  We study local cohomology of rings of global sections of sheafs on the
Alexandrov space of a partially ordered set. We give a criterion for a
splitting of the local cohomology groups into summands determined by the
cohomology of the poset and the local cohomology of the stalks. The face ring
of a rational pointed fan can be considered as the ring of global sections of a
flasque sheaf on the face poset of the fan. Thus we obtain a decomposition of
the local cohomology of such face rings. Since the Stanley-Reisner ring of a
simplicial complex is the face ring of a rational pointed fan, our main result
can be interpreted as a generalization of Hochster's decomposition of local
cohomology of Stanley-Reisner rings.
