  Generalizing the concepts of Stanley-Reisner and affine monoid algebras, one
can associate to a rational pointed fan the toric face ring. Assuming that this
ring is Cohen-Macaulay, the main result of this paper is to characterize the
situation when its canonical module is isomorphic to a fine graded ideal of the
toric face ring. From this result several algebraic and combinatorial
consequences are deduced in the situations where the fan may be related to a
manifold with non-empty boundary, or the fan is a shellable fan.
