The Existence of Pure Free Resolutions

Details The Existence of Pure Free Resolutions The Existence of Pure Free Resolutions

2007-09-11

arxiv.org/abs/0709.1529v5

Annales de l'institut Fourier, 61 (2011), no. 3, p. 905-926

Mathematics

Commutative Algebra

Dedicated to J\"urgen Herzog on the occasion of his sixty-fifth birthday, minor changes; NOTE: Title changed to: The Existence

Scientific paper

10.5802/aif.2632

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.

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