  It is known that the Frobenius algebra of the injective hull of the residue
field of a complete Stanley--Reisner ring (i.e. a formal power series ring
modulo a squarefree monomial ideal) can be only principally generated or
infinitely generated as algebra over its degree zero piece, and that this fact
can be read off in the corresponding simplicial complex; in the infinite case,
we exhibit a 1--1 correspondence between potential new generators appearing on
each graded piece and certain pairs of faces of such a simplicial complex, and
we use it to provide an alternative proof of the fact that these Frobenius
algebras can only be either principally generated or infinitely generated.
