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Codimension and projective dimension up to symmetry
Authors:Dinh Van Le  Uwe Nagel  Hop D Nguyen  Tim Römer
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
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.
Keywords:invariant ideal  monoid  polynomial ring  symmetric group  13A50  13C15  13D02  13F20  16P70  16W22
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