The Existence of Pure Free Resolutions

Abstract

Let d1,...,dn be a strictly increasing sequence of integers. Boij and Söderberg [arXiv:math/0611081] have conjectured the existence of a graded module M of finite length over any polynomial ring K[x_1,..., x_n], whose minimal free resolution is pure of type (d1,...,dn), in the sense that its i-th syzygies are generated in degree di. In this paper we prove a stronger statement, in characteristic zero: Such modules not only exist, but can be taken to be GL(n)-equivariant. In fact, we give two different equivariant constructions, and we construct pure resolutions over exterior algebras and Z/2-graded algebras as well. The constructions use the combinatorics of Schur functors and Bott's Theorem on the direct images of equivariant vector bundles on Grassmann varieties.

Dedicated to Jürgen Herzog on the occasion of his sixty-fifth birthday, minor changes; NOTE: Title changed to: The Existence of Equivariant Pure Free Resolutions

References in corpus (3)

• Graded Betti numbers of Cohen-Macaulay modules and the Multiplicity conjecture
• Pieri resolutions for classical groups
• The linear space of Betti diagrams of multigraded artinian modules

Cited by in corpus (32)

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• Stability patterns in representation theory
• Introduction to twisted commutative algebras
• Boij-Söderberg theory: Introduction and survey
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• Tensor complexes: Multilinear free resolutions constructed from higher tensors
• Jack polynomials as fractional quantum Hall states and the Betti numbers of the (k+1)-equals ideal
• Categorified duality in Boij-Söderberg Theory and invariants of free complexes
• Betti Numbers of Syzygies and Cohomology of Coherent Sheaves
• Representation stability for cohomology of configuration spaces in ${\mathbf{R}}^d$
• The linear space of Betti diagrams of multigraded artinian modules
• Filtering free resolutions
• The cone of Betti diagrams over a hypersurface ring of low embedding dimension
• Shapes of free resolutions over a local ring
• Betti numbers and shifts in minimal graded free resolutions
• Betti Numbers of Graded Modules and Cohomology of Vector Bundles
• Zipping Tate resolutions and exterior coalgebras
• Triplets of pure free squarefree complexes
• Supernatural analogues of Beilinson monads
• Towards Boij-Söderberg theory for Grassmannians: the case of square matrices
• Poset structures in Boij-Söderberg theory
• Foundations of Boij-Söderberg Theory for Grassmannians
• Three flavors of extremal Betti tables
• Questions about Boij-Söderberg theory
• Pure resolutions of vector bundles on complex projective spaces
• On decomposing Betti tables and $O$-sequences
• Cohomology of Coherent Sheaves and Series of Supernatural Bundles
• Setting the scene for Betti characters
• Bounds for the Multiplicity of Gorenstein algebras
• Local cohomology tables of sequentially almost Cohen-Macaulay modules
• New bounds on the toral rank with application to cohomologically symplectic spaces