Squarefree P-modules and the cd-index |
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Authors: | Satoshi Murai Kohji Yanagawa |
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Institution: | 1. Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka 560-0043, Japan;2. Department of Mathematics, Kansai University, Suita 564-8680, Japan |
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Abstract: | In this paper, we introduce a new algebraic concept, which we call squarefree P-modules. This concept is inspired from Karu's proof of the non-negativity of the cd-indices of Gorenstein* posets, and supplies a way to study cd-indices from the viewpoint of commutative algebra. Indeed, by using the theory of squarefree P-modules, we give several new algebraic and combinatorial results on CW-posets. First, we define an analogue of the cd-index for any CW-poset and prove its non-negativity when a CW-poset is Cohen–Macaulay. This result proves that the h-vector of the barycentric subdivision of a Cohen–Macaulay regular CW-complex is unimodal. Second, we prove that the Stanley–Reisner ring of the barycentric subdivision of an odd dimensional Cohen–Macaulay polyhedral complex has the weak Lefschetz property. Third, we obtain sharp upper bounds of the cd-indices of Gorenstein* posets for a fixed rank generating function. |
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Keywords: | cd-Index" target="_blank">cd-Index Flag f-vectors Stanley&ndash Reisner rings Regular CW-complexes Barycentric subdivisions |
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