Codimension and projective dimension up to symmetry |
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Authors: | Dinh Van Le Uwe Nagel Hop D Nguyen Tim Römer |
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Institution: | 1. Institut für Mathematik, Universität Osnabrück, 49069 Osnabrück, Germany;2. Department of Mathematics, University of Kentucky, 715 Patterson office tower, Lexington, KY, 40506-0027 USA;3. Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Hanoi, Vietnam |
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Abstract: | Symmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated. We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain. If the ideals are graded it is known that the codimensions grow eventually linearly. Here this result is extended to chains of arbitrary symmetric ideals. Moreover, the slope of the linear function is explicitly determined. We conjecture that the projective dimensions also grow eventually linearly. As part of the evidence we establish two non-trivial lower linear bounds of the projective dimensions for chains of monomial ideals. As an application, this yields Cohen–Macaulayness obstructions. |
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Keywords: | invariant ideal monoid polynomial ring symmetric group 13A50 13C15 13D02 13F20 16P70 16W22 |
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