  We consider simplicial complexes admitting a free action by an abelian group.
Specifically, we establish a refinement of the classic result of Hochster
describing the local cohomology modules of the associated Stanley--Reisner
ring, demonstrating that the topological structure of the free action extends
to the algebraic setting. If the complex in question is also Buchsbaum, this
new description allows for a specialization of Schenzel's calculation of the
Hilbert series of some of the ring's Artinian reductions. In further
application, we generalize to the Buchsbaum case the results of Stanley and
Adin that provide a lower bound on the $h$-vector of a Cohen-Macaulay complex
admitting a free action by a cyclic group of prime order.
