Cellular free resolutions for normalizations of toric ideals

For any toric ideal $I$ in a polynomial ring $S$, we provide a combinatorial description of a free resolution of the integral closure of the $S$-module $S\mathrm{/}I$. These new complexes arise from an extension of Bayer–Sturmfels’ theory of cellular free resolutions. As applications, we unify several constructions for a resolution of the diagonal embedding of a toric variety, and compare the locally free resolutions for toric subvarieties introduced by Hanlon–Hicks–Lazarev and Brown–Erman.