Dualizing Complex of a Toric Face Ring II: Non-normal Case

The notion of "toric face rings" generalizes both Stanley-Reisner rings and affine semigroup rings, and has been studied by Bruns, Romer, et.al. Here, we will show that, for a toric face ring $R$, the "graded" Matlis dual of a Cech complex gives a dualizing complex. In the most general setting, $R$ is not a graded ring in the usual sense. Hence technical argument is required.