Castelnuovo-Mumford regularity up to symmetry

Abstract

We study the asymptotic behavior of the Castelnuovo-Mumford regularity along chains of graded ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups. A linear upper bound for the regularity of such ideals is established. We conjecture that their regularity grows eventually precisely linearly. We establish this conjecture in several cases, most notably when the ideals are Artinian or squarefree monomial.

29 pages. Revised version, to appear in IMRN

References in corpus (9)

• FI-modules and stability for representations of symmetric groups
• GL-equivariant modules over polynomial rings in infinitely many variables
• Higher Lawrence configurations
• Syzygies of Segre embeddings and Delta-modules
• Equivariant Hilbert Series in non-Noetherian Polynomial Rings
• Bounded-rank tensors are defined in bounded degree
• FI- and OI-modules with varying coefficients
• Square-free Groebner degenerations
• Betti tables of monomial ideals fixed by permutations of the variables

Cited by in corpus (17)

• Betti numbers of symmetric shifted ideals
• An equivariant Hochster's formula for ${\mathfrak{S}}_n$-invariant monomial ideals
• Invariant chains in algebra and discrete geometry
• Symmetric subvarieties of infinite affine space
• Theorems of Carathéodory, Minkowski-Weyl, and Gordan up to symmetry
• Symmetric ideals of the infinite polynomial ring
• GL-algebras in positive characteristic I: the exterior algebra
• A Kruskal-Katona type result and applications
• Regularity of S_n-invariant monomial ideals
• Betti tables of monomial ideals fixed by permutations of the variables
• The semi-linear representation theory of the infinite symmetric group
• 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
• On monoids up to symmetry
• Equivariant algebraic and semi-algebraic geometry of infinite affine space
• Symmetric Ideals and Invariant Hilbert Schemes