Gorenstein rings through face rings of manifolds

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<jats:title>Abstract</jats:title><jats:p>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 <jats:italic>g</jats:italic>-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.</jats:p>

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