  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.
