Gorenstein rings through face rings of manifolds

The face ring of a homology manifold (without boundary) modulo a generic system of parameters is studied. Its socle is computed and it is verified that a particular quotient of this ring is Gorenstein. This fact is used to prove that the algebraic g-conjecture for spheres implies all enumerative consequences of its far-reaching generalization (due to Kalai) to manifolds. A special case of Kalai’s conjecture is established for homology manifolds that have a codimension-two face whose link contains many vertices.