TAILIEUCHUNG - Báo cáo toán học: "Reciprocal domains and Cohen–Macaulay d-complexes in Rd"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Reciprocal domains and Cohen–Macaulay d-complexes in Rd. | Reciprocal domains and Cohen-Macaulay d-complexes in Rd Ezra Miller and Victor Reiner School of Mathematics University of Minnesota Minneapolis MN 55455 USA ezra@ reiner@ Submitted Sep 9 2004 Accepted Dec 7 2004 Published Jan 7 2005 Mathematics Subject Classifications 05E99 13H10 13C14 57Q99 Dedicated to Richard P. Stanley on the occasion of his 60th birthday Abstract We extend a reciprocity theorem of Stanley about enumeration of integer points in polyhedral cones when one exchanges strict and weak inequalities. The proof highlights the roles played by Cohen-Macaulayness and canonical modules. The extension raises the issue of whether a Cohen-Macaulay complex of dimension d embedded piecewise-linearly in Rd is necessarily a d-ball. This is observed to be true for d 3 but false for d 4. 1 Main results This note begins by dealing with the relation between enumerators of certain sets of integer points in polyhedral cones when one exchanges the roles of strict versus weak inequalities Theorem 1 . The interaction of this relation with the Cohen-Macaulay condition then leads us to study piecewise-linear Cohen-Macaulay polyhedral complexes of dimension d in Euclidean space Rd Theorem 2 . We start by reviewing a result of Stanley on Ehrhart s notion of reciprocal domains within the boundary of a convex polytope. Good references for much of this material are 3 Chapter 6 10 Chapter 1 and 8 Part II . Let Q c Zd be a saturated affine semigroup that is the set of integer points in a convex rational polyhedral cone C R 0Q. Assume that the cone C is of full dimension d and EM and VR supported by NSF grants DMS-0304789 and DMS-0245379 respectively. Keywords reciprocity Cohen-Macaulay canonical module Matlis duality semigroup ring reciprocal domain THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2005 N1 1 pointed at the origin. Denote by F the facets subcones of codimension 1 of C. For each facet F 2 F let F x 0 be the associated facet inequality so that the .

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