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Gorenstein rings through face rings of manifolds

Part of: Polytopes and polyhedra Local rings and semilocal rings Arithmetic rings and other special rings

Published online by Cambridge University Press:  01 July 2009

Isabella Novik  and
Ed Swartz
Isabella Novik
Affiliation:
Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195-4350, USA (email: novik@math.washington.edu)
Ed Swartz
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA (email: ebs22@cornell.edu)
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Abstract

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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.

Keywords

g-conjecturessoclehomology manifolds and sphereslocal cohomology modules

MSC classification

Secondary: 13F55: Stanley-Reisner face rings; simplicial complexes 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 52B05: Combinatorial properties (number of faces, shortest paths, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 4 , July 2009 , pp. 993 - 1000
Copyright
Copyright © Foundation Compositio Mathematica 2009

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