  Via the BGG correspondence a simplicial complex Delta on [n] is transformed
into a complex of coherent sheaves on P^n-1. We show that this complex reduces
to a coherent sheaf F exactly when the Alexander dual Delta^* is
Cohen-Macaulay. We then determine when both Delta and Delta^* are
Cohen-Macaulay. This corresponds to F being a locally Cohen-Macaulay sheaf.
Lastly we conjecture for which range of invariants of such Delta it must be a
cone.
