  For a simplicial complex X on {1,2, ..., n} we define enriched homology and
cohomology modules. They are graded modules over k[x_1, ..., x_n] whose ranks
are equal to the dimensions of the reduced homology and cohomology groups.
  We characterize Cohen-Macaulay, l-Cohen-Macaulay, Buchsbaum, and Gorenstein*
complexes X, and also orientable manifolds in terms of the enriched modules. We
introduce the notion of girth for simplicial complexes and make a conjecture
relating the girth to invariants of the simplicial complex.
  We also put strong vanishing conditions on the enriched homology modules and
describe the simplicial complexes we then get. They are block designs and
include Steiner systems S(c,d,n) and cyclic polytopes of even dimension.
