Codimension and Projective Dimension up to Symmetry

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.

arXiv comment19 pages; revised version, to appear in Math. Nachr

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Cited by in corpus (13)

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• An equivariant Hochster's formula for ${\mathfrak{S}}_n$-invariant monomial ideals
• Invariant chains in algebra and discrete geometry
• Theorems of Carathéodory, Minkowski-Weyl, and Gordan up to symmetry
• GL-algebras in positive characteristic I: the exterior algebra
• Asymptotic behavior of symmetric ideals: A brief survey
• A Kruskal-Katona type result and applications
• GL-algebras in positive characteristic II: the polynomial ring
• A note on projective dimension over twisted commutative algebras
• Image closure of symmetric wide-matrix varieties
• Symmetric Ideals and Invariant Hilbert Schemes
• On monoids up to symmetry
• Equivariant algebraic and semi-algebraic geometry of infinite affine space