Asymptotic vanishing of homogeneous components of multigraded modules and its applications

In this article, we give a condition on the vanishing of finitely many homogeneous components which must imply the asymptotic vanishing for multigraded modules. We apply our result to multi-Rees algebras of ideals. As a consequence, we obtain a result on normality of monomial ideals, which extends and improves the results of Reid-Roberts-Vitulli, Singla, and Sarkar-Verma.