The Existence of Pure Free Resolutions

TL;DR

This paper proves the existence of pure free resolutions with specified degree sequences for modules over polynomial rings, providing explicit constructions in characteristic zero using equivariant methods.

Contribution

It establishes the existence of GL(n)-equivariant pure resolutions for any increasing degree sequence, extending prior conjectures and constructions.

Findings

Existence of pure resolutions in characteristic zero.

Two explicit equivariant constructions provided.

Extensions to exterior and Z/2-graded algebras.

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.