The Existence of Pure Free Resolutions
Abstract
Let d1,...,dn be a strictly increasing sequence of integers. Boij and S\"oderberg [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.
Cite
@article{arxiv.0709.1529,
title = {The Existence of Pure Free Resolutions},
author = {David Eisenbud and Gunnar Floystad and Jerzy Weyman},
journal= {arXiv preprint arXiv:0709.1529},
year = {2012}
}
Comments
Dedicated to J\"urgen Herzog on the occasion of his sixty-fifth birthday, minor changes; NOTE: Title changed to: The Existence of Equivariant Pure Free Resolutions