

We investigate Gr"obner bases of contraction ideals under some monomial homomorphisms. As an application of our theorem, we generalize the result of Aoki–Hibi–Ohsugi–Takemura and Hibi-Ohsugi. Using our results, one can provide many examples of toric ideals that admit square-free or quadratic initial ideals.