  In this article we construct a Koszul-type resolution of the p-th exterior
power of the sheaf of holomorphic differential forms on smooth toric varieties
and use this to prove a Nadel-type vanishing theorem for Hodge ideals
associated to effective \mathbb{Q}-divisors on smooth projective toric
varieties. This extends earlier results of Musta\c{t}\u{a} and Popa.
