On toric face rings

Following a construction of Stanley we consider toric face rings associated to rational pointed fans. This class of rings is a common generalization of the concepts of Stanley–Reisner and affine monoid algebras. The main goal of this article is to unify parts of the theories of Stanley–Reisner and affine monoid algebras. We consider (non-pure) shellable fan’s and the Cohen–Macaulay property. Moreover, we study the local cohomology, the canonical module and the Gorenstein property of a toric face ring.